# Extending harmonic principles in 12edo to 31edo. Part III: Temperament, vals, mapping, scales, functional harmony

$$
\def\monzo#1{\left[ \begin{matrix}#1\end{matrix} \right>}
\def\cov#1{\left< \begin{matrix}#1\end{matrix} \right]}
\def\braket#1#2{\left< \begin{matrix}#1\end{matrix}\right. \mid \left. \begin{matrix}#2\end{matrix}\right>}
$$
:::warning
:warning: Read this in [dark mode](https://hackmd.io/@docs/how-to-set-dark-mode-en#Set-View-page-theme).
:::
Previous [Part II: JI & back to basics](https://hackmd.io/@euwbah/extending-harmonic-principles-2).
Next [Part IIIa: Octave symmetries, application & summary](https://hackmd.io/@euwbah/extending-harmonic-principles-3a)

:::success
**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
:::

*[Dotted underline]: Abbreviation expansion or definition


[TOC]

### Intro

Imagine a world where the infinite continuum of notes were regarded as valid and usable &mdash; how would anyone ever construct instruments or give them names or teach music?

Well, up until the work and events of Zhu Zaiyu in 1584 (12edo), the Geneva convention 1939, IOS in 1955, and ISO in 1975 (A4 = 440hz), **that was the world**. Pitch was free, and it was a popularity contest of who could convince who to use which tuning and whatever names.

So far, we discussed how notes in Just Intonation (JI) are conceived, and the construction of notes, chords, and their names. We also know that JI gives us an infinitude of intervals.

We have discussed the Fundamental Theorem of Arithmetic and how it makes it impossible for intervals that use only primes 2 and 3 to equal another interval that uses only primes 5 and 7. Each prime lives in a different world, and all JI intervals can be decomposed into prime intervals.

We discussed how tonality is culturally constructed, free for all to interpret and invent.

Infinity is scary. On a physical level, it is not feasible to build instruments with infinite strings or buttons. Because of that, we **temper out commas**, which reduces the number of notes (which can still be a "smaller" infinity).

But having lesser notes is just one side of the story. Tempering enables magical coincidences and "musical puns" which are now so ingrained in the modern theory of harmony that we forget that these things &mdash; pluralism, symmetries, tritone substitution, diminished borrowing, the major scale, and so many other things &mdash; are facilitated by the commas we have chosen to **temper**.

One could believe it was a mere coincidence that we landed on today's tuning of 12edo. But we will soon understand that 12edo would have eventually be discovered sooner or later, because of how powerful the harmonic tools enabled by the temperaments of 12edo are.

After understanding the harmonic tools of our palette and coincidences, we can understand their principles and apply them to other tunings, which we will do in [Part IIIa](https://hackmd.io/@euwbah/extending-harmonic-principles-3a).

# Scales

First, I want to address a misconception of how scales, tonality, and chords are related. Many believe that these are all the same thing, and teach these constructs as a monolithic blob as in Schenkerian analysis, chord-scale theory or the Lydian Chromatic Concept of Tonal Organization. In this series, I want to show you how they are _really_ related, and for that we must dissect and understand the purpose of each in context:


A **scale** is a finite set of notes up to some interval of equivalence (usually octaves). The purpose of collecting them into a scale is so that we can reduce the infinitude of possible note choices to a finite number, only then we can name the notes, and teach their sounds to society. It is neither horizontal (in the passing of time) nor vertical (multiple notes simultaneously).

A **chord** is a vertical structure (all notes simultaneously, no passing of time) that does not necessarily require any interval of equivalence (unlike scales). My use of the word "chord" in this series is closer to what modern musicians would call a "voicing" or "orchestration". Chords in isolation cannot be consonant nor dissonant, they can only create sensations of concordance or discordance. Recall that the former is learned, the latter is intrinsic and, ideally, computable.

A **tonality** is a horizontal structure (experienced over time), with state, with no fixed set of notes (it can have infinite notes).

Both scales and chords are an abstraction over a **set of notes**.

The **clausulae** of transitioning from one state in the tonality to another forms important points in a scale.

The vertical analysis of a chord combined with an entrained perception of tonality creates the perception of **culturally informed consonance and dissonance**.

All of these three constructs are facilitated and only possible within the **temperaments** that were used to construct them.

:::warning
:warning: In the interest of simplicity, I have left out **rhythmic entrainment** and **form** in this model, but those are equally important to the perception of harmony as pitches themselves.
:::

## Where is major scale. What is major scale. Why is major scale.

With that in mind, let us look at the 7 nominals that we have obtained in Part II:


<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br16%2F27%7D%0A2%2F3_1%2F1_3%2F2_9%2F4_27%2F8_81%2F16_243%2F32" title="Xenpaper" frameborder="0"></iframe>

> :musical_note: Use headphones

Why do we care so much about these 7 notes that we give them these 7 alphabets? Why not more alphabets? Here's why:

* Historical: The Pythagorean culture of mathematical beauty created the systema teleion, and its [diatonic genus](https://www.youtube.com/watch?v=gsbuxnhqSVw) (for some parts/eras of ancient greece) are these note intervals you see above. However, the systema teleion did in fact assign more than 7 alphabets, but it understood that these alphabets were in multiples of 7 (or more accurately, in arrangments of the Hypaton, Meson, Diezeugmenon and Hyperbolaion tetrachords), after which Boethius notation in 6th century used the first 14 letters of the latin alphabet. The reduction down to 7 nominals is more modern.
> :pencil: The concept of a major scale with 7 notes as a single object didn't exist until after western harmony evolved from the hexachord system. And even prior to that, this Pythagorean major scale we have was conceived as two tetrachords (4-note sets) positioned a fourth apart. Each tetrachord had the intervals `1/1 9/8 27/16 4/3`. If you make two copies of those 4 notes, then position the last of those 4 notes at `1/1` and `3/2` respectively, you get (one of the) diatonic genus, which is also exactly what we got here.

* Primordial: The fifth is the second (octave-reduced) prime interval. If you were to discover the harmonic series on your own, in the order of the harmonics, this would be the first note that sounded different to you, because it actually introduces [multiple pitches of new harmonic content](https://hackmd.io/@euwbah/extending-harmonic-principles-2#It-matters-because-harmonics-affect-concordance) unlike the octave which only introduces one new frequency. Hence, it would be natural to want to repeat/stack the fifths of fifths.
* Physiological: Why stop at 7 (or 4 / 6 in the old systems)? Because of the fact that our basilar membranes in our inner ear can get confused when frequencies get too close which causes the perception of discordance. In the lower half of the human hearing range, the apotome `2187/2048` reaches a threshold called the [critical band](https://en.wikipedia.org/wiki/Critical_band), which means that we struggle to isolate different frequencies that are too close. Take this for example:

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A%23_can_you_hear_the_difference_between_60c_vs_20c%3F%0A%5B0c_60c_70c_100c%5D--.%0A%5B0c_20c_70c_100c%5D--.%0A%0A%23_this_is_the_difference._it_should_be_obvious_now.%0A60c-.20c-.%0A%0A%23_can_you_hear_it_now%3F%0A%5B0c_60c_70c_100c%5D--.%0A%5B0c_20c_70c_100c%5D--." title="Xenpaper" frameborder="0"></iframe>

This underlies the cultural preference for concordance. This is why we don't have the nominal alphabets H, I, J, K, ... and we'd rather rank those other notes as _**other**_ notes by using symbols # and b. (unless you're tuning in Bohlen-Pierce or German where B = B&flat; and H = B&natural;, see [the etymology fun fact](https://hackmd.io/@euwbah/extending-harmonic-principles-1?stext=32689%3A20%3A0%3A1736500262%3AS-R5uy) in Part I)

Note that for our major scale, we could have decided to use any 7 continuous fifths, e.g., F# to B#, instead of F to B, the choice of starting note was arbitrary.

Now we are motivated to collect these 7 notes and turn it into a scale. We do that by first reducing the octaves so that it all fits within one octave:

<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A4%2F3_1%2F1_3%2F2_9%2F8_27%2F16_81%2F64_243%2F128" title="Xenpaper" frameborder="0"></iframe>

And now we arrange it in ascending pitch order:

<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A1%2F1_9%2F8_81%2F64_4%2F3_3%2F2_27%2F16_243%2F128" title="Xenpaper" frameborder="0"></iframe>

> :musical_note: Notice how unsettling it feels for a scale to end there rather than going up one more note to the octave. This is the cultural entrainment of the Ti-Do clausula we have encountered in Part I.

This is our **major scale**, we currently call it that because the 2nd, 3rd, 6th and 7th degrees all have major qualities ([Recap](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Music-theory-crash-course-interval-names-in-classical-music-theory)), and the 4th and 5th are perfect, not augmented or diminished. As long as these 7 notes are present in any order or octave, we can consider it to be the C major scale. The decision to start this scale on the **root note** C and not on any other note is a completely arbitrary cultural convention and can be attributed to this property of the quality of the intervals. Throughout history, the _finalis_ (final resolution point) of a scale has been variable (e.g., modes, Gregorian modes, maqam).

:::info
:writing_hand: Setting 1/1 = C3, What are the note names & octaves of the above major scale? Explain your answer directly from the ratios as in Part II.

What if 1/1 = Eb3 instead? What would the notes be called then?
:::

Recall that the purpose of a scale is to serve as an abstraction &mdash; we just want to reduce the infinitude of notes down to a manageable few, then we can give names to notes (and the scale).

## The syntonic comma problem

If we fully commit to the scale diatonically (i.e, using only the 7 notes offered by the scale, their octaves, and nothing else), notice that we encounter an issue:

<iframe width="560" height="260" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B1%2F1_81%2F64_3%2F2_243%2F128%5D-%0A%5B9%2F8_4%2F3_27%2F16_2%2F1%5D-%0A%5B9%2F8_4%2F3_3%2F2_243%2F128%5D-%0A%5B1%2F1_81%2F64_3%2F2_2%2F1%5D-" title="Xenpaper" frameborder="0"></iframe>

:::info
:writing_hand: **Exercise**

Recall how to name [chords](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Chords) from Part II. If C = 1/1, what are the names of the above chords?

:::spoiler Answer
1. 3-limit `Cmaj7`, or `Cpythmaj7` (pythagorean major third 81/64)
2. 3-limit `Dm7`, or `Dpythmin7`
3. 3-lim `G7/D` (the bottom note is D). Writing something like `Dm6add4no5` or `Dm4add6no5` is also technically correct, but not the most succinct and not the best way to communicate [function](#Diatonic-Pluralism).
4. 3-lim `Cmaj` or `Cpmaj`. Adding the octave `2/1` of the root `1/1` doesn't mean we write `Cmaj8` (or `Cmajadd8`), because the octave doesn't matter in chord names.
:::

<!--:::-->
What's the issue? Now listen to this:

<iframe width="560" height="260" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B15c%7D%0A%5B1%2F1_5%2F4_3%2F2_15%2F8%5D-%0A%5B10%2F9_4%2F3_5%2F3_2%2F1%5D-%0A%5B9%2F8_4%2F3_3%2F2_15%2F8%5D-%0A%5B1%2F1_5%2F4_3%2F2_2%2F1%5D-" title="Xenpaper" frameborder="0"></iframe>

Did you hear the difference?

It's okay if you didn't, or if there was no preference, because the preference for concordance, and discernment of subtle differences in concordance is trained, not intrinsic. To be clear, the difference between the two examples is in the concordance of the intervals. The former example in 3-limit JI is less concordant than the latter in 5-limit. In the current western culture of music, concordance is preferred as a pragmatic choice because concordant pitches are easier to viscerally perceive and tune to than discordance, which comes in handy when singing pitches or tuning instruments.

> :warning: This isn't one of those [equal temperament vs JI videos](https://www.youtube.com/watch?v=6NlI4No3s0M) on YouTube where they show the difference between "out-of-tune" 12edo and "warm buzzy" JI. Both of these by definition, JI, because they are both using rational intervals.

Notice that we have cleared up some common misconceptions here:

* JI is not always more concordant than temperaments. Compare the first 3-limit JI example with its equivalent [tempered mapping](#Cents-mapping-of-14-comma-meantone) in 31edo:

<iframe width="560" height="420" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B1%2F1_81%2F64_3%2F2_243%2F128%5D-%0A%5B9%2F8_4%2F3_27%2F16_2%2F1%5D-%0A%5B9%2F8_4%2F3_3%2F2_243%2F128%5D-%0A%5B1%2F1_81%2F64_3%2F2_2%2F1%5D-%0A%7B31edo%7D%0A%5B0_10_18_28%5D-%0A%5B5_13_23_'0%5D-%0A%5B5_13_18_28%5D-%0A%5B0_10_18_'0%5D-" title="Xenpaper" frameborder="0"></iframe>

* Lower prime limit is not always more concordant than higher prime limit.

We can make the 3-limit major scale we started with more concordant by preferring 5-limit thirds over 3-limit ones. Let's work towards that goal:

> :pencil: Measuring concordance is deserves its own series. One of the simpler measures of concordance is [height](https://en.xen.wiki/w/Height), which is roughly based on the alignment of harmonics.

We start from this scale:


I have removed octave numbers as we assume octaves to be equivalent for now.

:::success
:pencil: Notice that in the [monzo](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Monzo) notation of the above diagram, the first prime component is left as a variable $x$, and that I have written fifths as the interval ratio `3/1` instead of `3/2`, because we are ignoring any [powers of 2](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Prime-interval-2-octave) in intervals.

This is what it means to assume **octave equivalence**.
:::

Let's start by optimizing as many 5-limit major thirds as we can. There are 3 major thirds present in the above C major scale, which are:

:::spoiler Answer
C-E, F-A, and G-B.
:::

By shifting scale degrees 3, 6 and 7 down by the syntonic comma `80/81`, we can make them the `5/4` major third of C, F and G respectively:


And when we do that, we get these notes:

<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B15c%7D%0A%5B1%2F1_5%2F4_3%2F2%5D-%0A%5B9%2F8_4%2F3_5%2F3%5D-%0A%5B9%2F8_4%2F3_3%2F2_15%2F8%5D-%0A%5B1%2F1_5%2F4_3%2F2_2%2F1%5D-" title="Xenpaper" frameborder="0"></iframe>

Well, that seems fine. Almost. Let's take a look at what's happening in the second chord:

<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B15c%7D%0A%5B9%2F8_4%2F3_5%2F3%5D-.%0A9%2F8_4%2F3_5%2F3_.%0A9%2F8_5%2F3_%5B9%2F8_5%2F3%5D-._%23_%3F%3F%3F" title="Xenpaper" frameborder="0"></iframe>

In the original 3-limit tuning, those three notes are `D F A`, with ratios relative to C being `9/8 4/3 27/16`. That `D` and `A` should have been a perfect fifth. However, after bringing `A` down a syntonic comma `81/80` to `A-`, we now have what's known as a **wolf fifth**, because it sounds like the discordant howling of a wolf:

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br4%2F3%7D%7B15c%7D%0A1%2F1_3%2F2_%5B1%2F1_3%2F2%5D--._%23_perfect_fifth%0A1%2F1_40%2F27_%5B1%2F1_40%2F27%5D--_%23_wolf_fifth" title="Xenpaper" frameborder="0"></iframe>

When two notes that are originally supposed to be concordant are detuned slightly because of a comma, we say that the two notes are **wolfing**.

> :pencil: There are also infinitely many other wolf fifths, such as the octave reduced stack of 11 fifths (interval formed between E# and C), which is wolfing because the spiral of fifths doesn't close resulting in the Pythagorean comma `531441/524288`. This example of the wolf fifth is a 5-limit version that arises due to the syntonic comma `81/80`.

Alright then, why not we move D down a syntonic comma to `10/9 = D-` as well, so that now both `D-` and `A-` can be a perfect fifth `3/2` apart instead of `40/27`?

<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B15c%7D%0A%5B1%2F1_5%2F4_3%2F2%5D-%0A%5B10%2F9_4%2F3_5%2F3%5D-%0A%5B10%2F9_4%2F3_3%2F2_15%2F8%5D-%0A%5B1%2F1_5%2F4_3%2F2_2%2F1%5D-" title="Xenpaper" frameborder="0"></iframe>


We get a wolf interval again in the third chord! Can you find it?

:::spoiler Answer
<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B15c%7D%0A%5B10%2F9_4%2F3_3%2F2_15%2F8%5D-.%0A10%2F9_4%2F3_3%2F2_15%2F8_.%0A10%2F9_3%2F2_10%2F9_3%2F2_%5B10%2F9_3%2F2%5D--._%23_wolf_fourth%0A9%2F8_3%2F2_9%2F8_3%2F2_%5B9%2F8_3%2F2%5D--._%23_perfect_fourth" title="Xenpaper" frameborder="0"></iframe>
:::

The reality of composing music in (low complexity) JI is that we cannot have a scale that satisfies all the following simultaneously:
1. Has finitely many notes
2. Does not have wolf/discordant intervals
3. Can support at least the most basic 5 limit functional harmony of tonic (I maj) dominant (V maj) and subdominant (IV maj) chords.

> :pencil: As an addendum to [Chords in Part II](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Chords), where I've hit the maximum word count, the use of **roman numerals** (I, ii, iii, IV, V, vi, vii) to denote chords is the same idea as using [scale degrees](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Scale-degree-notation). The roman numeral substitutes the root note of the chord as a scale degree. The roman numeral is usually in lower case for minor, and upper case for major.

Put in a few words, **there cannot possibly be any scales**! The very notion of a scale itself, which can reduce the infinitude of notes while maintaining concordant intervals throughout modulations of tonalities and keys, is a contradiction. This is probably why before the renaissance, the concept of a 7-note scale did not exist.

Instead, sets of notes were organized using composites of smaller structures (e.g., tetrachords & hexachords) that were concordant independently of each other. We will not be able to construct a usable 7-note scale until we solve this problem!

> :pencil: In most other musical cultures, notes are felt with respect to a single (or several) drone note(s), rather than across changing tonalities and root keys. For that reason, just intonation works fine in, for example, the [22 shrutis](https://www.youtube.com/watch?v=PQjWyBvLfqM) system of modern raga. The invention of temperament was necessary for the freedom of bass pitch, as opposed to a [constant iso-drone](https://en.wikipedia.org/wiki/Ison_(music)).

# Temperaments & Mapping

A temperament is a deliberate mistuning of prime intervals with goal of an interval being reduced to nothing, i.e., unison, `1/1`, 0 cents. When that happens, we say that that interval gets **tempered (out)**, or that the temperament makes the interval **vanish**.

Just as how each prime interval unlocks an infinite world of new intervals, each interval that gets tempered out unlocks new bridges between harmonic worlds that allows a musical culture to grow around a framework of **musical puns**/coincidences that would have been impossible otherwise.

:::success
:pencil: To motivate and give context for readers who are familiar with the modern theory of harmony, here is a quick glimpse of which temperaments enable which features of 12edo you are familiar with, along with which math tools can be used to formalize the cultural notions of using these harmonic tools:

| Harmonic tool | Temperament | Math tool | Principle |
| :--: | :--: | :--: | :--:
| Major scale & diatonic triads | [Meantone](https://en.xen.wiki/w/Meantone) | Exterior algebra | Scale, function, clausula, tonality
| Pluralism & secondary dominants | Meantone, [Augmented](https://en.xen.wiki/w/Augmented) | Exterior algebra, manifolds & fundamental group, homotopy type theory | Function, clausula, tonality
| Tritone substitution | [Compton/Pythagorean](https://en.xen.wiki/w/Compton_family), [Diachismic/Srutal](https://en.xen.wiki/w/Diaschismic_family), [Jubilismic](https://en.xen.wiki/w/Jubilismic_family) | Exterior algebra, group theory | Clausula, symmetry
| Diminished chords & diminished borrowing/substitution (Barry Harris), half-whole scale | [Dimipent](https://en.xen.wiki/w/Dimipent_family#Dimipent), Meantone, [Archy/Archytas clan](https://en.xen.wiki/w/Archytas_clan), Jubilismic, Diachismic | Exterior algebra, group theory | Clausula, symmetry
| Augmented chords, minor maj7th chords, 13#11 chords, augmented substitution (Coltrane changes), whole tone scale | [Marvel](https://en.xen.wiki/w/Marvel_family), [Mint](https://en.xen.wiki/w/Mint_family#Mint), Augmented, Meantone | '' | Clausula, symmetry
| Backdoor 2-5, blues, negative harmony, modes | All of the above | '' | All
| 12 tone serialism | All of the above | All of the above | All

We won't go into detail for each harmonic (or math) tool, but the goal is that at the end of this series, you will have the necessary foundation to click on those links and relate it back to the harmonic tools you are using, and, generalise the **principles** behind those tools to other tunings and temperaments, so that you can freely invoke the essence of today's culture of music, no matter which tuning you choose.
:::

Now we will explore how our current problem of wolf intervals in JI scales was tackled by Gaffurius (1496), Pietro Aron (1523), and made mathematically rigorous and standardized by Zarlino at the end of the 16th century. We begin in the world of **meantone temperament**.

## Meantone

:::success
:pencil: **Content overview**

In this section, we cover the motivation behind the meantone temperament and learn some technical tools, but weaved into a narrative format. If you wish to skip to the technical stuff, we will learn about

* Covectors, mappings & bra-ket dot product operation in [Mapping JI with cents: aka not tempering](#Mapping-JI-with-cents-aka-not-tempering)
* Solving system of linear equations & solving for prime intervals for the mapping of a temperament in [1/4-comma meantone as a mapping](#14-comma-meantone-as-a-mapping)
* How many intervals to temper to obtain an edo: [Rank-nullity theorem](#Rank-nullity-theorem-How-many-intervals-to-temper-out-before-it-becomes-an-edo)

Bear with me here, we'll get to the ***music*** after this section of the journey.

:::

Let's break apart this problem. On one hand, the prime interval `5/1` brings us up a 5-limit major third and two octaves. On the other hand, we go up by four `3/2` perfect 5ths, and we arrive at a 3-limit major third plus two octaves. This difference between the two types of major thirds (syntonic comma) is itself the cause of wolfing, because we have to choose between:


1. Using `D` in our C major scale, which makes `G D` in the `Vmaj` chord `G B- D` a perfect fifth, but `D` does not make a perfect fifth with `A-`, so we can't have a `ii-min` chord.
2. Using `D-`, which makes `D-` a perfect fifth with `A-` so we can have a proper `ii-min` chord, but now the `Vmaj` chord `G B- D-` has a wolf fifth.
3. Using `A` instead of `A-`, which gives us a proper fifth between `D` and `A`, but now the interval beteen `F` and `A` is no longer the concordant 5-limit M3 `5/4` but the discordant 3-limit M3 `81/64`, which means we can't have a proper `ii min` nor `IV maj` chord!

No matter what we do, there will be some interval **wolfing by the syntonic comma `81/80`**.

Now here's an idea: what if we can do sneakily **detune our prime intervals** so that `81/80` turns into nothing? For example, instead of using 3 as our prime interval, maybe we can use 2.999 or something. If the difference is small enough, it won't be that noticeable.

To get intuition on how we can detune the prime intervals, we start by drawing the lattice of the problem at hand:


<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A1%2F1_3%2F2_9%2F4_27%2F8_81%2F16_._%23_up_four_fifths%0A81%2F16_81%2F32_81%2F64_._%23_down_two_octaves%0A1%2F1_81%2F64_%5B1%2F1_81%2F64%5D-._%23_pyth_major_3rd%0A1%2F1_5%2F4_%5B1%2F1_5%2F4%5D-_%23_5-lim_maj_3rd" title="Xenpaper" frameborder="0"></iframe>

And next, we draw the lattice of the ideal situation, where E-4 and E4 becomes the **same note**, such that the `81/80` syntonic comma **vanishes**. We say that an interval vanishes in a temperament when it becomes effectively the same as the unison interval `1/1`.


> :pencil: No, Pythagoras' Theorem does not apply.

I have denoted each interval as unknowns $x, y, z$. Now, we are free to decide which intervals are set to what sizes such that the above diagram holds true.

The goal is that we want `E-` to end up in a place such that the distance between `C` and `D`, will be equal to the distance between `D` and `E-`, which is made possible by the fact that `E = E-`.


Then, `D` will exactly be the "average" of `C` and `E`, which is why this temperament is given the name **meantone**.

How do we decide which intervals get tuned to what? Well, that depends on which aspects of tuning you want to prioritize. Let's explore some options starting with...

### 1/4-comma meantone attempt

In this variant of meantone, we prioritize three goals.

1. The meantone property: the syntonic comma `81/80` should **vanish** and map to 0 cents.
1. Don't temper the octave. Octaves should stay at `2/1` otherwise we'll mess with octave equivalence.
2. We want the major third to always be precisely the 5-limit major third `5/4`.

This means that:
$$\begin{align*}y &= \frac{1}{2}, & z &= \frac{5}{4}\end{align*}$$
are fixed.

Remember that we add up intervals by [multiplying the ratios](https://hackmd.io/@euwbah/extending-harmonic-principles-2#So-what-do-logarithms-have-to-do-with-pitch). Since we want the interval $z$ to be equal to four copies of interval $x$ and 2 copies of interval $y$, we can write an equation:
$$z = x \cdot x \cdot x \cdot x \cdot y \cdot y = x^4 \cdot y^2 $$
Now we sub in the variables $y, z$:
$$\begin{align*}\frac{5}{4} &= x^4 \cdot \left(\frac{1}{2}\right)^2\\
x^4 &= \frac{5}{4} \cdot 2^2 = 5 \end{align*}$$
We are left with an equation with only one variable, $x$. Looking back at the diagram, $x$ stands for the interval of a fifth.

To solve for the size of the fifth that used to be `3/2`, we need to take the 4-th root:
$$x = \sqrt[4]{5} \approx 1.49534878\dots $$
That means our interval for $x$ is the ratio....

No, that is not a ratio, because it is not a rational number!

### Dealing with irrational intervals with mappings

In Part II, we have only learnt how to name and handle intervals which are presented as rational numbers. Recall that these numbers are of the form $n/d$, where $n, d \in \mathbb{Z}$ are integers.

However, surds (square/cube/n-th roots) can result in **irrational** numbers, meaning that these numbers cannot be presented in the form $n/d$.

What we have obtained for the 1/4-comma meantone fifth is not a mistake. That is indeed the size of the fifth in that temperament.

In Part II, we have only learnt to name notes based on their ratios, but if we cannot express our interval as a ratio, what do we even call the note?

For that, we'll need to introduce a concept called **mapping**.

We still want to use ratios to refer to what these notes _used to_ be, because the way we name notes depends on those ratios, even if they aren't written as ratios anymore.

However, instead of the prime decomposition of the interval mapping to... prime numbers, we can instead map them to other things. For example, I can map the first "prime" to whatever $\sqrt{2}$ is (a tritone), and the second "prime" to whatever $\pi$ is (almost an augmented fifth plus an octave), then

$$\monzo{3 & 5} \equiv \sqrt{2}^3 \cdot \pi^5$$

This solves the issue of naming these notes, because we can still clearly see according to the [monzo](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Monzo) notation, it's going 3 steps in the "octaves" direction and 5 steps in the "fifths" direction.

But doing it this doesn't give us a clue as to what the interval _sounds_ like. It's inhumane to have to pull off irrational number mental calculations every time we want to audiate (imagine the sound of) a note. We'll need to think of something better:

#### Cents

I have been using this unit for interval size throughout this series, however the motivation for this unit hasn't come up until now.

The cent is a unit of **perceived interval size**. 1200 cents make up one octave, and a cent is equally spaced in perception. If you recall Part II, that makes it a logarithmic unit, because [perception is logarithmic](https://hackmd.io/@euwbah/extending-harmonic-principles-2#What-are-intervals-Logarithms). Hence, when you add intervals up, you **add** their cents, instead of multiplying.

This makes it a very intuitive unit for measuring intervals because it's obviously easier to add than to multiply. The formula for converting an interval to cents is:

$$\text{cents} = 1200 \times \log_2(\text{interval multiplier})$$

:::spoiler Simplified mathematical explanation

A cent is an interval such that adding up 1200 of this interval will equal an octave. Let $c$ be the interval of one cent. We know that $2$ is interval for an octave. Hence, we can write:

$$
\begin{align*}
\underbrace{c \times c \times \dots \times c}_\text{1200 times} &= 2 \\
c^{1200} &= 2 \\
c &= 2^{1/1200}
\end{align*}
$$
To count how many cents is in an interval, we notice that if an interval `I` is equal to `x` number of cents, then:
$$
\begin{align*}
I &= \underbrace{c \times c \times \dots \times c}_\text{\(x\) times} \\
&= c^x \\
\log_c(c^x) &= \log_c(I) \\
x \log_c(c) &= \log_c(I) \\
x &= \log_{2^{1/1200}}(I) \\
x &= 1200 \times \log_2(I) \quad\blacksquare
\end{align*}
$$
:::

Notice this formula says $\text{interval multiplier}$, and not $\text{ratio}$. Ratios aren't the only intervals now, as we can have irrational numbers as ratios, like $\pi / e$, which sounds like:

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br440hz%7D%0A2718281828459045235360287471352%3A3141592653589793238462643383279---" title="Xenpaper" frameborder="0"></iframe>

> :pencil: $\pi, e$ are not integers, so $\pi/e$ was not a rational number.

:::info
:writing_hand: How many cents was that interval?

:::spoiler Answer
$$1200 \times \log_2(\pi / e) \approx 250.56 \text{ cents}$$
:::

<!--:::-->

#### Mapping JI with cents: aka not tempering

We can calculate the cents of prime intervals in JI using the above formula. Doing this, we obtain the following:

| prime interval | cents |
| :--: | :--: |
| 2 | 1200
| 3 | 1901.955
| 5 | 2786.313
| 7 | 3368.826
| 11 | 4151.318

and so on...

We can represent this mapping as a **covector mapping**:
$$\cov{1200¢ & 1901.955¢ & 2786.313¢ & 3368.826¢ & 4151.318¢}$$

> :pencil: A covector is like a function that turns vectors back into scalars (single-number values).
>
> More precisely, it is itself a n-dimensional vector in the space of linear maps that take n variables.

In Part II, we discussed how a monzo is a vector, which is like a coordinate in space, and that adding up intervals is the same as adding up vectors, because we are adding up the steps we take in each prime interval's direction.

We can put a monzo $\monzo{-2 & 0 & 1}$ and a mapping $\cov{1200¢ & 1901.955¢ & 2786.313¢}$ together in what's called [**bra-ket**](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation) notation to represent that we are applying the mapping to the monzo:

$$\left<\underbrace{\begin{matrix}1200¢ & 1901.955¢ & 2786.313¢\end{matrix}}_{\large \text{bra mapping}}\right. \ \Bigg|\ \left.\underbrace{\begin{matrix}-2 & 0 & 1\end{matrix}}_{\large\text{ket monzo}}\right>$$

Applying the mapping will let us evaluate the size of the interval like so:
$$
\require{color}
\begin{align*}
& \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1901.955¢ & \color{#00ffff} 2786.313¢}{\color{#ffaa00} -2 & \color{#aaff00} 0 & \color{#00ffff} 1} \\
&= {\color{#ffaa00} [1200¢ \cdot (-2)]} + {\color{#aaff00}[1901.955¢ \cdot 0]} + {\color{#00ffff} [2786.313¢ \cdot 1]} \\
&= 386.313¢
\end{align*}
$$
This operation where we sum up the products component-wise is known as the **dot product** of vectors in linear algebra.

:::spoiler :writing_hand: What is the name and ratio of the interval given by the above monzo?
It is a 5-limit major third, `5/4`.
:::

Intuitively, we know that we are taking -2 steps in the octave direction, and that octaves are 1200¢, which gives -2400¢. Then, we take 1 step in the prime interval 5 direction, which is 2786¢, which gives 2786¢. We add the two together and we have the size of that JI interval in cents.

### Adjusting to monzos

We'll need to get familiar with intervals being represented as monzos, as ratios will no longer be applicable in a temperament (intervals are irrational in a temperament).

Here is the tabulation of octave-reduced prime intervals in monzos:

| 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 |
| --: | :--: | -- | -- |:--:| :--: | :--: | :--:| :--: | :--: | :--:
| [1> | 1200.00 | Octave | 2-limit | 8 | C5 | NA | | | C5 = [1>
| [-1 1> | 701.955 | Perfect 5th | Pythagorean | 5 | G4 | NA | | | G4 = [-1 1>
| [-2 0 1> | 386.314 | Major 3rd | Classic | -3 | E-4 | 81/80 = [-4 4 -1> syntonic | `+` | `-` | E4 = `81/64` = [-6 4> | Down |
| [-2 0 0 1> | 968.826 | Subminor 7th | Septimal | <b7 | Bb<4 | 64/63 = [6 -2 -1> septimal | `>` | `<` | Bb4 = `16/9` = [4 -2> | Down
| [-3 0 0 0 1> | 551.318 | Superfourth | Undecimal | ^4 | F^4 | 33/32 = [-5 1 0 0 1> undecimal | `^` | `<` | F4 = `4/3` = [2 -1> | Up

And the table of 3-limit Pythagorean notes in scale degrees and monzos:

| | Diminished | Minor | Perfect | Major | Augmented |
|:--:|:--:|:--:|:--:|:--:|:--:|
| **Unison** | b1 = [11 -7> | :x: | 1 = [0> | :x: | #1 = [-11 7> |
| **2nd** | bb2 = [19 -12> | b2 = [8 -5> | :x: | 2 = [-3 2> | #2 = [-14 9> |
| **3rd** | bb3 = [16 -10> | b3 = [5 -3> | :x: | 3 = [-6 4> | #3 = [-17 11> |
| **4th** | b4 = [13 -8> | :x: | 4 = [2 -1> | :x: | #4 = [-9 6> |
| **5th** | b5 = [10 -6> | :x: | 5 = [-1 1> | :x: | #5 = [-12 8> |
| **6th** | bb6 = [18 -11> | b6 = [7 -4> | :x: | 6 = [-4 3> | #6 = [-15 10> |
| **7th** | bb7 = [15 -9> | b7 = [4 -2> | :x: | 7 = [-7 5> | #7 = [-18 12> |
| **Oct** | b8 = [12 -7> | :x: | 8 = [1> | :x: | #8 = [-10 7> |

Notice that an apotome has the monzo
$$\sharp := \monzo{-11 & 7}$$
and is added to the interval for sharps, and subtracted for flats.

:::info
:pencil: **Exercise**

Are there errors in the above tables?
:::

### 1/4-comma meantone as a mapping

We are finally ready to tackle quarter-comma meantone. Let's go back to our tempered lattice:


We have already calculated that:
$$x = \sqrt[4]{5}, \quad y = \frac{1}{2}, \quad z = \frac{5}{4}$$

Now we need the cents mapping for the prime intervals of the monzos, so that we can convert these notes' monzos to cents so we can identify the pitches of these notes. This is a rather involved 3-step process:

#### i. Obtain the cents of the unknown intervals $x, y, z$

Recall that we can get cents using the formula $c = 1200 \log_2(I)$, where $I$ is the interval multiplier:
$$
\begin{cases}
x = 1200 \log_2(\sqrt[4]{5}) \approx 696.57, \\ y = 1200 \log_2(1/2) = -1200, \\ z = 1200 \log_2(5/4) \approx 386.313
\end{cases}$$

#### ii. Obtain the monzos of the unknown intervals

Remember that the monzo is a vector, and vectors can be treated like positions in space, or movements between positions. In the case of $x, y, z$, we can imagine these intervals to be movements in space, and we want to find how far these move the note. Then, monzos of the notes themselves are like fixed positions in space.

Since $y$ takes us from the position `E6` to position `E5`, we subtract the position of E6 from E5 to find how far and where $y$ will move the note:


$$y = \texttt{E5} - \texttt{E6} = \monzo{-5 & 4} - \monzo{-4 & 4} = \monzo{-1 & 0}$$

Similarly, the monzo of $x$ moves us from `C4` to `G4`:


$$x = \texttt{G4} - \texttt{C4} = \monzo{-1&1} - \monzo{0&0} = \monzo{-1&1}$$

And the monzo of $z$ moves us from `C4` to `E-4`, not `E4`! Because $z$ originally pointed at `E-4`. This is important.


$$z = \texttt{E-4} - \texttt{C4} = \monzo{-2 & 0 & 1} - \monzo{0} = \monzo{-2 & 0 & 1}$$

> :pencil: Remember that the monzo implicitly contains infinite trailing zeroes, $\monzo{0} = \monzo{0 & 0 & 0 & \cdots}$, that's why when we subtract $\monzo{0}$ from $\monzo{-2&0&1}$, we treat the "missing" components as zeroes.

#### iii. Solve system of equations

To obtain the cents mapping, we need to obtain the cents of the individual prime intervals: $2 \equiv \monzo{1}$, $\ 3 \equiv \monzo{0&1}$, $\ 5 \equiv \monzo{0&0&1}$, $\ 7 \equiv \monzo{0&0&0&1}\dots$ and so on.

Before proceeding, notice that none of the intervals we considered any prime above 5, which means the temperament only concerns the 5-limit. While this doesn't mean that our choices of notes are restricted to 5-limit (we can still use higher primes as-is without tempering), it does mean that our calculations regarding the tempering itself can be restricted to just primes 2, 3 and 5.

This helps us to simplify what's going on.

If you're already familiar with linear algebra, you know what to do &mdash; you express this as an augmented matrix and run row operations on the matrix until you get the identity matrix on the left.

However, if you're not familiar with linear algebra, things are about to get *spicy*. Because we are going to do exactly just that, but don't worry, I will try to make this as visual as possible.

Gathering all the information we have so far, we have this:


The goal now is to isolate all three prime intervals, one at a time, until we know the cent value for all 3 of them.

Let's take the low hanging fruit first. Notice that $y$ is set to $\monzo{-1}$ and has the cent value of -1200. This immediately tells us that the monzo $\monzo{1} \equiv 1200¢$, because we are just reversing the direction we are going, which reverses the interval.

Algebraically, we can manipulate the monzo like so:
$$
\begin{align*}
\monzo{-1} &= -1200¢ \\
-1 \cdot \monzo{-1} &= -1 \times (-1200¢) \\
\monzo{1} &= 1200¢
\end{align*}
$$

And visually we flip the direction of the arrow:

Next, we realize that if we're able to remove the $\monzo{-1}$ from $x$, we will be able to isolate the next prime interval, since $\monzo{-1 & 1} - \monzo{-1} = \monzo{0 & 1}$. Well, we can remove -1 by adding 1, and we already know that $\monzo{1} \equiv 1200¢$, so:


Adding up the two correlated monzos and cent values, we find that the second prime interval comes in at 1896.57¢.

Similarly, notice that the monzo of $z \equiv \monzo{-2 & 0 & 1}$ will give us the third prime interval $\monzo{0 & 0 & 1}$ if we are able to add 2 octaves to it:


Thus, adding 2 of the associated cent value of $\monzo{1}$ to the cents of $z$ gives us 2786.31¢.

#### Cents mapping of 1/4-comma meantone

Now collect the cents of each prime interval into one place giving us the cents mapping of 1/4-comma meantone:

$$\text{1/4-comma meantone} := \cov{1200¢ & 1896.57¢ & 2786.31¢}$$

As for the cent values of the primes 7, 11, or higher, since they are not specified in the mapping, we assume them to be the defaults, which are cents of the prime intervals just intonation. Even though they're not explicitly written down, the cent values all prime intervals are still there, just like how there's infinite trailing unwritten zeroes at the end of a monzo.

#### Verifying the tone is mean using mapping

Recall the [**dot product** bra-ket operation](#Mapping-JI-with-cents-the-no-tempering-temperament) for evaluating the size of the interval using a monzo and a mapping.

We finally have obtained our cents mapping for 1/4-comma meantone, so now we can use the dot product to find the cents of intervals in this temperament.

Now we can verify if we have achieved our [tuning goals that we prioritized for 1/4-comma meantone](#14-comma-meantone-attempt)

1. The meantone property: the JI interval `81/80` should vanish under the temperament and map to 0 cents (unison interval `1/1`)

First we'll need to obtain the interval `81/80` in monzo form:

$$\frac{81}{80} = 2^{-4} \cdot 3^4 \cdot 5^{-1} \equiv \monzo{-4 & 4 & -1}$$

Then, we apply the bra-ket operator using the mapping we have obtained earlier:

$$
\begin{align*}
& \left<\underbrace{\begin{matrix}\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.57¢ & \color{#00ffff} 2786.31¢\end{matrix}}_{\large \text{1/4-comma meantone}}\right.\ \large\mid\ \left.\underbrace{\begin{matrix}\color{#ffaa00} -4 & \color{#aaff00} 4 & \color{#00ffff} -1\end{matrix}}_{\large\text{81/80}}\right> \\
&= {\color{#ffaa00} [1200¢ \cdot (-4)]} + {\color{#aaff00}[1896.57¢ \cdot 4]} + {\color{#00ffff} [2786.31¢ \cdot (-1)]} \\
&= -0.03¢ \approx 0¢
\end{align*}
$$

There was some rounding errors because we cut off some decimal places in the calculation of the mapping, but you can double check that `81/80 = [-4 4 -1>` should indeed map to 0.

2. The octave should still map to `2/1`

The monzo of an octave is $2 \equiv \monzo{1}$. We apply the same process:

$$
\begin{align*}
& \left<\underbrace{\begin{matrix}\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.57¢ & \color{#00ffff} 2786.31¢\end{matrix}}_{\large \text{1/4-comma meantone}}\right.\ \large\mid\ \left.\underbrace{\begin{matrix}\color{#ffaa00} 1 & \color{#aaff00} 0 & \color{#00ffff} 0\end{matrix}}_{\large\text{octave}}\right> \\
&= {\color{#ffaa00} [1200¢ \cdot 1]} + {\color{#aaff00}[1896.57¢ \cdot 0]} + {\color{#00ffff} [2786.31¢ \cdot 0]} \\
&= 1200¢
\end{align*}
$$

And we get back `1200¢`, which is a `2/1` octave by the definition of cents.

3. Both the two major thirds `81/64` and `5/4` should both map to `5/4`.

:::spoiler :writing_hand: Try it yourself first

First, we convert the `5/4` major third into monzo form $\monzo{-2 & 0 & 1}$, then we run the bra-ket mapping:

$$
\begin{align*}
& \left<\underbrace{\begin{matrix}\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.57¢ & \color{#00ffff} 2786.31¢\end{matrix}}_{\large \text{1/4-comma meantone}}\right.\ \large\mid\ \left.\underbrace{\begin{matrix}\color{#ffaa00} -2 & \color{#aaff00} 0 & \color{#00ffff} 1\end{matrix}}_{\large\text{5/4}}\right> \\
&= {\color{#ffaa00} [1200¢ \cdot (-2)]} + {\color{#aaff00}[1896.57¢ \cdot 0]} + {\color{#00ffff} [2786.31¢ \cdot 1]} \\
&= 386.31¢
\end{align*}
$$

Next, we convert the `81/64` major third into monzo form $\monzo{-6 & 4}$, and do the same:

$$
\begin{align*}
& \left<\underbrace{\begin{matrix}\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.57¢ & \color{#00ffff} 2786.31¢\end{matrix}}_{\large \text{1/4-comma meantone}}\right.\ \large\mid\ \left.\underbrace{\begin{matrix}\color{#ffaa00} -6 & \color{#aaff00} 4 & \color{#00ffff} 0\end{matrix}}_{\large\text{81/64}}\right> \\
&= {\color{#ffaa00} [1200¢ \cdot (-6)]} + {\color{#aaff00}[1896.57¢ \cdot 4]} + {\color{#00ffff} [2786.31¢ \cdot 0]} \\
&= 386.28¢
\end{align*}
$$

Now, we need to evaluate the cents of `5/4` to check that it is the same size:
$$1200 \log_2(5/4) \approx 386.31\dots$$

there is errors carried forward from of the truncated digits of the cents mapping, but otherwise, it is same interval.

:::

#### Notes in 1/4-comma meantone

Going back to the diatonic 3-limit major scale example that motivated this whole journey:

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B1%2F1_81%2F64_3%2F2_243%2F128%5D-%0A%5B9%2F8_4%2F3_27%2F16_2%2F1%5D-%0A%5B9%2F8_4%2F3_3%2F2_243%2F128%5D-%0A%5B1%2F1_81%2F64_3%2F2_2%2F1%5D-" title="Xenpaper" frameborder="0"></iframe>

We can now apply the 1/4-comma meantone temperament to this. First, we need to write all the ratios as monzos:

$$
\begin{cases}
1/1 \equiv \monzo{0} \\
9/8 \equiv \monzo{-3 & 2} \\
81/64 \equiv \monzo{-6 & 4} \\
4/3 \equiv \monzo{2 & -1} \\
3/2 \equiv \monzo{-1 & 1} \\
27/16 \equiv \monzo{-4 & 3} \\
243/128 \equiv \monzo{-7 & 5} \\
2/1 \equiv \monzo{1}
\end{cases}
$$
> :pencil: At this point, it's a good idea to have the first few powers of 2 and 3 memorized.

Afterwards, we apply the bra-ket operator on each one to obtain how many cents these intervals map to. I have added more decimal places to reduce rounding error. Since all of the above monzos only use the first two primes, we can ignore the prime-5 (3rd) component of the cents mapping

$$
\begin{cases}
1/1 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} 0} = 0¢ \\
9/8 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} -3 & \color{#aaff00} 2} = 193.1568¢ \\
81/64 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} -6 & \color{#aaff00} 4} = 386.3136¢ \\
4/3 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} 2 & \color{#aaff00} -1} = 503.4216¢ \\
3/2 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} -1 & \color{#aaff00} 1} = 696.5784¢ \\
27/16 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} -4 & \color{#aaff00} 3} = 889.7352¢ \\
243/128 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} -7 & \color{#aaff00} 5} = 1082.892¢ \\
2/1 \equiv \braket{\color{#ffaa00} 1200¢ & \color{#aaff00} 1896.5784¢ & \color{#00ffff} NA}{\color{#ffaa00} 1 & \color{#aaff00} 0} = 1200¢ \\
\end{cases}
$$

And finally, we replace the original ratios with the new cent intervals:

<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B0c_386.3136c_696.5784c_1082.892c%5D-%0A%5B193.1568c_503.4216c_889.7352c_1200c%5D-%0A%5B193.1568c_503.4216c_696.5784c_1082.892c%5D-%0A%5B0c_386.3136c_696.5784c_1200c%5D-" title="Xenpaper" frameborder="0"></iframe>

Congratulations! You have made and played your meantone.

### Why it is called 1/4-comma meantone

What we did in 1/4-comma meantone was to sacrifice only the tuning of the fifth to obtain the exact tuning of the `5/4` third.

If we look at the cents of the original `3/2` fifth: $1200 \log_2 (3/2) = 701.955$, and compare that with the cents our new fifth $696.5784$, we see that we have flattened it by
$$701.955¢ - 696.5784¢ = 5.3766¢$$
It doesn't seem like a lot, but because fifths are such concordant intervals, changing its tuning even by a little can be noticeable:

<iframe width="560" height="240" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A0c_696.5784c_%5B0c_696.5784c%5D-.%0A1%2F1_3%2F2_%5B1%2F1_3%2F2%5D-%0A%5B0c_696.5784c%5D--%5B1%2F1_3%2F2%5D--%5B0c_696.5784c%5D--" title="Xenpaper" frameborder="0"></iframe>

Consider the cents of the syntonic comma `81/80` that we have tempered out: $1200 \log_2(81/80) = 21.5063¢$, and notice that our fifth was flattened by 5.3766¢, which is:

$$\frac{5.3766¢}{21.5063¢} = \frac{1}{4}$$

of the syntonic comma. **Each fifth gets flattened a quarter of the syntonic comma, which is why we call this 1/4-comma meantone.**

### 12edo is also a meantone

What if we think 1/4-comma messes up the fifth by too much? We can always try detuning the fifth by smaller amounts, but that would also affect the size of our major third prime 5 interval!

For example, if we look at 12edo, the division of the octave into 12 equal parts where each edostep is exactly 100 cents, notice that we are detuning the fifth from $701.955¢$ to $700¢$, which is about $1.955¢$. Compared to the size of the syntonic comma, this is approximately $\frac{1}{11}$th of it. (We can call 12edo an approximately 1/11-th comma meantone. It is also called a 1/12 pythagorean comma meantone &mdash; we'll get to that later)

However, by tempering the fifth less in 12edo, we no longer have the pure `5/4` third like in 1/4-comma meantone:
$$\left<\underbrace{\begin{matrix}\color{#ffaa00} 1200¢ & \color{#aaff00} 1900¢ & \color{#00ffff} 2800¢\end{matrix}}_{\large \text{12edo}}\right.\ \large\mid\ \left.\underbrace{\begin{matrix}\color{#ffaa00} -2 & \color{#aaff00} 0 & \color{#00ffff} 1\end{matrix}}_{\large\text{5/4}}\right> = 400¢$$

And 400¢ compared to the pure $1200\log_2(5/4) \approx 386.3¢$ is a pretty long way off!

<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B15c%7D%0A%5B1%2F1_5%2F4%5D-%5B0c_400c%5D-%5B1%2F1_5%2F4%5D-%5B0c_400c%5D-" title="Xenpaper" frameborder="0"></iframe>

Hence, the [meantone temperaments](https://en.xen.wiki/w/Meantone_family) are a family of temperaments that involve messing with the tuning of the primes 3 and 5, with the common goal of making the **syntonic comma `81/80` vanish**.

### But a temperament (like meantone) is not always an edo

You may think that "temperament" and "edo" both mean the same thing. However, the definition of an edo is that it divides the octave into **equal intervals**, which means every interval in an edo must be representable as some integer number of **edosteps**: the size of the smallest non-unison interval in an edo.

If we go back to our 1/4-comma meantone mapping:

$$\text{1/4-comma meantone} := \cov{1200¢ & 1896.57\dots¢ & 2786.31\dots¢}$$

The number of cents that the prime interval 5 (major third + 2 octaves) is mapped to is the number of cents of the JI ratio `5/4` itself, and the number of cents for prime interval 2 (octave) is `2/1` itself. Because of the Fundamental Theorem of Arithmetic discussed in Part II, we already know that these two intervals will never be multiples of each other, and live in completely different worlds.

That is, no matter how many times we try to add up $1200\log_2(5) = 2786.31...¢$, we will never ever reach a multiple of an octave $1200 \log_2(2) = 1200¢$, which means we can keep going up by major thirds and down by N octaves and obtain infinitely many distinct notes within an octave, making it _**$\infty$ edo**_, which does not qualify as an edo.

### Rank-nullity theorem: How many intervals to temper out before it becomes an edo?

> tl;dr it is the number of primes you want to use in the JI subgroup minus one. Linear algebra people can reference the rank-nullity theorem, otherwise I have this plain English explanation here:

The answer depends on the [JI subgroup](https://hackmd.io/@euwbah/extending-harmonic-principles-2#The-JI-subgroup-of-a-tuning) (english: what prime intervals do you want to compose with?)

Specifically, if you want to compose with the primes `2.3.5`, as per 12edo music theory, then you have 3 primes. If you don't temper out any intervals, then each prime will be in its own axis, never intersecting with another prime. If you keep going forward one step at a time, will you ever move to the right? Never!

:::spoiler :nerd_face: :point_up:
If you are on a curved non-euclidean geometry like a sphere/hyperbola, Earth or space-time, actually yes, which is what a temperament is: it is folding a euclidean space on to itself so it becomes a non-euclidean geometry that is locally euclidean aka a topological manifold. Let's **not** go there.
:::

That means if you are using 3 untempered primes in your tuning, you actually need 3 **dimensions** to represent your note. The 2 can be your X axis, 3 can be Y and 5 can be Z.


However, an edo only lives in one dimension: the dimension containing the edostep. Every interval of an edo must be some number of edosteps, which is the whole point of dividing an octave equally. If your edostep is on the X axis, but you have some interval that the span of the X axis cannot reach, then it is not an edo anymore.


When you temper a comma, what's actually going on is you're creating an equivalence between two or more dimensions, and because of that, the total number of dimensions you need to represent your notes, which is the [**Rank**](https://en.xen.wiki/w/Rank_and_codimension), reduces by one. The meantone temperament tempering out the syntonic comma `81/80` created an equivalence between 4 steps in the prime-3 direction (4 fifths + 4 octaves), and 4 steps in the prime-2 direction plus 1 step in the prime-5 direction (1 major third + 6 octaves).


Why does this reduce the dimension by one? Well, imagine you're the snake in the snake game:

<iframe width="560" height="315" src="https://www.youtube.com/embed/DekS8Pgb1qc?si=KdLFo5mXOguAvI1I" 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 live in 2 dimensions: up-down, or left-right. What happens when you touch the edge? You lose.

Now imagine someone made a hack: instead of losing when you hit the left or right edge, you enter a portal. If you're on the right portal, you get teleported to the left portal and move one square up. If you're on the left portal, you teleport to the right portal and move one square down. This is what happens when you temper a comma.

You still have access to both dimensions, and you still can choose to move freely in either one. But fundamentally, if started from the bottom left of the screen, and you just kept on holding the right arrow key to go right, you would have traversed the entire game screen.

Now imagine you are the snake, looking forward (to the right side) the whole time. Instead of a wall, you see a portal in front of you, and through that portal you see the row of squares that is immediately to your left, but you see it in front of your path too. You could traverse the entirety of the map by simply going forward and backward, without having to make any 90&deg; turns, doesn't that make the map effectively one dimensional from your perspective?

This is the principle of the [**Rank-Nullity Theorem**](https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem).

Now instead of the snake game map, each coordinate in the map is a note.

What this means is, if you want to obtain an **edo**, you need to effectively change your perspective of the "game world" to a one dimensional one by installing portals, so that you can access all the notes only going forward/backward, without having to make any 90&deg; turns.

Each portal you install is a single interval tempered out, and each portal will reduce your effective dimension by one.

If you start with $N$ primes, you have $N$-dimensional snake game map. To end up with an effectively 1-dimensional game map, you will need $N-1$ portals installed, thus you need $N-1$ intervals tempered to obtain an edo from $N$ primes

> :bulb: For a formal introduction to the Rank-Nullity theorem from the xen math perspective, check out [Dave Keenan & Dougals Blumeyer's guide to RTT: 4. Exploring Temperaments](https://en.xen.wiki/w/Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT/Exploring_temperaments#Rank_and_nullity)

## What meantone enables for modern harmony

It's time to take a step back and remember why we're even doing all of this in the first place: :musical_note: <u>**MUSIC**</u> :musical_note: :stars:

We were (hopefully still) motivated by the concordance of the pure `5/4` major thirds, and in the process of maximizing the number of `5/4` major thirds in our scale, we encountered a problem in constructing the 7-note major scale. We could not settle on 7 fixed notes in 5-limit just intonation, because we had contradicting intervals between those notes that causes wolf intervals.

Because of that, in JI, we have to worry about which variant of which note to use to make it in tune:

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B15c%7D%0A%5B1%2F1_5%2F4_3%2F2_15%2F8%5D-%0A%5B10%2F9_4%2F3_5%2F3_2%2F1%5D-_%23_10%2F9_D-%0A%5B9%2F8_4%2F3_3%2F2_15%2F8%5D-_%23_9%2F8_D_natural%0A%5B1%2F1_5%2F4_3%2F2_2%2F1%5D-" title="Xenpaper" frameborder="0"></iframe>

And even if you were to accept that there can be two variants of a note, if you wanted to freely move around diatonic 7-note progressions, that variant would have a variant, and that variant would have its own variant, *ad infinitum*. This results in a problem known as a **comma pump**

<iframe width="560" height="315" src="https://www.youtube.com/embed/Tn0VZInlhoo?si=8xLD78pZIyNLfu-I" 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>

which can arise even in the simplest of progressions, as the above video demonstrates.

Now that we have made the syntonic comma **vanish**, we do not have to worry about our pitch drifting all over the place when we invoke the syntonic comma in our progressions, which means we can move the roots of our chords/bass notes freely.

Now, let's go into the specifics of how tempering the syntonic comma has impacted modern harmony.

### Diatonic Pluralism

In 12edo theory & culture, there is a general categorization of the different [sides of a tonality](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Tonality-as-a-state-diagram) into 3 primary functions:

* Tonic (T): The feeling of 'homeness' and resolution. The setting of the story
* Subdominant (S): The feeling of journeying away from home, but without much issue. The development of the story.
* Dominant (D): The feeling of having an issue. The climax of the story.

> :pencil: This is purely attributed to culture, nothing to do with calculations

Any of these three functions can move to any of the three functions, but the default culturally entrained tonality follows $S \to D \to T$:

<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B193.1568c_503.4216c_889.7352c_1200c%5D-_%23_S%3A_iim7%0A%5B193.1568c_503.4216c_696.5784c_1082.892c%5D-_%23_D%3A_V7%0A%5B0c_386.3136c_696.5784c_1200c%5D-_%23_T%3A_I" title="Xenpaper" frameborder="0"></iframe>

Because we are reducing all the chords down to a framework of only 3 categories, amongst the 7 diatonic triads (which are formed by stacking major or minor thirds on each of the 7 notes of the 7-note major scale), we categorize these triads like so:

<iframe width="560" height="400" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B0c_386.3136c_696.5784c%5D__________%23_I%3A___Ton%0A%5B193.1568c_503.4216c_889.7352c%5D___%23_ii%3A__Sub%0A%5B386.3136c_696.5784c_1082.892c%5D___%23_iii%3A_Ton%0A%5B503.4216c_889.7352c_1200c%5D_______%23_IV%3A__Sub%0A%5B696.5784c_1082.892c_'193.1568c%5D__%23_V%3A___Dom%0A%5B889.7352c_'0c_'386.3136c%5D________%23_vi%3A__Ton%0A%5B1082.892c_'193.1568c_'503.4216c%5D_%23_vii%3A_Dom%0A%5B'0c_'386.3136c_'696.5784c%5D_______%23_I%3A___Ton" title="Xenpaper" frameborder="0"></iframe>

* Tonic: `I = iii = vi`
* Subdominant: `ii = iv`
* Dominant: `V = vii dim`

If we apply the culturally entrained tonality $S \to D \to T$, we actually obtain numerous variations such as:

<iframe width="560" height="500" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A%5B503.4216c_889.7352c_1200c%5D_______%23_IV%3ASub%0A%5B696.5784c_1082.892c_'193.1568c%5D__%23_V%3A_Dom%0A%5B0c_'386.3136c_696.5784c%5D.________%23_I%3A_Ton%0A%0A%5B503.4216c_889.7352c_1200c%5D_______%23_IV%3A_Sub%0A%5B1082.892c_'193.1568c_'503.4216c%5D_%23_vii%3ADom%0A%5B889.7352c_'0c_'386.3136c%5D._______%23_vi%3A_Ton%0A%0A%5B193.1568c_'503.4216c_889.7352c%5D_%23_ii%3A_Sub%0A%5B696.5784c_1082.892c_'193.1568c%5D_%23_V%3A__Dom%0A%5B'386.3136c_696.5784c_1082.892c%5D_%23_iii%3ATon_(but_no_clausula)%0A%5B889.7352c_'0c_386.3136c%5D________%23_vi%3A_Ton%0A%5B'193.1568c_503.4216c_889.7352c%5D_%23_ii%3A_Sub%0A%5B696.5784c_1082.892c_193.1568c%5D__%23_V%3A__Dom%0A%5B0c_386.3136c_696.5784c_'0c%5D_____%23_I%3A__Ton" title="Xenpaper" frameborder="0"></iframe>

Sounds like music! But be careful &mdash; this doesn't mean that we can whack pluralisms with gleeful abandon. The `iii` minor chord did not sound resolved at all, even though it was classified as "Tonic"! The framework of pluralism is only an approximation, for the underlying mechanism of the [clausulae](https://hackmd.io/@euwbah/extending-harmonic-principles-1#The-modern-clausulae-of-semitones).

There are many explanations for why these categories are chosen, and with these triads. To me, the simplest understanding is based on the principle of applying [fifth-coloring](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Formulating-extension-notes-with-fifth-colorings) to the Fa-Mi and Ti-Do [clausulae](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Voice-Leading). Of course, this line of reasoning is just my own speculation based on personal experience.


> :pencil: These can also be understood using [PLR transformations](https://en.wikipedia.org/wiki/Neo-Riemannian_theory) as in the original neo-riemannian theory, but I don't think PLR is necessary for understanding modern tonality.

We'll deal with adding more notes soon, but for now let's walk through the above diagram and explain the principles behind pluralism for diatonic chords (i.e., chords using only 7 notes of the major scale)

#### Significance of a triad

Because of the concordance of octaves, we assume octaves, the first prime interval, to be the same note. In the principle of fifth-coloring, the concordance of the fifth, the second prime interval, makes the harmonic information of the notes similar.

Beyond the second prime interval, the number of aligned harmonics is no longer sufficient for the two notes of the interval to feel 'equal' in any way (although tunings that equate them as a period still do exist, such as [ED5/4s](https://en.xen.wiki/w/Ed5/4))

This means that the 3rd prime interval, `5/1` (or octave-reduced `5/4`) is the first prime that creates a unique distinguishable color, on its own, without having to stack multiple.

A triad is defined as any three note chord, but in the modern context it usually has the default interpretation as two notes a perfect 5th apart surrounding a note in the middle, which is usually a third (scale degree) from the root (bottom note of triad).

The 5-limit efficiently communicates color: we have the choice of the third being major `5/4` or minor `6/5`.


This highlights the primordial aspect of the triad that could possibly explain its cultural significance - it is the first structure that gives sufficient harmonic information and the notion of parity which enables it to be classified as either this or that, major or minor, while still maintaining a sufficiently high amount of concordance, and the two options for thirds are sufficiently spread apart in pitch (70.6¢ in JI). This makes the parity easily distinguishable.

#### Significance of the third of the triad

Hence, the third of the triad can be seen as a significant identifier of the harmonic information to be communicated, i.e. the major/minor parity.

#### Ti and Fa as thirds

We culturally construct our notion of tonality around clausulas. But we need an effective way to communicate the clausula we are preparing to execute.

Because of how effective the third of a triad is at adding harmonic information while maintaining concordance, the most efficient way to communicate harmonic information of a single note is to place it as the third of some triad.

Hence, we set Ti and Fa to be thirds of triads.

Restricted to the 7 diatonic notes, Ti is the major third of scale degree 5, and Fa is the minor third of scale degree 2.

This creates two triads, the triad associated with the plagal clausula, with scale degrees `2 4 6`, and the triad associated with the perfect clausula, `5 7 2 (or 9)`. We call this the subdominant and domianant respectively.

Then analyzing the resolution notes of the clausulae, Do and Mi already form the root and third of the major triad `1 3 5`. We call this the tonic.

The three triads `1 3 5` (tonic), `2 4 6` (subdominant), and `5 7 2` (dominant) form the three base triads:


#### Harmonic areas

From each of the three base triads above, we apply the principle of [fifths-coloring](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Formulating-extension-notes-with-fifth-colorings) to find other notes that will contribute similar harmonic information. This creates the harmonic areas that surround these triads that are similar in harmonic content.

This way, pluralism, the idea that different notes/chords can substitute for other notes to evoke similar function, can be understood as the extension of the significant thirds of triads that identify the important notes as per clausulae (Ti-Do, Fa-Mi) via fifths-coloring.

Apart from fifths-coloring, we also want dominant chords to have the Fa (4) of the Fa-Mi clausula, because the cultural notion of "Dominant" is that it should feel as far away from home as possible, so the more tension, the merrier. This is why we add the Fa to the Dominant harmonic area, even though it is not a fifth-coloring.

Here is the same diagram of the lattice simplified into chord notation relative to C:


#### Necessity of the meantone temperament

If the syntonic comma `81/80` did not vanish, the above diagram would not have made any sense, because the two Tonic harmonic areas will not be equal! The cultural concept of the three harmonic functions must only contain 3 harmonic areas, not 4, and it does not support two entirely different sets of tonics simultaneously.

This is why the meantone temperament, making `81/80` vanish, is a prerequisite step which allowed these specific theories of harmonic function and pluralism to evolve in western music culture.

### Chromaticism in extended pluralism

So far, we have conceived of pluralism amongst the 7 notes of our newly founded major scale in meantone temperament. But meantone alone still gives us an infinitude of notes, what shall we do with the rest of them?

In my opinion, 12edo has allowed for blatant chromaticism (opposite of diatonic) all over the place because of the sheer number of coincidences, temperaments, and symmetries falling into place which allow the use of mental gymnastics to justify just about any note choice. Chromaticism has become a forgotten art, that lost its roots from well founded principles, especially in the music of Bach.

Let's explore how pluralism supports the tasteful use of chromaticism (don't worry, there's more ways coming later).

In a nutshell, this is what I will refer to as **2nd-order pluralism**:

1. Start from the diatonic plurals:
* Tonic `I = iii = vi` (plus extensions, but following rules of clausula)
* Subdominant `ii = IV` (+ ext)
* Dominant `V = vii dim = V7` (+ ext)

2. Choose a tonic to resolve to, e.g. `Cmaj`, which will be our `I`. But choose a different tonic chord, like `Am = vi`, and think in terms of that root instead.

3. Now the dominant (and subdominant) chords can be chosen with respect with the **other** tonic chord, because it is **functionally equivalent** to the dominant of the initial tonic chord by one level of pluralism.

Confused? Here:


We choose to resolve to C maj. But instead of using scale degrees 5 = G and 7 = B, we decide to pretend that A = 1, because A is also a possible root to resolve to, since pluralism allows us to equate `A min = C maj`.

Now we are relative to A = 1, so the dominant chord `V maj` is now built off the 5th scale degree of A which is...

:::spoiler Answer
E
:::

and not forgetting the other option for the dominant `vii dim` (diminished chord on the 7th scale degree), `G# dim`.

To keep things simple, for subdominant chords, we relate the dominant 5 = E. Based on that, we get `ii = B min` and `IV = D maj` for subdominants. (We also could have taken things further and used 5 = G# and apply one more level of pluralism.)

According to this theory, we can start at any place in `SUBS`, and follow the arrows to `C maj`, so let's try:

<iframe width="560" height="240" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B31edo%7D%0A%5B5__15_23%5D__%23_Dmaj%3A_Sub_of_A%0A%5B10_20_28%5D__%23_Emaj%3A_Dom_of_A%0A%5B0__10_18%5D-_%23_C%3A____Ton" title="Xenpaper" frameborder="0"></iframe>

What? That was horrible!

#### Don't forget the principles

All of this functional equivalence stuff is just an abstraction over what's _really_ happening. We've gotten so carried away that we forgot two of the most fundamental ideas introduced in Part I: the principle of [**pitch memory**](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Pitch-memory-amp-semitone-cancellation-theory) and [**clausula**](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Voice-Leading).

First, there was no clausula to be found anywhere in the above progression. The "semitone" between B (Ti) and C (Do) is not a semitone because it was separated by an octave, and that doesn't count.

Next, the D major introduced D, F# and A to the memory. Next, E major introduced B and E, and its G# formed a clausula from A (Fa) to G# (Mi):


Right now we have the notes D, E, F#, G#, and B in our pitch memory.

But now the last chord C major hits and along comes a **G**.


Our ears are so confused. Do we have a G# to G clausula? Or a F# to G clausula? There is no current model of standard tonality entrained in culture that supports both happening **at the same time**. The only model is in enclosures of bebop, but I have a lot of counterarguments for the current understanding of them as well, which I will have to tackle in some future journal.

Simply put, we need to still pay homage to the essence of western harmony: pitch memory and clausula. We can fix this by adding just two notes:


<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B31edo%7D%0A%5B5__15_23%5D_____%23_Dmaj%3A_______Sub_of_A%0A%5B13_20_28_'10%5D_%23_Fdim(maj7)%3A_Dom_of_A%0A%5B0__10_18_'0%5D-_%23_C%3A____Ton" title="Xenpaper" frameborder="0"></iframe>

We add an F to the `E maj` chord, which is valid because of a (non-octave) symmetry we'll discuss later. The direct purpose of this F is to forget F#, so that now the F#-G clausula is no longer present. Because the conflicting F#-G clausula is gone, the G#-G clausula now stands perceivable.

Why did we chose to prioritize G#-G? Because we am borrowing from the Plagal tenorizans molle clausula of the soft modes in Renaissance music (aka b6 resolving to 5), which is ingrained into modern music culture. Even though this is #5 instead of b6, it is close enough that it registers as the same movement.

Also, we shifted E up an octave in order to prevent the clash between E and F. Clashes on clausula notes will bury the perception of the clausula due to excessive discordance, so shifting E up will actually improve the effectiveness of the F to E clausula between the second and third chords.

Adding F also has the additional consequence of contributing the Fa (F) to Mi (E) clausula, which is perfect, because Fa-Mi and Ti-Do works well together (perfect dominant 7th cadence).

Next, we also added a C an octave up, because the B in `E maj` did not resolve Ti-Do to C, because the C was an octave away!

Now let's take a look at what's left in our pitch memory at the end of the three chords: `C D E G 'C 'E`. A very consonant `C maj add9` chord.

:::warning
:warning: Pitch memory is a probabilty, so my treatment of it here as an binary recalled/forgotten dichotomy is incorrect. However, I find that even naive applications of it is effective.
:::

I wanted to give this example to highlight that **principles beat abstractions**. Abstractions like functional equivalences and pluralisms are ways to package what's going on under the hood, but in doing so, we may lose our bearings. This brings us to:

### The principles of pluralism

Motivated from the bad example we left off, and how we somehow managed to fix it, we need to find a way to generalize pluralism in a way that doesn't forget its roots.

What exactly about pluralism that makes it work, and why doesn't it work sometimes?

1. It works when the necessary clausula are still present, or when it happens to add proper clausula that contributes to the perception of the current tonality, not fight it.

2. It works because the fifth-colorings of pluralism add similar colors, which share common tones between chords.

To me, this means that pluralism isn't actually about the categorization of the 3 classes $S, D, T$, and the abstractions/transformations we can do.

Instead, pluralism boils down to:

1. Adding/substituting **similar-colored notes** to **existing culturally entrained tonalities**
2. in a way that **adds more clausulae** that **contributes to the tonality**
3. ensuring there are **no conflicting/contradicting clausulae**.

These principles transcend the meantone temperament, and can be applied to any tuning at all.

Let's show another example (still in meantone):


<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%7B31edo%7D%0A%5B25_12_'2%5D___%23_A%23min%2FE%23%3A_Sub_of_G%23%0A%5B7_15_23_'0%5D_%23_D%23dim7%3A___Dom_of_E%0A%5B10_18_'0%5D-__%23_C%2FE%3A______Ton" title="Xenpaper" frameborder="0"></iframe>


Here I am using more exotic clausula, `#1-1` of C (or `6-b6` of E which targets b9 of dominant of E), `#6-6` of C (or `#4-4` of E, targeting b7 of dominant of E), `#4-5` of C (or `7-1` of G, which can be seen as a fifth-coloring of `7-1` of C)

However, the principles are held: a system of tonality exists around these clausulae, and at each chord, the pitch memory agrees with the current state/side of the tonality, ending with the notes `C E G A` left in the pitch memory, a plural tonic of C borrowing from the relative minor.

If you haven't yet realized, all the above examples were either in 31edo or 1/4-comma meantone, and 31edo almost exactly equal to 1/4-comma meantone (we'll get here). Everything we've discussed in this section is immediately applicable to the goal of this series: to take harmonic principles from 12edo and apply them to 31edo and beyond.

## Symmetries (and edos)

So far, meantone has given us the major scale, and has made the cultural construct of harmonic functions and pluralisms possible.

However, one thing it hasn't yet given us is the guarantee that we can make octaves out of intervals that do not use the prime interval 2 (octaves).

For example, in 1/4-comma meantone, our major third is equal to the JI `5/4`, which means we still run into this problem:

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A1%2F1_5%2F4_25%2F16_125%2F64-.%0A1%2F1_125%2F64_%5B1%2F1_125%2F64%5D-" title="Xenpaper" frameborder="0"></iframe>

Three major thirds is still _almost_, but not, an octave.

Also, four fifths in meantone always equal to whatever our thirds are plus two octaves, which means if our thirds don't add up to octaves, then neither will our fifths in meantone.

Recalling the [rank-nullity theorem](#Rank-nullity-theorem-How-many-intervals-to-temper-out-before-it-becomes-an-edo), notice that we've only been dealing with the JI subgroup `2.3.5` here (we haven't explored 7, 11 yet). We started with the 3 dimensions of the JI subgroup, we already made 1 "portal" which is the meantone temperament, so now we're at rank-2 (2 dimensions worth of notes), which means if we tempered **one more linearly-independent comma** on top of the syntonic comma we already did for meantone, we would end up with an edo (rank-1).

### edos the hard way: tempering commas

Our new tuning goal now is that we want 3 major thirds to equal an octave. Otherwise, this couldn't have existed:

<iframe width="560" height="315" src="https://www.youtube.com/embed/vKWrrI6DPCg?si=Bv0yOU2ug6GZdmQe" 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>

Nor this:

<iframe width="560" height="315" src="https://www.youtube.com/embed/TQSyRXRuk6Y?si=DD8-YwVNaGfDjrtd&amp;start=236" 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>

What's the interval of this comma that we want to temper?

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br32%2F27%7D%0A1%2F1_5%2F4_25%2F16_%5B125%2F64_125%2F128%5D-.%0A1%2F1_125%2F128_1%2F1_125%2F128" title="Xenpaper" frameborder="0"></iframe>

:::spoiler Answer
`128/125` (Commas are usually written as positively sized intervals that go up, not down)
:::

This comma is the [diesis](https://en.xen.wiki/w/128/125) (plural dieses), also known as the augmented comma, because it is the only thing preventing us from having a perfectly symmetrical augmented chord that doesn't have issues with octave-displacements and inversions.

I'm going to gloss over some math quickly, just for those who want to see it. Don't worry, we have the easier way later, though this is already quite easy compared to what we did earlier for meantone.

First, we break down the diesis as a monzo: $128/125 \equiv \monzo{7 & 0 & -3}$.

We want to temper out the diesis so that it **vanishes**. Thus, our prime mappings need to be such that $\monzo{7&0&-3}$ maps to 0 cents.

By inspection, we want going up 7 octaves to equal going down 3 steps in the prime-5 direction.

We're definitely not going to change the cents of an octave, because we still want octaves (though [tempered octave tunings are well established](https://en.xen.wiki/w/TOP_tuning), they're out of scope for us now).

Thus, 7 octaves will be fixed at $7 \times 1200 = 8400¢$.

Which means we want to set:

$$
\begin{align*}
0¢ &\equiv \monzo{7 & 0 & -3} = \monzo{7} + \monzo{0&0&-3} \\
0¢ &\equiv \monzo{7} + 3 \cdot \monzo{0&0&-1} \\
\implies -8400¢ &\equiv -3 \cdot \monzo{0&0&1} \\
\implies \monzo{0&0&1} &\equiv 2800¢ \quad\blacksquare
\end{align*}
$$

Setting the prime interval 5 to 2800¢ will give us an **augmented** temperament, which makes three `5/4 = [-2 0 1>` mapped major thirds now equal to an octave.

Next, we still need to obtain the cents mapping for the prime interval 3. We also still want it to be meantone.

This means $81/80 \equiv \monzo{-4&4&-1}$ must map to nothing as well. Since we already know our cent sizes for primes 2 and 5, we can plug these in:

\begin{align*}
0¢ &\equiv \monzo{-4&4&-1} = -4 \cdot \monzo{1} + 4 \cdot \monzo{0&1} - \monzo{0&0&1}\\
\implies 4 \cdot 1200¢ + 2800¢ &\equiv 4 \cdot \monzo{0&1} \\
\implies \monzo{0&1} &\equiv 1900¢ \quad\blacksquare
\end{align*}

This means we have this mapping:

$$\text{mystery map} := \cov{1200¢ & 1900¢ & 2800¢}$$

If we make a major scale with this:

<iframe width="560" height="390" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%0A0c____%23_%5B0%3E%0A200c__%23_%5B-3_2%3E%0A400c__%23_%5B-2_0_1%3E_or_%5B-6_4%3E%0A500c__%23_%5B2_-1%3E%0A700c__%23_%5B-1_1%3E%0A900c__%23_%5B-4_3%3E_or_%5B0_-1_1%3E%0A1100c_%23_%5B-3_1_1%3E_or_%5B-7_5%3E%0A1200c_%23_%5B1%3E" title="Xenpaper" frameborder="0"></iframe>

**Congratulations, you have made 12edo**.

### edos the easy way: Constructing directly

What we just did was combine the meantone and augmented temperaments by simultaneously tempering 2 different commas starting from a rank-3 (requiring 3 dimensions of note names) JI subgroup to obtain a rank-1 tuning (1 dimension of note names) via the [rank-nullity theorem](#Rank-nullity-theorem-How-many-intervals-to-temper-out-before-it-becomes-an-edo). That rank-1 tuning contains an octave, which by definition, makes it an edo.

Xen math speak aside, we can also skip all that and start from the edo directly!

> :pencil: There are benefits for tunings to **not** be edos, because they give more range of expression, though executing it is hard in practice.

The first immediate simplification that edos offer is that we don't have to measure interval sizes in cent values anymore, nor deal with ratios (though we're still implicitly dealing with them through monzos). Instead, we can refer directly to the **size of edosteps**.

Notice that $1 \text{ edostep} \equiv 1200¢/\text{edo}$, which is the definition of edos.

So, we can convert cents to edosteps: $\text{edosteps} = \text{cents} \cdot \text{edo}/1200$

Then, we can convert prime intervals directly into edosteps, recalling the formula for cents of an interval:
\begin{align*}
\text{cents} &= 1200 \log_2(\text{interval}) \\
\text{edosteps} &= \frac{\text{edo}}{1200} \cdot 1200 \log_2(\text{interval}) \\
\text{edosteps} &= \text{edo} \cdot \log_2(\text{interval}) \quad\blacksquare
\end{align*}
Let's say we are in 12edo, we can obtain the prime interval mappings in terms of edosteps:
\begin{align*}
\monzo{1} &\equiv 12 \log_2(2) = 12 \\
\monzo{0&1} &\equiv 12\log_2(3) = 19.01955 \approx 19 \\
\monzo{0&0&1} &\equiv 12\log_2(5) = 27.863 \approx 28 \\
\monzo{0&0&0&1} &\equiv 12 \log_2(7) = 33.688 \approx 33 \\
&\quad \vdots
\end{align*}
Notice that we have to **round to the nearest edostep**, otherwise it wouldn't be an equal division of the octave anymore.

Using this, instead of a cents mapping, we can directly use edosteps as an interval size, to obtain this edo mapping:

$$\cov{12&19&28&33&\cdots}$$

> :nerd_face: :point_up: Also called a _**patent val**_. **Patent** as in "the default one", because we can also intentionally mistune edosteps to second-best (or nth-best) rounded approximations using [warts](https://en.xen.wiki/w/Val#Warts_and_generalized_patent_vals); and **val** as in "a linear sum of [p-adic valuations](https://en.wikipedia.org/wiki/P-adic_valuation)". Though, that's not necessary knowledge to know how to apply this.

Notice that if we don't specify if prime-limit for the JI intervals we are considering, the above mapping must map **infinite primes**, because we have infinite dimensions to "quantize" to edosteps.

Using this mapping, we can convert JI ratios (in monzo form) to directly to edosteps, instead of cents:

$$\left<\underbrace{\begin{matrix}\color{#fa0} 12 & \color{#af0} 19 & \color{#0ff} 28 & \color{#f3f} 33\end{matrix}}_{\large \text{12edo}}\right.\ \large\mid\ \left.\underbrace{\begin{matrix}\color{#fa0} 0 & \color{#af0} 2 & \color{#0ff} 0 & \color{#f3f} -1\end{matrix}}_{\large\text{9/7}}\right> = {\color{#af0}2(19)} {\color{#f3f}-1(33)} = 4 \setminus 12$$

Which tells us that the septimal supermajor third `9/7` gets mapped as 4 edosteps in 12edo.

:::info
:writing_hand: **Exercise**

Compute how many edosteps of 31edo the ratio `9/7` gets mapped to.

:::spoiler Answer
\begin{align*}
\monzo{1} &\equiv 31 \log_2(2) = 31 \\
\monzo{0&1} &\equiv 12\log_2(3) \approx 49 \\
\monzo{0&0&1} &\equiv 12\log_2(5) \approx 72 \\
\monzo{0&0&0&1} &\equiv 12 \log_2(7) \approx 87 \\
\therefore \text{31edo} &:= \cov{31&49&72&87&\dots} \\
\therefore \braket{31&49&72&87}{0&2&0&-1} &= 2(49) - 1(87) = 11\setminus 31 \quad\blacksquare
\end{align*}
:::

<!--:::-->
Recall the [naive attempt at converting between tunings back in Part I](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Why-not-just-approximate-frequencies).

**We have finally solved the issue of inconsistent intervals in an edo by:**

1. Finding the intended JI interval underlying the culture of music & the tuning
2. Finding the mapping of the edo we want
3. Using that mapping on the underlying JI interval from step 1

### Using null spaces to find which commas are tempered out

When constructing an edo directly using its edosteps mapping instead of tempering out commas, we don't already know what commas are being tempered out, which means we can't utilize the harmonic tools facilitated by the temperaments offered by the edo. Thus, when taking the [easy way](#edos-the-easy-way-Constructing-directly), we still need to manually work out what temperaments are available to us.

Recall what it means for an interval/comma to be tempered out: it vanishes, meaning it maps to 0 size, i.e. 0 edosteps.

Finding out commas of an edo becomes the question of finding out which monzos map to 0. For example, in 12edo we have the map $\cov{12&19&28}$. We want to solve all the possible $x,y,z$ such that:

$$\braket{12&19&28}{x&y&z} = 0 \setminus 12$$

Well, actually, there are **infinite** such values of $x,y,z$, but because of the rank-nullity theorem, we only need to find 2 linearly-independent monzos $\monzo{x&y&z}$, as tempering 2 commas is sufficient to bring a rank-3 `2.3.5` JI subgroup down to a rank-1 edo (as long as we don't temper the octave, otherwise there is no more _o_ to be _ed_-ed).

What that also implies is that all other commas of 12 edo can be represented in terms of those two commas that we found. The commas that are tempered out, forms its own vector space, and the dimension of that space is equal to how many linearly independent commas we tempered out. This space is called the **null space** in linear algebra.

Since `2.3.5` is only 3-dimensional, we don't need any fancy math tools because it's tractable by trial and error:

$$\braket{12&19&28}{-4&4&-1} = 0 \qquad \braket{12&19&28}{19&-12&0} = 0$$

However, if we want to be precise, we need to [solve for the null-space](https://en.xen.wiki/w/Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT/Exploring_temperaments#Nullspace) and deal with [saturation/torsion](https://en.xen.wiki/w/Saturation,_torsion,_and_contorsion), and from that we obtain the [comma basis](https://en.xen.wiki/w/Comma_basis).

> :pencil: The goal of this series is to show the links between the big ideas in xen theory and the entrainment of present-day musical culture, but the actual mathematical procedure is out of scope of this writing. It's encouraged to check out those links.

However, if you are working in a common edo, you can find the list of tempered intervals on the xenwiki, e.g. for 34edo:


> :pencil: If you code, check out https://github.com/xenharmonic-devs/temperaments, which uses [Clifford algebras](https://en.wikipedia.org/wiki/Clifford_algebra) via [ts-geometric-algebra](https://www.npmjs.com/package/ts-geometric-algebra) to work these out. Really cool stuff.

### What is a symmetry?

Here is a square:


And here is the same square rotated 90 degrees


That is a symmetry.

A symmetry, in its most general definition, is when you **do some action to an object, and it looks like nothing happened**. The language of symmetries is formalized in [Group Theory](https://www.maths.gla.ac.uk/~mwemyss/teaching/3alg1-7.pdf).

:::warning
:warning: [We continue talking about symmetries in Part IIIa](https://hackmd.io/@euwbah/extending-harmonic-principles-3a). We've hit the word count here.
:::

## Support me

Thank you for reading! I am open for queries, lessons and collaboration. Reach me [@euwbah](https://www.instagram.com/euwbah/) on instagram.

[](https://www.youtube.com/@euwbah)

[](https://github.com/sponsors/euwbah)

## 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
*[equave]: The more general notion of an octave - notes an equave apart are considered & named as the same note, but an equave may not be 2/1
*[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
*[meantone]: Temperament that splits the major third into two equal-sized tones by tempering out 81/80.
*[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
*[vanish]: an interval, usually a small comma, disappears and becomes the 1/1 unison interval because of a temperament

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

equave
: The more general notion of an octave - notes an equave apart are considered & named as the same note, but an equave may not necessarily be `2/1`

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

meantone
: Temperament that splits the major third into two equal-sized tones by tempering
out 81/80.

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

vanish
: an interval, usually a small comma, disappears and becomes the 1/1 unison interval because of a temperament