# Extending harmonic principles in 12edo to 31edo. Part II: JI & back to basics
$\def\monzo#1{\left[ \begin{matrix}#1\end{matrix} \right>}$
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:warning: Read this in [dark mode](https://hackmd.io/@docs/how-to-set-dark-mode-en#Set-View-page-theme).
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### Abstract
Previous: [Part I: Cultural entrainment](https://hackmd.io/@euwbah/extending-harmonic-principles-1)
Next: [Part III: Temperament, vals, mapping, scales, functional harmony](https://hackmd.io/@euwbah/extending-harmonic-principles-3)
The goal of this Part II is to introduce just intonation by revisiting the fundamentals of music. We explore
* What is a note
* What is an interval
* What are chords
* How to name them
Part I threw us in the deep of tonalities in a foreign tuning system — now we are motivated to get a new perspective on the fundamental elements of music theory that we all take for granted and overlook: note names, chord names, tunings, functional equivalences. In order to understand how to extend music to other tunings, we'll need to first revisit the very basics, but the xen way.
This will be written assuming zero music theory knowledge, and will be suitable for all readers. For readers with no theory background, starting this part first before reading Part I is recommended.
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**Legend**
----
:writing_hand: Exercise for the reader.
:pencil: Additional notes
:warning: Disclaimers & warnings
:bulb: Further reading & ideas
:musical_note: Information about [xenpaper](https://luphoria.com/xenpaper) itself
Dotted underline: Mouse over for abbreviation expansion or definition. Alternatively, you can view the [Glossary](#Glossary) at the bottom of the article
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*[Dotted underline]: Abbreviation expansion or definition
[TOC]
# II. Just Intonation (JI)
## The harmonic series
We begin in the realm of the most fundamental and tangible: physics. When any physical compressible/flexible object (air, string) vibrates primarily in one dimension/axis, we obtain the harmonic series:
<iframe width="560" height="315" src="https://www.youtube.com/embed/dVkO3MfaU7s?si=1smib4kDyrazw2yN&start=2" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
The rarefaction of air in a wind instrument is primarily in the longitudinal axis, and vibration of a string on a stringed instrument is primarily transverse. These are standing waves:
<iframe width="560" height="315" src="https://www.youtube.com/embed/7cDAYFTXq3E?si=rXIfS3iIjIosQcoi" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
The harmonic series stems from the principle of harmonic motion and the [harmonic oscillator](https://en.wikipedia.org/wiki/Harmonic_oscillator) in physics:
<iframe width="560" height="315" src="https://www.youtube.com/embed/py3EWLKQaMs?si=fd5CwA_fWgHHglmU" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
Each point in a standing wave can be modeled with a harmonic oscillator. There is an inverse correlation between the length of the string/length of the wind channel and the frequency. If we halve the length of the string, the frequency will double.
Even if we don't alter the length of the string, there will be different **modes of resonance** naturally present in the oscillations of the string:
<iframe width="560" height="315" src="https://www.youtube.com/embed/5qUouwW-m2s?si=a4kIv0Rax0WDblXn" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
The modes of resonance will naturally divide the length of the wave's period into integer multiples (1, 2, 3, ...), which will also multiply the frequency by integers:
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%0A100hz_200hz_300hz_400hz_500hz_600hz_700hz_800hz%0A900hz_1000hz_1100hz_1200hz_1300hz_1400hz_1500hz_1600hz" title="Xenpaper" frameborder="0"></iframe>
**This is the harmonic series**. If the fundamental frequency $f_0$ (aka root note) is 100 Hz, then all the integer multiples of 100 Hz are called the **harmonics** of the fundamental.
> :pencil: **Partials** are constituent frequencies (including the fundamental) that may not be integer multiples (i.e., inharmonic frequencies), whereas **harmonics** must be integer multiples.
>
> **Overtones** are all other partials excluding the fundamental itself. In a harmonic timbre, the first overtone is the second harmonic.
This phenomena was known of since at least **3300 BC** in Sumerian writings (and independently, the Pythagoreans in 600 BC), on an instrument called the *monochord* (one string), and is one of the most fundamental universal aspects of music made with instruments which oscillated primarily on one axis (e.g. strings, air tubes).
<iframe width="560" height="315" src="https://www.youtube.com/embed/9W4jr4CSohI?si=o2umCxdOGxFfNmsb" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
> :pencil: But there are musical cultures based on **inharmonic** instruments, like Gamelan, which use metallophones (like gongs) that produce [circular, spherical, or other harmonics](https://www.youtube.com/watch?v=Ziz7t1HHwBw) that are not one-dimensional. In these musical cultures, tuning works completely differently.
### It matters because harmonics affect concordance
When a string is plucked or wind instrument is played, the various modes of resonance occur simultaneously. The unique collection of harmonics at different amplitudes (loudness) allows us to identify **timbre**. Timbre lets us distinguish between instruments, human voices, and vowels in speech.
<iframe width="560" height="315" src="https://www.youtube.com/embed/VRAXK4QKJ1Q?si=0DAAh5BvA8VZ5B5f&start=25" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
<iframe width="560" height="315" src="https://www.youtube.com/embed/VnC8I3d2MXQ?si=2lwnYlTTM-a4VBEM&start=6" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
<iframe width="560" height="315" src="https://www.youtube.com/embed/-Zf0YEh262o?si=DuoksNquNesMeJ_G" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
But our goal is to write microtonal music — not analyze timbre. So why does this matter for us now?
When a "single note" is played, it is usually not just one single frequency. Within each note are theoretically infinite frequencies extending up the harmonic series, and the gestalt of these notes is what's perceived as the "single note" by the human brain. The only timbre that truly comprises one frequency is the sine wave. Even then, nonlinearities in the air, speakers, and the sense of hearing itself will distort the perception of the sine wave such that harmonics will **still be present**.
Now when we play two "single notes", we're not merely playing two pure sine wave frequencies at once, which would sound like this:
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)(osc%3Asine)%0A400hz_600hz_%5B400hz_600hz%5D" title="Xenpaper" frameborder="0"></iframe>
Instead, usually we are listening to instruments that are not sine waves, which would mean the actual list of frequencies we are hearing would instead be something more like this:
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)(osc%3Asine)(env%3A2128)%0A%5B400hz_800hz_1200hz_1600hz_2000hz_2400hz_2800hz_3200hz%5D%0A%5B600hz_1200hz_1800hz_2400hz_3000hz_3600hz_4200hz_4800hz%5D%0A%5B400hz_800hz_1200hz_1600hz_2000hz_2400hz_2800hz_3200hz_600hz_1200hz_1800hz_2400hz_3000hz_3600hz_4200hz_4800hz%5D" title="Xenpaper" frameborder="0"></iframe>
...where "one note" comprises all the frequencies of its harmonics. For the sake of not destroying your computer/ears, I've only included the first 8 harmonics.
Notice what happens when we play "two notes" at once. Some of the harmonics of the first note happen to align with the harmonics of the second note and some frequencies `1200hz 2400hz 3600hz ...` are made louder by the alignment of harmonics.
This number 1200 is in fact the **lowest common multiple** of the two fundamental frequencies 400 and 600.
Notice what happens if we pick two different notes instead. Here we choose 400 and 450 Hz:
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)(osc%3Asine)(env%3A2128)%0A%5B400hz_800hz_1200hz_1600hz_2000hz_2400hz_2800hz_3200hz%5D%0A%5B450hz_900hz_1350hz_1800hz_2250hz_2700hz_3050hz_3600hz%5D%0A%5B400hz_800hz_1200hz_1600hz_2000hz_2400hz_2800hz_3200hz_450hz_900hz_1350hz_1800hz_2250hz_2700hz_3050hz_3600hz%5D" title="Xenpaper" frameborder="0"></iframe>
Amongst the 8 harmonics that were listed above, none of them coincide! It's not until `3600hz` (the 9th harmonic of 400 Hz and the 8th harmonic of 450 Hz) that we see the first alignment of harmonics. Again, 3600 is the lowest common multiple of 400 and 450.
Compare this with the previous example, where the 3rd, 6th, 9th, ... harmonic of 400 Hz aligns with the 2nd, 4th, 6th, ... harmonic of 600 Hz.
This means there is much less harmonic information present when `[400hz 600hz]` is played compared to `[400hz 450hz]`, because whenever two harmonics align, instead of two different frequencies' worth of information, only one frequency perceived. The harmonics aligning much more often in `[400hz 600hz]` is one of the fundamental reasons why it sounds simpler: there is **less information**!
The amount of harmonic information is one of the contributing factors of **concordance**. This is the principle of Euler's [*gradus suavitatis*](https://www.youtube.com/watch?v=B6Dvfv_ASVg), which correlates the lowest common multiple to discordance. The more often harmonics align, the less harmonic information there is in the stimuli, the simpler the chord is perceived. The spectrum of concordance-discordance refers to an objective computable complexity of a sound. Concordance is an abstract concept because perception is never objective, however we can still develop models around this.
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:warning: Concordance is not the same as consonance. Consonance is subjective and influenced by cultural and environmental factors.
:::
Although concordance is abstract, you can get a rough idea of what concordance means like this: **Given a chord, or a part of a song, how confident are you in transcribing (identifying) the notes accurately?** But you must assume you don't already know the chords/notes used in the song, otherwise existing knowledge will cause subjectivity because familiarity affects perception.
> :pencil: Besides the *gradus suavitatis* notion of concordance, there's also other psychoacoustic factors that can be modelled to contribute to the perception of concordance, such as sum and difference tones (Tartini tones/combination tones), virtual fundamental, harmonic entropy/information and spectral entropy/information.
### It matters because concordance is part of the cultural entrainment of western tonality
Concordance still means nothing on its own. In [Part I](https://hackmd.io/@euwbah/extending-harmonic-principles-1), I have already explained why [tonality is fundamentally cultural](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Tonality-is-fundamentally-cultural). Also, Gamelan music is the perfect counterargument for why one cannot simply assume that concordance is the fundamental rule of music. Noise music is also another counter argument, because it lives on a whole other realm of function of which concordance is not a part of.
However, the tonalities in the modern western music tradition can be traced back to
1. European (Viennese, German, Italian, etc...) schools of music, which can be traced back to
2. key figures of the late Renaissance like Johannes Tinctoris and Gioseffo Zarlino, which can be traced back to
3. Ars Nova with figures such as Philippe de Vitry and Guillaume de Machaut, which can be traced back to
4. the Notre-Dame school of music, which can be traced back to
5. Guido of Arezzo in the period of the High medieval music period and peers (whom the Guidonnian hand is attributed to, and solmization: Ut Re Mi Fa Sol La, precursor of solfege), which can be traced back to
6. the treatises musica enchiriadis and scolica enchiriadis in the 9th century, which can be traced back to
7. Byzantine music, which can be traced back to Archytas (400 BC), who is a
8. Pythagorean (600 BC), whose lineage birthed the
9. Systema Teleion which is based on
10. Mathematical principle of concordance defined as using multiples of ratios (as in $x:y$) for intervals between frequencies being the 'pure music'
And because of the lineage of western music being based on rational ratios and the mathematical notion of concordance, we find a correlation between the concordance of intervals and how often such intervals are used in modern western music.
Without the context of phrasing and a prior established tonality, there is a tendency for more concordant structures to be perceived as the tonic/resolution point over less concordant structures.
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1%2F2)%7Br330hz%7D%0A%5B1%2F1_45%2F32%5D_%5B1%2F1_4%2F3%5D._%23_resolving%0A%7Br310hz%7D%0A%5B1%2F1_5%2F4%5D_%5B15%2F16_4%2F3%5D._%23_going_out%0A%7Br398hz%7D%0A%5B1%2F1_45%2F32%5D%5B1%2F1_11%2F8%5D._%23_resolving_(%3F)%0A%7Br370hz%7D%0A%5B1%2F1_13%2F11%5D%5B1%2F1_6%2F5%5D._%23_resolving_(%3F)" title="Xenpaper" frameborder="0"></iframe>
Of course, cultural entrainment of a tonality, or short term pitch memory, or rhythmic/phrasing cues can skew perceptions such that listeners perceive otherwise. E.g.,
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(bpm%3A100)%0A3%2F2_7%2F4_%5B1%2F1_2%2F1%5D_%0A%5B1%2F1_5%2F4_7%2F4_7%2F3%5D-.%0A%5B1%2F1_5%2F4_7%2F4_7%2F3%5D-.%0A%5B1%2F1_5%2F4_7%2F4_7%2F3%5D-.%0A%5B1%2F1_5%2F4_7%2F4_7%2F3%5D_7%2F4_2%2F1%0A%5B1%2F1_4%2F3_5%2F3_7%2F3%5D-.%0A%5B1%2F1_4%2F3_5%2F3_7%2F3%5D-.%0A%5B1%2F1_4%2F3_5%2F3_7%2F3%5D-.%0A%5B1%2F1_4%2F3_5%2F3_7%2F3%5D_11%2F4_21%2F8%0A%5B1%2F1_5%2F4_7%2F4_7%2F3%5D-%5B1%2F1_5%2F4_2%2F1%5D.%5B1%2F1_5%2F3_2%2F1%5D%5B7%2F8_7%2F4%5D%0A%5B1%2F1_2%2F1%5D--" title="Xenpaper" frameborder="0"></iframe>
Here, what is perceived to be the tonic (home) side of the tonality is objectively more discordant than the other side. This relies on the cultural entrainment of blues form, the repetition to build familiarity, and the phrasing that relies on simultaneously establishing the "home" tonality and the "One" of the rhythm/form using a pickup.
## What are intervals? Logarithms?
An interval is the distance between pitches. The question is, how do we measure and give names to these distances?
[Weber-Fechner law](https://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law) states that much of human perception is **logarithmic** in nature, and this applies to the perception of frequencies as well. Instead of jumping straight to highschool algebra, let's intuit logarithms with a thought experiment:
Close your eyes and imagine you are in a room with a thousand lit candles. Now imagine one candle was snuffed out. Well, if you hadn't known that one candle was snuffed out, you wouldn't have thought that anything changed at all!
<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:1000hz--_999hz--" title="Xenpaper" frameborder="0"></iframe>
Now there are 999 lit candles. One more gets snuffed out. Still nothing really changed
<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:999hz--_998hz--" title="Xenpaper" frameborder="0"></iframe>
You keep going. Now there are 100 lit candles left. One more candle starts flickering and dies away...
<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:(osc%3Afmsquare16)_100hz--_99hz--" title="Xenpaper" frameborder="0"></iframe>
And now the difference between 100 candles vs 99 candles barely noticeable (use headphones!).
You keep going till there are 2 candles left. One more goes out, and the impact of that was strongly felt, because the amount of light was halved!
And the last candle goes out, and the impact of that was infinite, because now there is only complete darkness.
The perceived amount of dimming when a certain candle is snuffed out is different depending on how many candles were lit then. You didn't **feel** the number of candles that was snuffed out, you **felt** the **percentage** or **ratio** of candles that got put out. Going from 1000 to 999 candles only had a reduction of candles by 0.1%, which is most likely not even humanly perceivable. But going from 2 candles to 1 candle, will have as much as an impact as going from 1000 to 500 candles, because what would be perceived was that the amount of light was **halved**.
In the above image stolen from Wikipedia, compare the left column with the right column. In both columns, the top and bottom images differ by 10 dots. If I told you that the numbers were wrong for the left column — the top square has 20 dots and the bottom has 10. Well, you would be quick to say I'm wrong, because it is obvious that the lower square has more dots.
However, what if I told you that for the right column, the top has 120 dots and the bottom has 110. Would you have known?
We say that human perception is **logarithmic** because when we take the logarithm of a number, we are turning it from a scale of absolutes to a scale of relatives. If the percentage change of the input is constant, the change in the output of the logarithm of the input maintains constant. The logarithm doesn't care about the absolute change, 100 compared to 200 is the same amount of relative change as 10 trillion compared to 20 trillion.
> :pencil: Fun math fact: the derivative of the natural logarithm: $\frac{d}{dx} \ln x = 1/x$ can be intuitively understood using the above thought experiment.
The base-2 logarithm of 2 is 1, $\log_2(2) = 1$. If we double 2, we get 4, $\log_2(4) = 2$, the logarithm has increased by 1. If we double that again, we get 8 $\log_2(8) = 3$, the logarithm has increased yet again by 1, even though we added 4 to the number this time. We double yet again $\log_2(16) = 4$, and yet again $\log_2(32) = 5$. Each time, we are adding twice as much to the number as before, but because the relative change to the input is always the same (we are always doubling the previous input), so the logarithm will always increase by a constant amount, 1.
The base of the logarithm $n$, denoted with the subscript in $\log_n$ refers to the amount needed to be multiplied to increase the output of the logarithm by 1. E.g., in log base 5, we have to multiply by 5 to increase the output by 1:
$$\log_5(5) = 1,\quad \log_5(25) = 2, \quad \log_5(125) = 3$$
### So what do logarithms have to do with pitch?
When we experience intervals (the distance between frequencies), we are not experiencing the absolute difference in Hertz (hz, vibrations per second). We can only experience the relative ratio of Hz between the two frequencies.
**An interval is the relative ratio between frequencies.**
In fact, semitones, octaves, and cents, are all linearly correlated to the the logarithm of frequencies, not the frequencies themselves!
:::success
:bulb: **How to convert between frequencies, ratios, cents, octaves, and semitones**
$$\text{ratio} = f_1 / f_2$$
is the ratio interval between the lower note $f_2$ and higher note $f_1$, where $f_1, f_2$ are frequencies in Hz.
$$
\begin{align*}
\text{octaves} &= \log_2(\text{ratio}) \\
\text{cents} &= \text{octaves} \times 1200 \\
\text{semitones} &= \text{octaves} \times 12 \\
\text{ratio} &= 2^{\text{octaves}}
\end{align*}
$$
:::
For example, consider the major third interval. What we call the "5-limit major third" has the designated ratio `5/4`, which means that the ratio between the higher and lower frequencies is 5 is to 4. For example, `400hz : 500hz` is a major third:
<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:(1%2F2)_400hz_500hz_%5B400hz_500hz%5D" title="Xenpaper" frameborder="0"></iframe>
The perfect fifth interval is designated `3/2`, so `300hz : 200hz` is a perfect fifth, and so is `600hz : 400hz`:
<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:(1%2F2)_200hz_300hz_%5B300hz_200hz%5D%0A400hz_600hz_%5B400hz_600hz%5D" title="Xenpaper" frameborder="0"></iframe>
Now what if I want to take the fifth `400hz : 600hz`, and add a major third to the fifth? Because pitch is logarithmic, we cannot just add the frequencies of each one. We have to multiply the fifth `600hz` by the **ratio** of the major third interval, so $5/4 \times 600 = 750 \text{ hz}$ is the addition of the two intervals, which in 12edo theory is called the major seventh.
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1%2F2)_400hz_600hz_%5B400hz_600hz%5D.%0A600hz_750hz_%5B600hz_750hz%5D.%0A%5B400hz_600hz_750hz%5D-" title="Xenpaper" frameborder="0"></iframe>
Notice that it doesn't matter whether we add the fifth first, or the major third first: $500 \times 3/2 = 750 \text{ hz}$ too.
:::success
:exclamation: **The point is...**
If we want to add a frequency by a musical interval, we multiply by the ratio of that interval. Similarly, if we want to subtract a frequency by a musical interval, we divide by the ratio of that interval.
:::
## Decomposing intervals into prime harmonics & naming notes
But how do we even know the assigned ratios of intervals? And how do we know what to call these intervals?
We'll answer the first question in this section.
The [Fundamental Theorem of Arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic) is a cornerstone of mathematics that states that for any integer (whole number), a decomposition of it into multiples of **prime numbers exists**, and that decomposition **uniquely identifies every whole number**.
**Prime numbers** are any integers 2 and above that cannot be divided without remainder except by itself or 1. 2 is prime, because trying to divide 2 by any other number besides 1 or 2 will result in a fraction. 6 is not prime, because it can also be divided by 2 or 3.
A prime decomposition/factorization goes like this. Let's say we want to decompose 250 into primes. We have to figure out using some [divisibility tests](https://byjus.com/maths/divisibility-rules/) or by experience/intuition to see which numbers can divide 250 without remainder. 250 is an even number because it ends with an even number 0, which means we can divide it into 2.
$$ 250 = 125 \times 2 $$
Now, we see that 125 is divisible by 5, because it ends with a 5.
$$ 125 \times 2 = 5 \times 25 \times 2 $$
And 25 is again divisible by 5,
$$ 5 \times 25 \times 2 = 5\times 5 \times 5 \times 2 $$
Both 5 and 2 are prime, so we stop here.
Hence, the prime factorization of 250 is $2 \times 5 \times 5 \times 5$, or we can write more succinctly using exponents/powers: $2 \times 5^3$.
### How does prime factorization relate to music?
As discussed in the last section, [an interval is a (positive) ratio](#What-are-intervals-Logarithms) between frequencies, and in just intonation, these ratios are rational (fractions). A rational number is constructed from two integers, the numerator and denominator, usually denoted as $n/d$:
$$\frac{\text{numerator}}{\text{denominator}} = \frac{n}{d}$$
For musical intervals in JI, $n, d$ are always positive integers, (which we write as $n, d \in \mathbb{N}$, meaning $n$ and $d$ belong to the set of natural numbers).
For any JI interval, we can decompose $n, d$ into their prime decompositions:
$$
\begin{cases}
n = 2^{p_2} \times 3^{p_3} \times 5^{p_5} \times \cdots \\
d = 2^{q_2} \times 3^{q_3} \times 5^{q_5} \times \cdots
\end{cases}
$$
For example, the major third `5/4` that we discussed before is decomposed as:
$$
\begin{cases}
n = 5 = 2^0 \times 3^0 \times 5^1 \times 7^0 \times \cdots \\
d = 4 = 2^2 \times 3^0 \times \cdots
\end{cases}
$$
> :pencil: Any natural number to the power of zero is equal to 1, and multiplying by 1 does nothing to the number (we call it the multiplicative identity)
Next, we note that dividing by a number is the same as multiplying by reciprocal of it. For example,
$$\frac{3}{5} = 3 \times 5^{-1}$$
And that powers of powers are multiplied. For example:
$${(3^2)}^5 = 3^{2\times5} = 3^{10}$$
Putting this all together, this is how we decompose the major third interval `5/4`:
$$\frac{5}{4} = 5^1 \times (2^2)^{-1} = 2^{-2} \times 5^1 $$
Putting this back into music, recall that we add intervals by multiplying, and subtract intervals by dividing. The above decomposition of `5/4` means:
* Divide by the number `2` twice, because it has the power of negative 2. This means we lower the pitch by two copies of the interval associated with `2`. `2` is associated with the octave, so $2^{-2}$ means we go **down two octaves**.
* Multiply by the number `5` once. `5` is associated with the interval of a major third plus two octaves.
Putting these together, we get that the major third interval `5/4`, is the interval we get where we go down two octaves, then go up a major third and two octaves. Great.
### The JI subgroup of a tuning
That didn't seem like it did anything! However, the important takeaway here is that **we don't need to memorize every number, fraction, and their corresponding interval!** By decomposing an interval down to its [prime intervals](#The-prime-intervals), **we only need to learn the intervals of prime numbers**.
Because we are working within a fixed number of notes, e.g., 12 or 31 edo, we don't need many prime intervals to fully capture the essence of the tuning. To represent 31edo, we only need 2, 3, 5, 7, and 11, and for 12edo we only need 2, 3 and 5, though 7 and 11 are still useful for the blues tonalities. Of course, we can use more, but they become redundant because of the [tempered commas (Part III)](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Temperaments-amp-Mapping).
In xen math, we call the collection of primes (or prime powers) that we consider for a tuning/temperament the **JI subgroup** of the tuning. We write it using dots: `2.3.5` is the conventional JI subgroup for 12edo and `2.3.5.7.11` is for 31edo. We are free to decide which primes we want to use or ignore.
Because of the Fundamental Theorem of Arithmetic, each prime in the JI subgroup will unlock an infinite set of new pitches/intervals that are unique to that prime number. Every prime yields a set of intervals that only *lives in the world of that prime*. A prime number in the JI subgroup is like a completely new axis in a coordinate system. For example, the prime 2 can map to the up-down dimension, 3 can map to left-right, and 5 can map to front-back. You can't go up if you only keep going right. Unfortunately, for higher primes, you must imagine an abstract higher-dimensional space. Maybe this is one of the contributing factors as to why 12edo was settled on.
Here is a visualization of the 3-dimensional `2.3.5` JI subgroup by mannfishh, where the brightness of the squares is also its own dimension (2), that way the other 2 dimensions (3 and 5) can fit into a 2-dimensional video:
<iframe width="560" height="315" src="https://www.youtube.com/embed/jg8038krbTY?si=xgITymHFKm1YI6Zd" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
This is the essence of what a JI subgroup entails musically speaking. In a later section, if time allows, we may discuss group theory and the actual definition of a subgroup.
## The prime intervals
| Prime interval | Cents | Interval name | Prime limit classifier name |
| --: | :--: | -- | -- |
| 2 | 1200.00 | Octave | 2-limit |
| 3 | 1901.96 | Perfect 5th + octave | Pythagorean / 3-limit|
| 5 | 2786.31 | Major 3rd + 2 octaves | Classic / 5-limit |
| 7 | 3368.83 | Subminor/harmonic 7th + 2 octaves | Septimal/7-limit |
| 11 | 4151.32 | Superfourth + 3 octaves | Undecimal/11-limit |
As we discussed in the previous section, each prime number unlocks a new world/dimension of intervals. Let us explore each world one at a time.
### Prime interval 2: octave
<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:(1)_100hz_200hz_400hz_800hz_1600hz" title="Xenpaper" frameborder="0"></iframe>
The octave is the second harmonic. Doubling the frequency increases the pitch by an octave. This is why the 12th fret of a guitar is positioned exactly at the middle of the string (and if your guitar has 24 frets, the 24th would be 1/4 of the way from the bridge to the nut).
:::info
:writing_hand: **Exercise**
Recall the formula for converting an interval ratio to the number of octaves it spans:
$$\text{octaves} = \log_2(\text{ratio})$$
Based on the intuitive [explanation of logarithms](#What-are-intervals-Logarithms) in the above section, can you explain this formula?
:::
If we limit ourselves to using only the prime number 2 for our JI intervals (i.e., 2-limit JI), notice that we won't get any other notes except octaves of the same note. Octaves are the simplest interval there is, it is the first prime. Two notes an octave apart have a high concordance because apart from the lower note's fundamental, every harmonic is aligned!
For this reason, western music theory assumes a theory of **octave equivalence**, where notes that are octaves apart are still regarded functionally the same. Of course, this isn't actually true as we have seen in the previous part how octaves do matter in the perception of tonalities.
We notate the default tuning note A4 = 440hz as the note A in the 4th octave. The octave number is relative to the octaves displaced from the tuning note, so if we are tuning to A4 = 440:
$$
\begin{align*}
& \vdots \\
\texttt{A3} &= 440 \div 2 = 220 \text{ Hz} \\
\texttt{A4} &= 440 \text{ Hz} \\
\texttt{A5} &= 440 \times 2 = 880 \text{ Hz} \\
& \vdots
\end{align*}
$$
### Prime interval 3: fifth + octave
The next prime interval 3 corresponds to the interval of a fifth plus an octave:
<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:(1%2F2)_100hz_300hz_900hz_2700hz" title="Xenpaper" frameborder="0"></iframe>
Using what we've learned before, we can subtract an octave from this by dividing by the octave interval `2`.
$$\underbrace{3}_\text{fifth + oct} \div \underbrace{2}_\text{octave} = \underbrace{\frac{3}{2}}_\text{fifth}$$
Thus, `3/2` is the perfect fifth interval, as expected. The act of reducing by octaves to obtain an interval that is less than an octave wide is called **octave-reduction**. We can call `3/2` the **octave-reduced 3rd harmonic**.
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)_400hz_600hz_%5B400hz_600hz%5D-%0A%7Br400hz%7D_1%2F1_3%2F2_%5B1%2F1_3%2F2%5D-%0A2%3A3._3%3A2." title="Xenpaper" frameborder="0"></iframe>
> :musical_note: There is a slight difference when ratios are written as `3/2` vs `2:3` vs `3:2` in xenpaper and also in xen nomenclature used in this series of articles.
>
> * The fraction `3/2` refers to a single note, which is tuned 3/2 times the frequency of whatever the tuning note is, which in the above example is 400hz.
> * The ratio `2:3` refers to two notes (a dyad), the first note corresponding to `2` units of the ratio, which is tuned to the tuning note at 400hz, and the second note is 3 units of the ratio, so it is `3/2` the frequency of the tuning note.
> * The ratio `3:2` also refers to a dyad, but the first note that is `3` units corresponds to the tuning note, so the second note of `2` units is `2/3` of the tuning note, a fifth **lower**.
Adding this prime number now gives us the _"circle"_ of fifths.
<iframe width="560" height="240" src="https://luphoria.com/xenpaper/#embed:(1)_1%2F1_3%2F2_9%2F4_27%2F16_81%2F64_243%2F128_729%2F512_2187%2F2048_6561%2F4096_19683%2F16384_59049%2F32768_177147%2F131072_531441%2F524288-%0A1%2F1_531441%2F524288_1%2F1_531441%2F524288_%23_circle_%3F%3F%3F" title="Xenpaper" frameborder="0"></iframe>
I say _"circle"_ because in fact, it was never originally a circle! If we keep going up or down in `3/2` fifths, no matter how hard we try, because of the Fundamental Theorem of Arithmetic, every prime decomposition is unique to that number, which means no power of two (stacked octaves) can align with any power of three (stacked octaves + fifths), so we will never reach back at our starting point again!
<iframe width="560" height="315" src="https://www.youtube.com/embed/-6VEu64x4zU?si=wkaCuADLv-ZV4AjQ" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
Unlike in 12edo, where we go up 12 fifths and arrive back at the same note (assuming octave-equivalence), in JI, fifths never repeat and should instead be represented with a spiral of fifths.
#### The nominals & Pythagorean accidentals
We're only on our second prime number and we have theoretically infinite octaves times infinitely many fifths. How do we name all these notes?
We start with the musical alphabets $$\texttt{C D E F G A B}$$ These are called the **nominals**, the fundamental notes without accidentals.
The octave numbering goes up/down when we wrap around the alphabet. E.g.,
$$\texttt{... A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 ...}$$
However, the fifths don't relate to the nominals in this order. Instead the fifths uses this ordering:
> :pencil: The above diagram is called a JI lattice, where moving one step in a dimension/axis represents one step of a interval (usually prime).
What happens above B? We start from F again, but now every note has a sharp sign
$$\dots \texttt{E B F}\sharp\texttt{ C}\sharp\texttt{ G}\sharp\texttt{ D}\sharp \dots$$
Or if we need to go below F, we loop back around to B, but add a flat sign:
$$\dots \texttt{C F B}\flat\texttt{ E}\flat\texttt{ A}\flat\texttt{ D}\flat\texttt{ G}\flat\texttt{ C}\flat\dots$$
The sharp sign refers to the act of going up 7 fifths (i.e. $(3/2)^7 = 2187/128$) and subtracting by 5 octaves ($2^5 = 32$) so that the interval is reduced to less than an octave, the size of about a semitone. If we do the math, that all adds up to inteval:
$$\left(\frac{3}{2}\right)^7 \div 2^5 = \frac{2187}{2048} \approx 113.69\text{ cents}$$
This interval is `2187/2048` and is called the **apotome**.
$$\sharp = \frac{2187}{2048}$$
Similarly, the flat sign is just the apotome but downwards, `2048/2187`.
$$\flat = \frac{2048}{2187} = \sharp^{-1}$$
We also have double accidentals:
$$
\begin{align*}
\flat\mkern-2.4mu\flat &= \flat \times \flat = \frac{4194304}{4782969} \\[1em]
\texttt{x} &= \sharp \times \sharp = \frac{4782969}{4194304}
\end{align*}
$$
These accidentals ♭ ♯ are called the **Pythagorean accidentals**, because they live in the 3-limit world of fifths. There are also triple, quadruple, quintuple sharps or flats, obtained by combining double/single sharp/flat symbols.
There's still one thing to address. If we keep going up in fifths, the octave numberings will change. Every time we go past C (e.g., the fifth from A to E will pass by C in the alphabetical nominal order: A B **C** D E), the octave number should go up, regardless of accidentals attached to the note:
There's a fixed octave offset pattern of `+0 +1 +1 +2 +2 +3 +3` following the order of fifths $\texttt{F C G D A E B}$.
:::info
:writing_hand: **Exercise** Pythagorean intervals
Using positive/negative powers of $3/2$ to raise/lower a note by fifths, and powers of 2 to raise/lower by octaves, write down all the ratios for the spiral of fifths from Fb to B#, with all the notes in the 4th octave, relative to the tuning note $\texttt{A4} = 1/1$
E.g., the note `E4` is `2/3` relative to A4, the note `Bb4` is `256/243` relative to A4.
:::
:::info
:writing_hand: **Exercise** Apotomes and limmas
3-limit JI is also known as Pythagorean tuning, as the cultural importance of the fifth was attributed to Pythagoras. There are two main Pythagorean semitones, the first is the semitone between two adjacent nominals, such as E-F or B-C. This was called the limma. The other type of semitone we have already encountered, it is the semitone interval obtained by raising a note by a sharp/flat, which is the apotome `2187/2048`.
What is the ratio of a limma? Do there exist pairs of notes which are a limma apart where both notes have different accidentals?
:::spoiler Answer
`256/243`. Yes, e.g., `C` and `Db` are a limma apart.
:::
<!--:::-->
### Prime interval 5: major third + 2 octaves & what does major third even mean?
The third prime is 5, which corresponds to a major third plus 2 octaves:
<iframe width="560" height="315" src="https://www.youtube.com/embed/oUcimJsovsI?si=Fwz4GuFingRZ1fA0" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
Recall that in [How does prime factorization relate to music?](#How-does-prime-factorization-relate-to-music), we reduced this by two octaves to obtain the major third `5/4`.
In 12edo, a major third is `4\12` (4 steps of 12edo), which means that three of them will add up to an octave. But as with fifths, because of the Fundamental Theorem of Arithmetic, no power of 5 will align with a power of 2, so we will never end up back again at the same note no matter how many thirds we stack:
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)_1%2F1_5%2F4_25%2F16_125%2F64_._1%2F1_2%2F1_.%0A2%2F1_125%2F64_2%2F1_125%2F64_%0A%23_3_major_thirds_don't_add_up_to_an_octave" title="Xenpaper" frameborder="0"></iframe>
We can obtain the difference between 3 stacked thirds `125/64` and an octave `2/1` by dividing them, which gives us the interval `128/125`. This is called the diesis (we'll encounter this in 31edo as `1\31` later), which is the difference between 3 major thirds and an octave.
Now the pressing issue here is this: We already have a major third from the 3-limit world, which is `81/64`. When we decompose this interval into primes, we get $81/64 = 2^{-6} \times 3^4$, which translates to going up the prime interval `3/1` four times, then going down `2/1` six times.
After you work it out, it translates to going up 4 fifths, and up $4 - 6 = -2$ octaves, that is, down 2 octaves (we had to subtract 6 because the prime interval 3 is a fifth *plus an octave*).
For example, if we start on the note `C3`, and go up 4 fifths and down 2 octaves, we get:
$$\texttt{C3} \to \texttt{G3} \to \texttt{D4} \to \texttt{A4} \to \texttt{E5} \to \texttt{E4} \to \texttt{E3}$$
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)_%7Br16%2F27%7D%0A1%2F1_3%2F2_9%2F4_27%2F8_81%2F16_%23_up_4_fifths%0A81%2F32_81%2F64-._%23_down_2_octaves%0A1%2F1_81%2F64_%5B1%2F1_81%2F64%5D" title="Xenpaper" frameborder="0"></iframe>
Those familiar with music theory will be wondering: isn't the interval `C3` to `E3` already a major third? But now we're saying `5/4` is also a major third... Why are there two major thirds?
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1%2F2)_%5B1%2F1_81%2F64%5D-_%5B1%2F1_5%2F4%5D-%0A%23_why_are_there_two_major_thirds%3F" title="Xenpaper" frameborder="0"></iframe>
What do we even mean by major third? Is there some kind of threshold of cents/interval size for certain names? Well, some people do use cents and bounding regions to decide interval names, but there is a way to constructively define these names:
#### Music theory crash course: interval names in classical music theory
> This guide does not assume any background in music theory, but even if you do know what these words mean, this section might give you a new perspective, because I will introduce these familiar terms **entirely without the use of 12edo**.
In modern day western music theory, we name intervals by a system of cardinal numbers: Unison, second, third, fourth, fifth, sixth, seventh, eighth/octave, ninth, etc... (Instead of "first" we usually say "unison"). In short, we can write root/1st, 2nd, 3rd, 4th, 5th, ... as well.
These cardinal numbers don't actually mean anything on its own. The interval distance of a 'second' is not always the same as some other 'second', and in certain tunings, sometimes a 'seventh' can be exactly the same as a 'sixth'.
The only information these cardinal numbers give is the relative distance between **nominals** (the musical alphabets). For example, if we start on D:
* D - unison
* E - second
* F - third
* G - fourth
* A - fifth
* B - sixth
* C - seventh
* D - octave
But if we start on E instead, then:
* E - unison
* F - second
* G - third
* A - fourth
* ... you get the idea.
Back in [Prime interval 3: Nominals & Pythagorean accidentals](##The-nominals-amp-Pythagorean-accidentals), recall that the nominals were named and tuned according to the spiral of fifths.
Using that information, we can work out the interval ratios for thirds. We've already worked out the interval between C3 and E3 just now, which is `81/64`. Does that mean a third is always `81/64`? Let's verify this by checking the interval between D3 and F3, which is also a third:
We can obtain F3 from D3 using this lattice:
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)_%7Br4%2F3%7D%0A1%2F1_2%2F3_4%2F9_8%2F27_%23_down_3_fifths%0A16%2F27_32%2F27-._%23_up_2_octaves%0A1%2F1_32%2F27_%5B1%2F1_32%2F27%5D" title="Xenpaper" frameborder="0"></iframe>
The third interval between `D3-F3` is `32/27` ≈ 294.13¢, which is not the same as `81/64` between `C3-E3`!
If we repeat this process starting on each nominal, we realize there are only two different sizes of thirds, regardless of octave:
* C-E, F-A, and G-B are `81/64` thirds, which are larger, so we call them **major**
* D-F, E-G, A-C, and B-D are `32/27` thirds the smaller third, so we call them **minor**.
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B1%2F1_81%2F64%5D___%23_I_maj___CE%0A%5B9%2F8_4%2F3%5D_____%23_ii_min__DF%0A%5B81%2F64_3%2F2%5D___%23_iii_min_EG%0A%5B4%2F3_27%2F16%5D___%23_IV_maj__FA%0A%5B3%2F2_243%2F128%5D_%23_V_maj___GB%0A%5B27%2F16_2%2F1%5D___%23_vi_min__AC%0A%5B243%2F128_9%2F4%5D_%23_vii_min_BD" title="Xenpaper" frameborder="0"></iframe>
> :pencil: The terms "major" and "minor" are words used to describe **qualities**, and are heavily [semantically overloaded](https://en.wikipedia.org/wiki/Semantic_overload), because these qualities can describe not only intervals and chords but also tonalities, as we have already seen in the previous section.
Notice how the distribution of sizes of thirds are pretty even, 3 major thirds and 4 minor thirds amongst the nominals.
However, for unisons and octaves, there really is only ever one octave. The size of an octave doesn't change depending on which note you start at. For that reason, we use the word **perfect** to describe octaves and unisons.
Doing this procedure for fifths, we find that:
* C-G, D-A, E-B, F-C, G-D, A-E are all `3/2`
* but B-F is `1024/729`
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B1%2F1_3%2F2%5D_______%23_CG%0A%5B9%2F8_27%2F16%5D_____%23_DA%0A%5B81%2F64_243%2F128%5D_%23_EB%0A%5B4%2F3_2%2F1%5D_______%23_FC%0A%5B3%2F2_9%2F4%5D_______%23_GD%0A%5B27%2F16_81%2F32%5D___%23_AE%0A%5B243%2F128_8%2F3%5D___%23_BF" title="Xenpaper" frameborder="0"></iframe>
Wait... didn't we establish earlier that the definition of [fifth](#Prime-interval-3-fifth--octave) is `3/2`?
Well, we did, but remember that fifths added a sharp every time we reached B and have to wrap around back to F. So `B-F#` would be the actual `3/2` fifth here, not `B-F`.
Since almost all fifths are the same, we call the `3/2` the **perfect** fifth. Although, it is not as perfect as octaves or unisons, because of that exception.
For the smaller fifth `1024/729` between B and F (where `1/1` is B and `1024/729` is F), we don't really want to call it a _minor_ fifth, because the word "minor" implies that the dual opposite "major" must exist. Instead, we call it the **diminished** fifth. The perfect fifth of B is F#, but since the fifth here is `B-F`, it is missing a #, which means the fifth is being shrunk by an apotome, `2187/2048`. We can sanity check this:
$$\underbrace{\frac{3}{2}}_\text{F#} \div \underbrace{\frac{2187}{2048}}_\text{without #} = \underbrace{\frac{1024}{729}}_\text{is F}$$
Hence, the word **diminished** means **shrinking by one apotome**.
Similarly, we can enumerate all the fourths:
* C-F, D-G, E-A, G-C, A-D, B-E are all `4/3`, we call them **perfect fourths**
* F-B is `729/512`, is an **augmented fourth**, because it is bigger.
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B1%2F1_4%2F3%5D_______%23_CF%0A%5B9%2F8_3%2F2%5D_______%23_DG%0A%5B81%2F64_27%2F16%5D___%23_EA%0A%5B4%2F3_243%2F128%5D___%23_FB%0A%5B3%2F2_2%2F1%5D_______%23_GC%0A%5B27%2F16_9%2F4%5D_____%23_AD%0A%5B243%2F128_81%2F32%5D_%23_BE" title="Xenpaper" frameborder="0"></iframe>
Compared to the perfect fourth starting on F, which should go to Bb, not to B, the F-B fourth is **augmented by an apotome**, because the flat in Bb is being removed by the addition of a sharp (yes, that is how it works — sharps cancel flats and vice versa). We can sanity check this:
$$\frac{4}{3} \times \frac{2187}{2048} = \frac{729}{512}$$
For seconds, sixths and sevenths, much like the thirds, we also find that there are only two sizes:
* Seconds
* C-D, D-E, F-G, G-A, A-B are `9/8` major seconds
* E-F, B-C are `256/243` minor seconds (aka limmas)
* Sixths
* C-A, D-B, D-F, G-E are `27/16` major sixths (equal to an octave minus a minor third)
* E-C, A-F, B-G are `128/81` minor sixths (equal to an octave minus a major third)
* Sevenths
* C-B, F-E are `243/128` major sevenths (equal to an octave minus a minor second)
* D-C, E-D, G-F, A-G, B-A are `16/9` (equal to an octave minus a major second)
:::info
:writing_hand: **Exercises**
**Question 1**
Work out the ratios for the above dyads (2-note chords) formed by the seconds, sixths and sevenths going up the scale of nominals relative to C, and type them into xenpaper, following the previous xenpaper examples.
**Question 2**
The 7-note scale formed by going up the 7 nominals is called the **C major scale**, because the unison note is set to C, and the quality of the second, third, sixth and seventh are all **major**, and the fourth and fifth are **perfect**. However, this not a reason to believe that the major scale is the "most correct" scale. This is only a cultural convention.
The word **diatonic** is an adjective meaning "transposing up/down according to a major scale instead of fixed interval ratios".
C-E-G being transposed to D-F#-A is not a diatonic transposition. However, C#-E-G transposed to D-F#-A is. **Which major scale does this belong to? Do not use 12edo in your explanation and explain using ratios.**
:::
So far amongst the 7 diatonic notes of the major scale, we have seen that:
* unison and octaves can only be perfect
* fourths can be perfect or augmented
* fifths can be perfect or diminished
* seconds, thirds, sixths, sevenths can be major or minor
However, we can go beyond 7 notes by extending systematically by these rules
* For unisons, octaves, fourths and fifths, the augmented and diminished modifiers can be used (and repeatedly stacked, e.g., triple augmented, or double diminished) to refer to additions/subtractions of apotomes (# = `2187/2048`) from the base interval.
* For seconds, thirds, sixths and sevenths, the augmented and diminished modifiers can extend (repeatedly) above major and below minor respectively, also adjusting by apotomes.
Putting all of this together, we can tabulate the ratios and note names that form the respective intervals with respect to the note C4:
| | Diminished (d) | Minor (m) | Perfect [P] | Major [M] | Augmented [A]|
|:--:|:--:|:--:|:--:|:--:|:--:|
| **Unison** | Cb4 = 2048/2187 | :x: | C4 = 1/1 | :x: | C#4 = 2187/2048 |
| **Second** | Dbb4 = 524288/531441 | Db4 = 256/243 | :x: | D4 = 9/8 | D#4 = 19683/16384 |
| **Third** | Ebb4 = 65536/59049 | Eb4 = 32/27 | :x: | E4 = 81/64 | E#4 = 177147/131072 |
| **Fourth** | Fb4 = 8192/6561 | :x: | F4 = 4/3 | :x: | F#4 = 729/512 |
| **Fifth** | Gb4 = 1024/729 | :x: | G4 = 3/2 | :x: | G#4 = 6561/4096 |
| **Sixth** | Abb4 = 262144/177147 | Ab4 = 128/81 | :x: | A4 = 27/16 | A#4 = 59049/32768 |
| **Seventh** | Bbb4 = 32768/19683 | Bb4 = 16/9 | :x: | B4 = 243/128 | B#4 = 531441/262144 |
| **Octave** | Cb5 = 4096/2187 | :x: | C5 = 2/1 | :x: | C#5 = 2187/1024 |
We can use the abbreviations d, m, P, M, A prefixing a number to succinctly refer to an interval. E.g., d5 = diminished fifth, M2 = major second, AA3 = doubly augmented third.
If the interval is wider than an octave, we can use ninths, tenths, elevenths, etc... to refer to seconds, thirds, fourths, etc... respectively shifted up an octave, e.g. M9 = major ninth = major second + 1 octave. m25 = minor 25th = minor 3rd + 3 octaves.
:::info
:writing_hand: **Exercise**
Have I made a mistake in the above table?
:::
Honestly, this system of naming is clearly inefficient, inconsistent and should be abolished. However, I have no choice but to use these words as over time, these words have gained extra semantic meaning. "Major third" to the modern musician may no longer just about the interval `81/64` itself, but also hint at the possible tonality state(s) we are at, and even emotional associations. Revisit the [section on tonalities in Part I of this series](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Tonality) for better context.
#### So what does that have to do with having two different major thirds?
The purpose of the previous section was two-fold: to reintroduce conventional interval names in basic music theory without the use of 12edo, and to set in stone that the major third `81/64` is regarded as the "original" primordial major third, at least within the context our JI notation system, but also in history.
The `81/64` major third is used in the [diatonic genus](https://en.wikipedia.org/wiki/Genus_(music)#:~:text=Ptolemy%27s%20ditonic%20diatonic%2C) of the [systema teleion](https://en.wikipedia.org/wiki/Musical_system_of_ancient_Greece), which is the [ancestor of the major scale](#It-matters-because-concordance-is-part-of-the-cultural-entrainment-of-western-tonality) in modern music theory.
So, when we say that `5/4` is also a major third. What we really mean is that `5/4` is the **_new_** major third, not **_the_** major third. In Figure 1, page 109 (PDF page 57), of [History of Consonance and Dissonance](https://www.plainsound.org/pdfs/HCD.pdf) by James Tennney, it tabulates the evolution of specific intervals being perceived as consonant or dissonant over diffferent eras of music. (Recall that consonance, "pleasant-sounding"-ness, is subjective, unlike concordance)
Thirds were not perceived as consonant until the 12th century. There are various hypotheses as to why, but the main one was that the musicians of that time were not using the `5/4` major third, but instead the `81/64` one, which is objectively more discordant:
<iframe width="560" height="150" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A4%3A5--._64%3A81--" title="Xenpaper" frameborder="0"></iframe>
#### How to name the "other" major third
To distinguish between the two major thirds, we can use the interval's prime-limit to categorize it.
`81/64` is called the 3-limit major third, also known as Pythagorean major third, because its highest prime number base in its decomposition is 3, and tuning in fifths (3-limit) is attributed to Pythagoras (even though other cultures developed this first).
`5/4` is called the 5-limit major third, also known as the classic major third, because of its popularity as a consonant interval in the "classical" periods of music (renaissance onwards).
The difference between these two intervals is:
$$\frac{81}{64} \div \frac{5}{4} = \frac{81}{80}$$
which is known as the **syntonic comma**. In **Part I: 31edo tonalities**, the reference to syntonic comma is the same as this one. We will highlight the importance of commas, or more importantly, the lack of commas, in [Part III: Temperaments](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Temperaments-amp-Mapping).
> :pencil: **comma** comes from κόμμα: (noun) a piece that is cut off; chaff.
There is a standardized way to name these "other" intervals/notes outside of the 3-limit/Pythagorean ones:
1. Find the "nearest" (:pencil:) 3-limit interval/note
2. Find the **comma**, which is an interval of the difference between the nearest 3-limit interval and the actual higher-limit interval that we want
3. Define an accidental that adjusts by the comma, such that if we append the comma's accidental to the 3-limit interval/note, we get the higher-limit interval that we are looking for
> :pencil: We only pick the 3-limit note within 6 up or down of the root note of the interval. Otherwise, there are infinitely many 3-limit Pythagorean notes — even after you've found the "nearest" 3-limit note to the note that you want to name, there will always be one 3-limit note that is nearer still.
>
> For example, relative to C, the 3-limit note E = 81/64 is around 408 cents from C, and the 3-limit note Fb = 8192/6561 is around 384 cents from C. The `5/4` major third that we want to name is around 386 cents, which makes Fb closer to it than E. However, do we really want to call the `5/4` major third triad `C Fb+ G`? Even then, there will be yet another 3-limit note that will be closer to `5/4` than Fb, like Axxx (A sextuple sharp), which is 387.9 cents. Imagine a world where the C major triad reads `C Axxx- G`!
In this case, the nearest 3-limit note is `81/64`, and the comma is the syntonic comma `81/80`. Specifically, we have do go down from the 3-limit E4 = `81/64` by multiplying by the reciprocal `80/81`. If we set C4 = `1/1`, we can draw the diagram:
The `81/80` interval is written as `+`, and its reciprocal `80/81` is written as `-`. This is the text-based [HEWM (Helmholtz-Ellis-Wolf-Monzo) notation](http://www.tonalsoft.com/enc/h/hewm.aspx), which makes it easier to type and communicate notes in JI.
In notation, the most common JI notation would be HEJI (Helmholtz-Ellis Just Intonation). For this comma in HEJI, an up or down arrow is attached to a Pythagorean (♭ ♯) or natural (♮) accidental. Refer to https://w3c.github.io/smufl/latest/tables/extended-helmholtz-ellis-accidentals-just-intonation.html for what they look like.
#### For 12edo, we stop here
We already had more than enough notes for 12edo in the 3-limit, it had the whole circle of 12 fifths and infinitely more.
However, it's common to associate 12edo with the 5-limit [JI subgroup](#The-JI-subgroup-of-a-tuning) `2.3.5`, because the 5-limit major third is the consonant major third in history, and in 12edo, we generally regard major thirds as consonant:
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A0%5C12_4%5C12_7%5C12_%5B0_4_7%5D--" title="Xenpaper" frameborder="0"></iframe>
We use the 5-limit world to gain access to the notion of consonant major thirds, and usually detemper (convert from non-JI to JI) the consonant thirds, sixths, and sevenths in 12edo using 5-limit intervals. We'll get to this in Part III.
### Prime interval 7: subminor 7th + 2 octaves
The interval of the (octave-reduced) 7th harmonic, `7/4`, goes by many possible names. It is significantly flatter than the other two minor 7ths we have. The 3-limit min 7th `16/9` stands at 996.08¢, 5-limit min 7th `9/5` is 1017.59¢, but this interval `7/4` is only 968.82¢, nearly half a semitone flatter than `9/5`:
For this reason, the 7-limit m7 is usually called subminor 7th, especially if the tuning system we're working in supports both the normal minor 7th and this 7-limit minor 7th. We can also call this the **septimal minor 7th**.
> :pencil: The modifiers **sub** and **super** can be used to generally categorize intervals that are sufficiently far away from the 3-limit ones, though currently, there is no consensus for how far away exactly.
<iframe width="560" height="315" src="https://www.youtube.com/embed/q3olHellHzM?si=koK_ISlm-41zn_qL" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
Here is the JI lattice comparing the three different sevenths.
:::info
:writing_hand: Write out the ratios relative to C and enter them in xenpaper to hear the notes in above lattice.
:::spoiler Hint
<iframe width="560" height="390" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A1%2F1_%5B1%2F1_4%2F3%5D_4%2F3_%5B4%2F3_16%2F9%5D._1%2F1_16%2F9_%5B1%2F1_16%2F9%5D--._%23_3-limit_m7%0A1%2F1_%5B1%2F1_6%2F5%5D_6%2F5_%5B6%2F5_9%2F5%5D._1%2F1_9%2F5_%5B1%2F1_9%2F5%5D--._%23_5-limit_m7%0A1%2F1_5%2F4_6%2F4_7%2F4_%5B1%2F1_5%2F4_6%2F4_7%2F4%5D_._1%2F1_7%2F4_%5B1%2F1_7%2F4%5D--._%23_7-limit_m7%0A%0A%5B1%2F1_16%2F9%5D-%5B1%2F1_9%2F5%5D-%5B1%2F1_7%2F4%5D-" title="Xenpaper" frameborder="0"></iframe>
:::
<!-- ::: hackmd editor bug -->
#### Notating the prime interval 7
The standard procedure for primes 5 and higher are all the same.
1. Pick the nearest related 3-limit note, within 6 sharps or flats of the base note
2. Find the comma (the discrepancy between the 3-limit note and the target 7-limit note)
3. Append an accidental corresponding to that comma to the 3-limit note.
In this case, the nearest Pythagorean note will be the minor 7th. Relative to C = `1/1`, that would be Bb = `16/9`. The comma between `16/9` and `7/4` is:
$$ \frac{16}{9} \div \frac{7}{4} = \frac{64}{63}$$
`64/63` is known as the **septimal comma**. In HEWM notation, we write `>` for `64/63` and `<` for `63/64`.
> :pencil: `<` and `>` look like the number 7. 7 ~ septimal. The accidental for this comma in HEJI also looks like a 7.
Thus, to notate the octave-reduced 7th harmonic, relative to C, we write:
$$\underbrace{\frac{16}{9}}_{\LARGE \texttt{Bb}} \times \underbrace{\frac{63}{64}}_{\LARGE\texttt{<}} = \underbrace{\frac{7}{4}}_{\LARGE\texttt{Bb<}}$$
### The prime interval 11: superfourth + 3 octaves
<iframe width="560" height="315" src="https://www.youtube.com/embed/E3AGo_bOagA?si=XO23yf8pRwLtuZWe" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
> You can find pseudoscience online claiming that this harmonic cures cancer lol
We can reduce this by 3 octaves to get the interval `11/8`, which comes in at 551.31¢ (almost halfway between a fourth and sharp fourth in 12edo). Because this is sharper than a perfect fourth, we call this an **undecimal superfourth**.
The closest sane 3-limit note to a superfourth is the perfect fourth `4/3`, and we can calculate the comma:
By going up by the comma `33/32` it takes us from the fourth to the undecimal superfourth.
In HEWM, we write the comma `33/32` as `^` and `32/33` as `v`. The HEJI notation can be found in the above video by mannfishh, and uses the standard quartersharp `+` and quarterflat `d` accidentals.
### Scale degree notation
Currently, we have been using ratios to denote relative intervals between notes, and note names to denote absolute pitches.
However, ratios can be unwieldy and hard to read. We can use scale degree notation to have the best of both worlds - the succinctness of naming absolute pitches, and the relativity of the interval so that we are not fixed to any one particular note or key.
#### HEWM scale degree notation
If we want the scale degree of an interval, we write the interval relative to C, then substitute the nominals C, D, E, F, G, A, B with the scale degrees (numbers) 1, 2, 3, 4, 5, 6, 7 respectively. For intervals an octave or wider, we can use 8, 9, 10, etc... The **accidentals are written before the number**, instead of in note names where the accidentals go after the nominal. Apart from that, accidentals don't have to be in any particular order, though it would be good formatting to put the Pythagorean accidental last, and arrange the accidentals in prime-limit order.
For example, the interval `11/8` relative to C is written `^F` in HEWM.
Then, we just replace `F` with `4`, which means the scale degree notation for `11/8` is `^4`. If we take it up an octave, we just increase the number by 7:
<iframe width="560" height="300" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A%23_interval_%7C_scale_degree%0A1%2F1________%23_1%0A11%2F8_______%23_%5E4%0A11%2F4_______%23_%5E11%0A11%2F2_______%23_%5E18" title="Xenpaper" frameborder="0"></iframe>
Now, no matter which note is the root/tuning note, we can always refer to the `11/8` of the root as `^4` instead of having to use the ratio. E.g., if `D4` is `1` (the root), then `^4` of `D4` refers to the `11/8` of `D4`, which is...
:::spoiler Answer
G^4
:::
:::info
:writing_hand: **Exercise: Scale degrees**
What are the absolute note names for the...
1. `2` of `F4`
2. `2` of `B4`
3. `b7` of `Gb4`
4. `#4` of `Db4`
5. `-3` of `E4`
6. `-7` of `B+3`
7. `++<^^b12` of `D-->vv3`
What are the scale degrees for the following ratios?
8. 4/3
9. 9/4
10. 729/512
11. 1024/729
12. 5/4
13. 32/25
14. 125/64
15. 7/1
16. 49/32
:::spoiler Answers
1. G4
2. C#5
* C#4 is wrong because we went past from B to C so the [octave number increments](#The-nominals-amp-Pythagorean-accidentals).
3. Fb5
* E5 is wrong. It is enharmonically equivalent in 12edo, but E is not equal to Fb in an infinite number of other tuning systems.
* Fb4 is wrong, the octave number increments.
4. G4
5. G#-4
6. A#4
* A#+-4 is technically correct, but is not good because the + `81/80` and - `80/81` comma accidentals cancel out.
7. Ab4
* The comma accidentals of the tuning note and the scale degree perfectly cancel out. Hence, we can ignore all of them, simplifying to the `b12` of `D3`. 12 is an octave above 5, so we can simplify to `b5` of `D4`.
8. 4
9. 9
* 2 is wrong, the interval is displaced up one octave.
10. #4
11. b5
12. -3
* 3 is wrong, there is a syntonic comma adjusted down from the Pythagorean major third.
13. ++b4
* We can simplify this ratio by breaking it down to $\frac{32}{25} = 2 \times \frac{4}{5} \times \frac{4}{5}$. We know that `4/5` is the reciprocal of `5/4`, so it is interval of going down by a 5-limit major third. `2` brings us up an octave. Hence, this is the interval of up and octave to scale degree `8`, down a classic major third to `b6+`, and down another one to `b4++`.
14. ---#7
* Same principle as the 13., we factorize $\frac{125}{64} = \left(\frac{5}{4}\right)^3$.
15. <b21
* This is the 7th harmonic `7/4` which is `b7<`, but displaced 2 octaves higher.
16. <<b6
* We can factorize $\frac{49}{32} = \frac{7}{4} \times \frac{7}{4} \times \frac{1}{2}$, which translates to going up two octave-reduced 7th harmonics from `1` to `<b7` to `<<b13`, then down an octave to `<<b6`.
:::
<!--:::-->
#### Functional Just System (FJS)
Another system of using scale degrees to name interval ratios is the [Functional Just System](https://misotanni.github.io/fjs/en/index.html). We don't use it in this guide, so I will not go through it, but it's beginning to gain traction and is worthwhile learning.
The gist of it is this: instead of using symbols that you have to memorize for each comma between the 3-limit and the prime interval, you write the prime number of the comma as a superscript or subscript. The 3-limit base intervals are based on the classical interval names, just as we've seen before: **d**iminished, **m**inor, **P**erfect, **M**ajor, and **A**ugmented.
### Summary of prime intervals
Here is a tabulation of the octave-reduced prime harmonics we have encountered:
| Interval | Cents | Octave-reduced interval | Prime limit classifier name | Scale degree | Note relative to C4 | Comma | HEWM up comma | HEWM down comma | Associated 3-limit note (rel. to C4) | Comma direction for prime |
| --: | :--: | -- | -- |:--:| :--: | :--: | :--:| :--: | :--: | :--:
| 2 | 1200.00 | Octave | 2-limit | 8 | C5 | NA | | | C5 = `2/1`
| 3/2 | 701.96 | Perfect 5th | Pythagorean | 5 | G4 | NA | | | G4 = `3/2`
| 5/4 | 386.31 | Major 3rd | Classic | -3 | E-4 | 81/80 syntonic | `+` | `-` | E4 = `81/64` | Down |
| 7/4 | 968.83 | Subminor 7th | Septimal | <b7 | Bb<4 | 64/63 septimal | `>` | `<` | Bb4 = `16/9` | Down
| 11/8 | 551.32 | Superfourth | Undecimal | ^4 | F^4 | 33/32 undecimal | `^` | `<` | F4 = `4/3` | Up
Here is a table of 3-limit Pythagorean notes relative to C4:
| | Diminished (d) | Minor (m) | Perfect [P] | Major [M] | Augmented [A]|
|:--:|:--:|:--:|:--:|:--:|:--:|
| **Unison** | Cb4 = 2048/2187 | :x: | C4 = 1/1 | :x: | C#4 = 2187/2048 |
| **Second** | Dbb4 = 524288/531441 | Db4 = 256/243 | :x: | D4 = 9/8 | D#4 = 19683/16384 |
| **Third** | Ebb4 = 65536/59049 | Eb4 = 32/27 | :x: | E4 = 81/64 | E#4 = 177147/131072 |
| **Fourth** | Fb4 = 8192/6561 | :x: | F4 = 4/3 | :x: | F#4 = 729/512 |
| **Fifth** | Gb4 = 1024/729 | :x: | G4 = 3/2 | :x: | G#4 = 6561/4096 |
| **Sixth** | Abb4 = 262144/177147 | Ab4 = 128/81 | :x: | A4 = 27/16 | A#4 = 59049/32768 |
| **Seventh** | Bbb4 = 32768/19683 | Bb4 = 16/9 | :x: | B4 = 243/128 | B#4 = 531441/262144 |
| **Octave** | Cb5 = 4096/2187 | :x: | C5 = 2/1 | :x: | C#5 = 2187/1024 |
There are infinitely many prime intervals. If you want more intervals, you can visit the [Gallery of Just Intervals](https://en.xen.wiki/w/Gallery_of_just_intervals) on xenwiki.
## Monzo
So far, both the names and ratios for intervals have seen were unwieldy. It can seem like there are lots of numbers, symbols, and names to memorize. Instead of dealing with huge numbers and ratios, we can use the **monzo**.
The monzo is a way of representing ratios using the [prime decompositions of an interval](#How-does-prime-factorization-relate-to-music).
Let's consider the interval $$\frac{13475}{13824} = 2^{-9} \cdot 3^{-3} \cdot 5^2 \cdot 7^2 \cdot 11^1$$
:::info
:writing_hand: Relative to C4, how do we write that note in HEWM?
:::spoiler Answer
This is done in 2 steps. First we add up the contribution of fifths `3/2` and octaves `2/1` of each prime. Next, we notate the commas for primes 5 and above.
* From $2^{-9}$, we have -9 octaves
* From $3^{-3}$, we have -3 fifths and -3 octaves
* From $5^2$, we have +8 fifths, because the associated 3-limit major third is `81/64`, which is obtained by going up 4 fifths and down 2 octaves. The prime interval 5 is a major third plus 2 octaves, so the up and down 2 octaves cancel out. Thus, we go up 4 fifths for each of the associated 3-limit note `81/16` of prime interval `5/1`, and we stack two of these for a total of +8 fifths.
* From $7^2$, we have -4 fifths and +8 octaves, because the associated 3-limit note for `7/4` is `16/9`, adding back the 2 octaves we reduced from `7/4` we get `64/9` to represent prime interval `7/1`. This translates to -2 fifths and +4 octaves for each prime interval `7/1`. There are two of these, so -4 fifths and +8 octaves.
* From $11^1$, we have -1 fifth and +4 octaves. The associated 3-limit note for `11/1` is a fourth plus three octaves, which is equal to up -1 fifth + 4 octaves
Summing all these up, we get the base 3-limit note of $-9-3+8+4 = 0$ octaves and $-3+8-4-1 = 0$ fifths, which means the base 3-limit note is in fact just C4 itself.
Then, we add the commas for all the primes. From the two 5th harmonics, we get `--` (each 5th harmonic contributes a syntonic comma **down**). From the two 7th harmonics, we get `<<`, and from the one 11th harmonic we get `^`.
Thus, the note name for `13475/12824` is `C--<<^4`.
:::
<!-- ::: -->
:::info
:writing_hand: What is the size of this interval in cents?
:::spoiler Answer
$$
\begin{align*}
\text{cents} &= 1200 \times \log_2(13475/13824) \\
&\approx -44.267
\end{align*}
$$
:::
<!--:::-->
Because the prime numbers can be listed in a standardized increasing order:
$$2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$$
We can instead write the above ratio using only powers of the primes:
$$\monzo{-9 & -3 & 2 & 2 & 1 & 0 & 0 & 0 & \cdots} \equiv 2^{-9} \cdot 3^{-3} \cdot 5^2 \cdot 7^2 \cdot 11^1 \cdot 13^0 \cdot 17^0 \cdot 23^0 \cdots $$
Because there are an infinite number of primes, the actual monzo has infinite numbers in it! But all the numbers corresponding to prime numbers that are above the prime-limit of the interval will be 0, because we don't have any of these prime factors in our interval. So, we can write in short-form and omit all the trailing zeroes:
$$\monzo{-9 & -3 & 2 & 2 & 1} \equiv 2^{-9} \cdot 3^{-3} \cdot 5^2 \cdot 7^2 \cdot 11^1 $$
which considers only the `2.3.5.7.11` JI subgroup.
#### Monzo is a vector in a vector space
This notation is quintessential for Part III: Temperaments. The monzo is actually a **vector** in a **vector space**. For our purposes we can understand vector spaces like how we understand coordinates in 2D/3D space.
If we want to describe an object's position in 3D, we can say where it is along the X, Y, and Z coordinates, which could correspond to the left-right, up-down, front-back axes respectively. Each axis in a vector space is a unique direction — no amount of going right will ever move you upwards. Sounds familiar?
Recall how the Fundamental Theorem of Arithmetic was applied in the decomposition of intervals into prime intervals. No amount of multiplying by one prime will equal of any other prime.
We can visualize each prime number as its own unique direction. Multiplying by a prime interval as moving one step forward in the _direction_ of the prime, and dividing by that same prime interval will move you one step backward.
The number of steps taken associated to each direction are called **components**.
#### A vector space is closed under addition
Notice that when we add intervals, we multiply them, but multiplying two intervals is actually mathematically the same as adding their prime powers up. For example, if we want to add the perfect 4th `4/3` to the perfect 5th `3/2`:
$$\frac{4}{3} \times \frac{3}{2} = 2$$
We get the octave. However, we can also do imagine these intervals as movements in space.
The perfect 4th can be thought of moving 2 steps forward in the 2-direction, and -1 steps (backward) in the 3-direction:
$$\frac{4}{3} = 2^2 \cdot 3^{-1} \equiv \monzo{2 & -1}$$
And the perfect 5th can be thought of moving -1 step in the 2-direction and 1 step in the 3-direction:
$$\frac{3}{2} = 2^{-1} \cdot 3^{1} \equiv \monzo{-1 & 1}$$
To add these intervals, we can add these movements up component-wise! Meaning that we simply add up the number of steps taken for each p-direction:
$$
\begin{align*}
\frac{4}{3} \times \frac{3}{2} &= \left(2^2 \cdot 3^{-1}\right) \cdot \left(2^{-1} \cdot 3^1\right) && \equiv \monzo{2 & -1} + \monzo{-1 & 1} \\
&= 2^{2-1} \cdot 3^{-1+1} && \equiv \monzo{(2-1) & (-1+1)} \\
&= 2^1 \cdot 3^0 = 2 &&\equiv \monzo{1 & 0} \quad\blacksquare
\end{align*}
$$
> :pencil: In math-speak, when we say "a vector space is closed under addition", we mean that we can take two vectors, add them together, and we get a vector that still lives inside the same vector space (and not get sent into some other arbitrary dimension or mathematical object). This is true, we have a monzo/interval, we add another monzo/interval, and we get back a monzo/interval that lives within the same prime-limit.
#### The unison interval is the identity vector
Notice what happens when we add the unison interval `1/1` to the perfect fifth `3/2`:
$$\frac{3}{2} \times \frac{1}{1} = \frac{3}{2}$$
Absolutely nothing!
In monzo notation, we can write:
$$\frac{1}{1} = 2^0 \cdot 3^0 \cdot 5^0 \cdot 7^0 \cdots \equiv \monzo{0 & 0 & 0 & 0 & \dots} = \monzo{0}$$
#### We can repeatedly add intervals by multiplying monzos by scalars
A **scalar** is a single numerical value, not a vector, as opposed to a monzo which is a **vector** with zero or more components which give information about how many steps to take in which direction.
Now consider the interval obtained by going up 3 major seconds (also known as a tritone, because a major second is also called a tone, and there are 3 of them). To do this, we have to multiply the major second interval `9/8` by itself 3 times:
$$\left(\frac{9}{8}\right)^3 = \frac{9}{8} \times \frac{9}{8} \times \frac{9}{8} = \frac{729}{512}$$
The monzo for `9/8` is $\monzo{-3 & 2}$. We can also add up 3 of the same monzos to accomplish the same thing:
$$
\monzo{-3 & 2} + \monzo{-3 & 2} + \monzo{-3 & 2} = \monzo{-9 & 6} \equiv 2^{-9} \cdot 3^6 = \frac{729}{512}
$$
To reduce repetition, we can also write it as a scalar multiplication of a monzo:
$$3 \monzo{-3 & 2} = \monzo{-3 \cdot 3 & 2 \cdot 3} = \monzo{-9 & 6}$$
## Chords
A chord is a collection of one or more notes.
### 12edo is the easy way out
There is misconception we take for granted in 12edo where we assume chords imply tonality (e.g., the diatonic triads, ii-V-Is, etc...), and that chords, scales and tonalities are related.
They are related to some very far-fetched extent, but the logic behind the relation has been lost to time, and one of the goals of this series of articles is to connect all the dots to bring that fundamental relation into the limelight. The western culture of tonality developed independently of chord/chord-scale theory. The fact that tonalities, functions, scales and chords happen to (partially) coincide in 12edo is a coincidence that is made possible by the commas that are tempered in 12edo, which we will visit in Part III.
Now, we have to see chords as just sets of notes, free of any extra semantic information from cultural entrainment. The definition of tonality was covered in Part I, and we can use chord names to describe certain sets of notes in one side/state of a tonality, but recall that a tonality is not about sets of notes, but about the set of **movements** between **states**, and the notes themselves have room for interpretation.
Hence, this final section of Part II will only cover naming conventions of chords. However, this series rarely use chord names without directly naming their ratios, so if you're in a crunch, this section is optional reading.
I will present my ideal of how the chords we are familiar with in 12edo are named and understood in the context of JI. This is a system that is nearly identical to jazz/pop chords, but from the xen perspective. Readers without theory background will walk away understanding chord names the same way those with theory background do.
### Naming chords as a ratio
First, the simplest most direct way to name a chord. We have already encountered this notiation before: `4:5:6` means that the first note is 4 units of frequency, the second note is 5 units, and third is 6 units:
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A4%3A5%3A6--" title="Xenpaper" frameborder="0"></iframe>
:::info
:writing_hand: **Exercise**
Set A4 = 440hz. The first note of the chord is set to the tuning frequency 440hz.
What are the note names of this chord? What are the monzos of the chord with respect to A4?
:::spoiler Answer
$$
\begin{cases}
\texttt{A4} &= \monzo{0} \\
\texttt{C#-5} &= \monzo{-2 & 0 & 1} \\
\texttt{E5} &= \monzo{-1 & 1}
\end{cases}
$$
:::
<!-- :::-->
However, this way of naming chords does not tell you what the frequency of the notes are, or what root does the chord build from. For that, I like to use the @ symbol to clarify:
$$\texttt{4:5:6 @ 440 Hz}$$
states that the first note with 4 units of frequency is equal to 440 Hz. By the ratio, 5 units of frequency will be 550 Hz, and 6 units will be 660 Hz.
### Chord names in contemporary western music
If you are a programmer and read regex, here you go:
```js
let chordRegex = /^([A-Ga-g])(bb|b|#|x|)(.*?)(2|3|4|5|7|9|11|#11|13|15|#15)?((?:(?:(?:bb|b|#|x)?(?:10|1?[1-9]))|[Dd][Ii][Mm]|o|O|\u{1D698}|\u{25CB}|\u{1D3C}|\u{25E6}|\u{00B0}|[Hh][Dd][Ii][Mm]|0|\u{00D8}|\u{00F8}|\u{2205}|\u{2300}|\u{1D1A9}|[Ss][Uu][Ss]|[Aa][Uu][Gg]|\+|[Aa][Dd][Dd][b#]*1*[0-9]|[Nn][Oo][b#]*1*[0-9]|[Aa][Ll][Tt]|[()])*)$/u;
```
https://regex101.com/r/3khmml/1
Otherwise, let me give you an example **megachord** that will contain all 7 components of a chord, and slowly break down each part of it.
Don't worry, you will never see anything like this in practice.
#### i. Root
The root note of the chord tells you which note to build it from. Because octave-equivalence is assumed in western harmony, there's no need to write which octave the note is in. When using contemporary chord notation, none of the notes will have a deterministic octave.
In the megachord example, our root note is C. However, it means nothing until we set a tuning note. For simplicity's sake, let's set C4 = 400 Hz.
#### ii. Bass note
The bass note tells you which note to put at the bottom of the chord. Sometimes, it is a note that is already contained in the chord as per the chord notation (in the megachord example, it is). In these cases, we can understand the chord as an **inversion** (which means the ordering of the notes of the chord are rotated), though in modern contexts, the concept of inversions don't matter as much anymore.
The bass note can also be a new note not already in the chord, in that case, we just add the bass note at the bottom verbatim.
#### iii. Quality
Contrary to the internet and most of the world, I would claim there are only 3 primary chord qualities: Major, Minor and Dominant (for the *syntactical* role of quality description)
The quality of the chord tells 2 key pieces of information. What the second [extension note](#iv-Extension-fifth-coloring) of the chord is, and whether or not you have to change the 4th extension note of the chord (if it exists). The quality and extension number together will give the initial set of notes for the chord, which I call the extension notes, from which other alterations are performed later.
| Quality | 2nd extension note (3) | 4th extension note (7) |
| :--: | :--: | :--:
| Major / Maj / M / Δ | Major 3rd | unchanged |
| minor / min / m / - | Minor 3rd | unchanged |
| Dominant / dom / blank (no symbol) | Major 3rd | Minor 7th |
> :pencil: In older chord notation, the `+` symbol may also mean Major quality, but this symbol is semantically overloaded and should not be used.
> :pencil: The word "quality" is semantically overloaded. The quality of a chord is not the same as the quality of an interval in classical interval nomenclature (i.e., diminished, minor, perfect, major, augmented).
In the megachord example, I have purposely left a blank circle to emphasize that there is no symbol for the quality, which means it is a dominant quality chord.
However, there is still one more thing to clarify. Which major/minor 3rd/7th do we use? Do we use the `81/64` Pythagorean major third or `5/4` classic major third?
The default these days is to assume thirds and sevenths are 5-limit unless otherwise stated.
Otherwise, we can also specify exactly what kind of third or seventh we want by qualifying it with prime-limit or names. Here are a list of examples. I have given the notes in scale degrees and ratios:
| Quality 2.0 | 2nd extension (3) | 4th extension (7) |
| :--: | :--: | :--: |
| pythagorean major / pmaj | `81/64` | unchanged |
| (septimal) supermajor / smaj | `9/7` = 3> | unchanged |
| (septimal) harmonic dominant / h / ! | `5/4` = 3- | `7/4` = b7< |
| (septimal) subminor / smin | `7/6` = b3< | unchanged |
| (undecimal) neutral | `11/9` = b3^ (n3, neutral 3rd) | unchanged |
| primodal-6 minor-dominant | `7/6` = b3< | `11/6` = b7^ (neutral 7th) |
> :pencil: The interval qualifier **neutral** is used for intervals almost halfway between major and minor. It is only applicable to 2nds, 3rds, 6ths, and 7ths.
This is a free-for-all and there's barely any standardization at this time, you can make up whatever qualities you want based on the 3 primary qualities. The main rules are:
* For the 2nd extension note, the major third is generally wider than the minor third.
* For the 4th extension note, it is usually unchanged unless the quality is dominant. The change to the 4th extension note will usually make it flatter than usual. The dominant chords also generally use some kind of major 3rd for the 2nd extension note, but you can make exceptions.
#### iv. Extension (fifth-coloring)
The first step of forming a chord is to work out its extension notes using the root, quality and extension number.
The extension number (indirectly) tells you how many extension notes there are. A chord only has one extension number. It is always either placed immediately after the quality of the chord (or root note if the quality symbol is dominant/nothing), or after a combo symbol.
It is always an **odd number** and **defaults to 5** if not specified.
If an **even** number is placed where the extension should be, that number is **not the extension**, and the chord will have the default extension 5.
E.g., in the chord name `C69` (which can be written more precisely as `C(6, 9)`), we first understand that there will almost never be any numbers in the chord above 20, so we do not interpret `69` as a single number but two separate numbers. The first number 6 is not odd, so the chord will have the default extension 5. Even though 9 is odd, it did not appear immediately after the quality or combo symbol, so it is also not the extension of the chord.
The number of extension notes in the chord is given by:
$$
\text{no. ext. notes} = \frac{x+1}{2}
$$
That is, if the extension is 1, 3, 5, 7, 9, 11, ..., then there are 1, 2, 3, 4, 5, 6, ... extension notes respectively. In standard practice, the extension does not go above 13. In our megachord example, the extension number is 13, which means we should have 7 extension notes by the end of this step.
##### Formulating extension notes with fifth-colorings
In the original conception of the extension (borrowed from figured bass notation), the number stood for the scale degree of the highest note in the extension series. However, that notion changed when George Russell's [Lydian Chromatic Concept of Tonal Organization](https://www.academia.edu/87115509/The_Lydian_chromatic_concept_of_tonal_or_1) (LCCOTO) and fellow chord-scale theorists have tried to reason about chords and scales, though the developments in this field of xen music-math that you are currently reading about were probably not yet known to them.
I don't agree with the theories of LCCOTO, but it is a step towards a concept underlying the modern usage of chord extensions which I call **fifth-coloring**:
[Earlier in this part](#It-matters-because-harmonics-affect-concordance), a discussed that two notes (with harmonic timbre) an octave apart have almost all their harmonics aligned except for the fundamental of the lower note, which motivates the entrainment of **octave-equivalence**.
Well, if two notes are a perfect 5th `3/2` apart instead, then an alignment occurs 1 of every 3 of the lower note's harmonics and 1 of 2 of the higher note's harmonics (2 of 5 between the two notes). This is the second highest density of harmonic alignment you can get for intervals within an octave.
Thus, amongst dyads, the fifth is the second most concordant interval within an octave, other than the octave itself.
Presently, we think that octaves are so concordant that they deserve the same name (though this is not the case in the original Gamut/Ut Re Mi, c. 11th century). So, if octaves get the **same name**, we can think of fifths, the second most concordant interval, as "**similar names**". But that doesn't really make sense, so I say that fifths have **adjacent hues**:
This is the motivating idea: even if we add fifths to our existing notes in the chord, we would not increase the complexity/harmonic information in the chord by a lot because of its relatively high harmonic alignment. Thus, we can add the fifths of existing notes to make it sound lusher/fuller without completely changing the nature of the chord:
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A%5B1%2F1______________5%2F4%5D___________%23_major_3%0A%5B1%2F1_3%2F2__________5%2F4%5D___________%23_major_5%0A%5B1%2F1_3%2F2__________5%2F4_15%2F8%5D______%23_major_7%0A%5B1%2F1_3%2F2_9%2F4______5%2F4_15%2F8%5D______%23_major_9%0A%5B1%2F1_3%2F2_9%2F4______5%2F4_15%2F8_45%2F16%5D%23_major_(%23)11%0A%5B1%2F1_3%2F2_9%2F4_27%2F8_5%2F4_15%2F8_45%2F16%5D%23_major_13" title="Xenpaper" frameborder="0"></iframe>
Though, we have to be careful, the more fifths we go up, the further we get from the starting hue, so we cannot over do it.
The **extension notes** are **fifth-colorings** of the root (1st extension note) and third (2nd ext. note) of the chord, which are set by the [quality](#iii-Quality). We stack fifths alternating between the root and third, changing the hues of the original 2 notes slightly:
For the **dominant quality**, we stack fifths using the first two extension notes as usual. Once we stack till the desired number of notes, change the 4th extension note as required by the quality.
> :warning: In common practice, the extension number is used to represent scale degrees, and the above explanation of extensions is a necessary liberty I took to better generalize chord construction in JI. For major chords, it is still common to use `#11` and `#15` to denote the extension numbers 11 and 15, because those are the scale degrees of the final extension note (in a meantone temperament).
Going back to the megachord example:
We already know that the root is C, and the quality is dominant. According to the dominant quality, we know that the first two extension notes are `1/1 = C` and `5/4 = E-` relative to the root C.
The extension is 13, so stack three more fifths to `C` and two more fifths to `E-`, giving the 7 extension notes, in order:
$$\texttt{C E- G B- D F#- A}$$
Then, because the dominant quality requires the 4th extension note to be set to a minor 7th, we change `B-` to `Bb+`, which is the default 5-limit minor 7th. Thus, the final set of extension notes of this chord is:
$$\texttt{C E- G Bb+ D F#- A}$$
:::warning
:warning: There is debate on whether the 11 (6th ext. note) of dominant chords should default to a sharpened or natural 11.
The general guideline I use is: In all cases, the dominant chord uses a sharpened 11th, just as you have seen. But, if the dominant chord's extension is 11 itself, we use either the natural 11th (or `+11`, for `6/5` minor-based tonalities), and in most cases, omit the 3 (2nd ext. note) because it clashes. This makes it so that a chord like `C11` can be a shorthand for `C9sus4`.
To explicitly state that a dominant chord should be extended until a `#11` and no more, we can clarify by writing `C9#11` instead. Do **not** write `C#11`, because that looks like the root note is C#.
:::
#### v. Add-alts
These are a series of numbers referencing the extension numbers (or scale degrees) that **add** to, or **alt**er notes of a chord after the main set of extension notes are constructed. Usually, these numbers have accidentals attached to them, but we consider the accidental and number separately.
##### Alterations
If the add-alt is an extension number of an existing extension note, and no prior alteration was done on that number, then it represents an **alteration** of the extension note corresponding to the number.
For example, in the megachord has the add-alts `b9` and `^11`. Both `9` and `11` are odd numbers that are part of the extension series 1, 3, 5, ..., 13. Hence, these add-alts reference the existing notes in the set of extension notes, and are **alterations**.
Hence, instead of the note corresponding to extension number 9, which is the 5th extension note `D`, we alter it to b9, so it becomes `Db` instead.
And instead of the note at extension number 11, `F#-`, we alter it to the ^11, which means it changes to:
:::spoiler Answer
F^
:::
##### Additions
Otherwise (e.g., add-alt is **even**, or **not part of the extensions**), then the add-alt tells us to **add** that scale degree to the chord verbatim.
In the mega chord, we have the add-alt `#-15`. 15 is not part of our extension series. Thus, we add the scale degree `#-15` relative to the root `C`, which is `C#-`. Although the scale degree uses `15`, we don't necessarily have to place this note exactly 2 octaves above the root.
In case we need to make the additions clearer/more explicit, we can prepend `add` in front, e.g., `add#-15`. If we do that, the scale degree will never be read as an alteration.
Some chords use the `maj` prefix to denote additions with no accidentals (e.g., `Cdim7(maj7)`). This `maj` is not the chord quality (because it didn't immediately follow the root of the chord symbol), but is instead a modifier that means "♮" (natural, no accidental) for the add-alt. However, this is not recommended for chords in non-12 music because there is no consensus on whether to use the 3 or 5-limit notion of `maj`.
#### vi. Removals
A removal removes extension notes. In the example, `no3` tells us that we remove the note corresponding to the extension number 3, which is the 2nd extension note `E-`.
The removal does not need to specify the accidental, even if the extension note has accidentals, e.g., the 11th `F#-` of the megachord.
#### vii. Combos
There are several special symbols that are used as shorthand for combinations of qualities, add-alts and removals. I call these **combos**. Many also consider these "chord qualities", however, I find that it makes the framework of constructing chords much less elegant.
Combos, however, have the special syntactial property where they can be written before the extension number, which is probably why combos are confused with qualities.
| Combo symbol | Name | What it means |
| :--: | :--: | :--: |
| aug / + | Augmented | **Append add-alt** `--#5` (or `-#5 `or `#5`, depending on context & tuning) |
| dim / o / ° | Diminished | **Force quality** to minor. Mutually exclusive with other qualities. Append the add-alts: `++b5`, and `++bb7` (if extension 7 is present). Depending on tuning, inversion, and context, can also mean `+b5`, and `++bb7` or any variation of commas.
| hdim / ø / Ø / m7b5 | Half-diminshed | **Set quality** to minor only if dominant & no symbol. **Set extension** to at least 7. **Append add-alt** `++b5` (or any variation of commas)
| sus2 / sus4 | Suspended 2nd/4th | **Append add-alt** `2` / `4`. **Append removal** `no3`
| sus X | suspended 4th | If 2 or 4 not specified, or a different number `X`, defaults to `sus4`, and the trailing number `X` is either the extension or another add-alt.
| alt | Altered dominant | **Force quality** to dominant. Mutually exclusive with other qualities. **Append add-alts** (one or more from the following): `b5, #5, b9, #9`. `alt` is **mutually exclusive** with add-alt `#11` (or any variations of commas)
> :pencil: The words "augmented" and "diminshed" are semantically overloaded, and these shouldn't be confused with the qualities of intervals.
#### Cracking the code
The first step is to construct the extension notes.
We begin with a dominant quality chord with extension 13 over the root C. However, notice that the **diminshed combo** `o` is present, which **forces our quality to minor** instead. This means we have to recalculate our extension notes for the chord with the minor quality:
This gives the 7 extension notes to start with: `C Eb+ G Bb+ D F+ A`.
Next, we apply the combos in order.
* `+` appends the add-alt `--#5`
* `o` appends the add-alts `++b5 +++bb7` (assuming a baseline of 5-limit harmony)
* `Ø` appends the add-alt `++b5`, we already did, and the extension is already more than 7.
* sus4 means to add 4 and remove the note corresponding to extension 3, `Eb+`.
Next, we apply the add-alts:
* `b9` changes `D` to `Db`
* `^11` changes `F+` to `F^`
* `#-15` adds the note `C#-`
And the removal `no3` doesn't change anything because `sus4` already applies `no3`.
Putting everything together, we have:
* Root: C
* Extension: 13
* alterations: `--#5 +++bb7 b9 ^11`
* additions: `++b5 4 #-15` (`++b5` is an addition because 5 is already altered in `--#5`)
* removal: `3`
* bass: `F^`
This gives us the final chord:
$$\texttt{F^ C G#-- Bbb+++ Db Gb++ F C#-}$$
:warning: **VOLUME WARNING**
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br%609%5C12%7D%0A%5B216%2F125_25%2F16_16%2F3_135%2F128_11%2F16_512%2F243_1%2F1_72%2F25%5D-----" title="Xenpaper" frameborder="0"></iframe>
:::info
:writing_hand: **Exercise**
C3 is set to 1/1. Which ratios in the above sound example correspond to which notes? What are the octaves of these notes?
:::spoiler Answer
* 11/16: F^2
* 1/1: C3
* 25/16: G#--3
* 216/125: Bbb+++3
* 512/243: Db4
* 72/25: Gb++4
* 135/128: C#5
* 72/25: F5
:::
<!--:::-->
# Part III: Temperaments
[Part III: Temperament, vals, mapping, scales, functional harmony](/3EBTY7Q1SJCXJvO6qDPAlA)
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## Glossary
*[apotome]: The semitone interval contributed by a sharp or flat. In JI, this is 2187/2048
*[cents]: 1/100ths of a 12edo semitone
*[cent]: 1/100ths of a 12edo semitone
*[comma]: Small tuning discrepancy between intervals
*[detemper]: Reinterpreting an interval from a tempered tuning system as a JI interval
*[detempering]: Reinterpreting an interval from a tempered tuning system as a JI interval
*[diatonic]: Acccording to the 7 notes of the major scale, instead of fixed intervals.
*[diesis]: One step of 31edo, or 128/125 in just intonation
*[dieses]: Plural of diesis, steps of 31edo or 128/125
*[discordance]: The part of dissonance that is objectively identifiable, but with no inherent favor or preference for concordance or discordance
*[concordance]: The opposite of discordance
*[ECT]: European classical tradition
*[edo]: Equal division of the octave
*[edos]: Tunings which are based on equal divisions of the octave
*[12edo]: 12 equal divisions of the octave
*[31edo]: 31 equal divisions of the octave
*[HEWM]: Helmholtz-Ellis-Wolf-Monzo notation for JI
*[HEJI]: Helmholtz-Ellis Just Intonation notation
*[integer]: Whole number. Not fractional, nor decimal point, nor irrational
*[integers]: Whole numbers. Not fractional, nor decimal point, nor irrational.
*[JI]:Just Intonation
*[limma]: The semitone interval that is between two white keys. In JI, this is 256/243
*[octave-reduced]: Octaves are subtracted until the interval fits within an octave.
*[otonal]: contained the overtone/harmonic series, as opposed to utonal
*[nominals]: The musical alphabet C, D, E, F, G, A, B without accidentals
*[nominal]: A musical alphabet C, D, E, F, G, A, B without accidentals
*[sesquitone]: An interval approximately 1.5 semitones wide
*[utonal]: contained in the reciprocal of the harmonic series
apotome
: The semitone interval contributed by a sharp or flat. In JI, this is 2187/2048
cents
: 1/100ths of a 12edo semitone
concordance
: the opposite of discordance
detempering
: Reinterpreting an interval from a tempered tuning system as a JI interval
diatonic
: According to the 7 notes of the major scale, instead of fixed intervals
diesis
: One step of 31edo, or 128/125 in just intonation
dieses
: Plural of diesis, steps of 31edo or 128/125
discordance
: The part of dissonance that is objectively identifiable, but with no inherent favor or preference for concordance or discordance
ECT
: European classical tradition
edo
: Equal divisions of the octave
HEWM
: Helmholtz-Ellis-Wolf-Monzo notation for JI
HEJI
: Helmholtz-Ellis Just Intonation notation
integer
: A whole number. Not fractional, irrational or with decimal point.
JI
: Just Intonation
limma
: The semitone interval that is between two white keys. In JI, this is 256/243
nominals
: The musical alphabets C, D, E, F, G, A, B without any accidentals
octave-reduced
: Octaves are subtracted until the interval fits within an octave.
otonal
: contained the overtone/harmonic series, as opposed to utonal
sesquitone
: An interval approximately 1.5 semitones wide
utonal
: contained in the reciprocal of the harmonic series
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