# Extending harmonic principles in 12edo to 31edo. Part I: Cultural entrainment

:::warning
:warning: Read this in [dark mode](https://hackmd.io/@docs/how-to-set-dark-mode-en#Set-View-page-theme).
:::

Next part: [Extending harmonic principles in 12edo to 31edo. Part II: JI & back to basics](https://hackmd.io/@euwbah/extending-harmonic-principles-2)

### Abstract

An opinionated guide/reflection journal written like a textbook of my ideal music curriculum. The goal of this journal is to distill the essence of harmony of the European classical tradition (ECT), and use that to extend into microtonality/xenharmony while maintaining the culture and familiarity of present-day harmony. This is written from the perspective of answering a deceptively simple question that I had 9 years ago: **How do I convert 12edo notes into 31edo?**

This document is a reflection of my journey of finding bigger questions to answer smaller ones.

I also want to bridge a gap in music theory (and xen/microtonal theory). I wanted to write something down that, hopefully, serves as a highway between topics — something to link the basics of musical history, to the basics of music theory, to the basics of xenharmonic math. Something beginner-level, yet not shying away from linking big ideas and pointing to further readings. A **general** place to start, yet heavily opinionated with a precise way to see how things connect.

Even though this guide concerns 31edo, I've only formally introduced the notation in [Part II](https://hackmd.io/@euwbah/extending-harmonic-principles-2) and tuning system all the way back in [Part III](https://hackmd.io/@euwbah/extending-harmonic-principles-3). The main principles are intended to be understood in a tuning-agnostic way.

However, here in Part I, I will still make plenty of references to 31edo and just intonation intervals (and 12edo). Part I motivates the writings of the subsequent parts, so it would be good to get a basic understanding first by either skipping to those sections when necessary, or by using some YouTube video or online resource. This guide is meant to be read in a non-linear fashion, almost like an FAQ, with the top-down narrative focused on motivating principles rather than giving necessary background info first.

However, if you wish to read in an order where prerequisite information is given first, I recommend reading in this order:

1. [Part II](https://hackmd.io/@euwbah/extending-harmonic-principles-2)
2. Part I (this part) until before the section on [31edo tonalities](#31edo-tonalities)
3. [Part III](https://hackmd.io/@euwbah/extending-harmonic-principles-3)
4. [Part IIIa](https://hackmd.io/@euwbah/extending-harmonic-principles-3a)
5. [31edo tonalities](#31edo-tonalities)

:::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]

## Why not just approximate frequencies?

For converting from 12edo to 31edo, the naive approach I first thought of was to just find the closest frequency approximation. We'll explore this idea for a short while, and find out why this didn't work out in the end.

The tetrad of `A+` (A augmented) consists of the notes:

<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:%5B0%5C12_4%5C12_8%5C12_12%5C12%5D---_%23_A_C%23_E%23_A" title="Xenpaper" frameborder="0"></iframe>

> :musical_note: Sound examples are provided with notation on xenpaper (the original domain is no longer hosted, there is a mirror at https://luphoria.com/xenpaper).
>
> The notation `x\y` (backslash, not forward slash) means `x` steps of `y` edo, counting from some fixed tuning note (A3 = 220Hz by default). In the case of 12edo, each step is a semitone.
> The square brackets [] groups notes to be played together as a chord.
>
> :warning: Occasionally, xenpaper would spontaneously crash if the article is left on for too long. Just refresh the page if this happens.

For edos (equal division of the octave), we convert these into frequencies using the formula:

$$ f_0 \times 2^{x/y} $$

where $f_0$ is the frequency of the tuning note (e.g., A4 = 440 Hz), $x$ is the number of edosteps from the tuning note, and $y$ is the edo. E.g., C#4 is 4 semitones (edosteps of 12edo) above A3, which is tuned at 220 Hz. Hence,

$$ C\sharp = 220 \times 2^{4/12} \approx 277.18 \text{ Hz} $$

Calculating the chord `A+` from 220 Hz, we get:

<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:%5B220hz_277.18hz_349.23hz_440hz%5D---" title="Xenpaper" frameborder="0"></iframe>

You can verify that it sounds exactly the same as the above example.

We turn these frequencies back into edosteps of any arbitrary edo with this formula, which gives the number of edosteps from the tuning note $f_0$. If $f$ is the frequency we want to convert,

$$ \text{EDO} \times \log_2(f / f_0) $$

is the exact number of edosteps. This usually results in a decimal number that we have to round off.

For example, The C#4 (`4\12`) of 12edo (277.18 Hz), relative to A3 (220 Hz), in 31edo would be

$$ 31 \times \log_2(277.18 / 220) \approx 10.33 \backslash 31 $$

> :pencil: Recall, the notation `x\y` means `x` steps of `y` edo.

Hence, if we round to the nearest edostep of 31edo, C#4 in 12edo would be 10 steps above A3, assuming we set A3 of both tunings to the same 220 Hz.

Using this formula, we find that the closest approximation of the chord `A+` is:

<iframe width="560" height="140" src="https://luphoria.com/xenpaper/#embed:%5B0%5C31_10%5C31_20%5C31_31%5C31%5D---" title="Xenpaper" frameborder="0"></iframe>

### What's the problem?

`A+` in 12edo has the edosteps 0, 4, 8, 12. Each note is `4\12` from the previous. However, in 31edo, the first two intervals are `10\31` apart, but the last one (between E#4 and A) is `11\31`.

This may not be a problem for you, but this was the inconsistency that started a long series of questions. Do you use scales to justify the choice of notes? If so, what even is a scale? Can a scale preserve its essence when changing tunings? Do you convert to just intonation (JI) as an intermediate step before converting to 31edo? If so, which major third do I use: `9:7`, `16:13`, `5:4`, `81:64`? Do you approximate each interval separately so that major thirds in 12edo are always realized as `10\31` in 31edo? ...

:::info
:writing_hand: Try to think of ways to resolve this problem, or argue why this discrepancy may not be a problem at all. What are the factors you're considering?
:::

:::success
:bulb: One popular way to generalize note choices across tunings is with **Regular Temperament Theory** (see [Dave Keenan & Douglas Blumeyer's guide to RTT](https://en.xen.wiki/w/Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT)).

RTT can be seen as an evolution of [MOS](https://en.xen.wiki/w/MOS_scale) (moments of symmetry) where the generalization of note choices is motivated by scales. Specifically, using the assumption that scales should only have two step sizes, large and small. However, when converting the major scale into 22edo (or most other tunings), we see this is not always what we want. While MODMOS makes exceptions, RTT generalizes even further.

We will only briefly cover a tiny bit of RTT in the section about monzos and vals, but since we're dealing with edos only (rank-1 temperaments, i.e., every note of the tuning system can be obtained by using one generator, the edostep), most of RTT is overkill.
:::

# 1. Cultural entrainment

Cultural entrainment, in the context of this writing, refers to the learned understanding of music stored in long term memory, which can be realized as habits, familiarity, and opinionated preference. Almost all of art is culturally entrained, but there is still science and empirical facts that determines the realm of what may even be perceivable. If something is not humanly perceivable _at all_ (not even by narrative, concept, or technological assistance) in the finished work, why should we analyze it?

My philosophy is that there is merit to studying the overlap, of how cultural entrainment and the sciences influence each other over the course of history. Although the evolution of music is not predictable, a scientific approach to music can still narrow down the realm of possibilities to a smaller infinity, making it possible for artists to imagine and construct bizarrely novel _xen_ music of an imagined world evolved from the current tradition, rather than as a standalone work completely isolated from reality.

The laws of nature gives a palette of infinities, and time picks a color for humans.

## "Voice Leading"

Usually, _"voice leading"_ is about treating the individual notes between chords as independent voices of different parts, but the intended result of _"voice leading"_ is that the distance between chord voicings is reduced so it sounds "smooth". I haven't seen this term defined rigorously and consistently amongst online music theory content, so I want to clarify my definition of "voice leading" in this section.

The idea of "voice leading" is rooted in history &mdash; the music of ECT stems from writing for voices (plainchant, organum, clausulae, motet, ...), and eventually a system of [cadences comprising clausula](https://www.earlymusicsources.com/youtube/cadences) is developed.

<iframe width="560" height="315" src="https://www.youtube.com/embed/jaCRUdxTRSM?si=UrbRyS98LN_tberE" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

### The modern clausulae of semitones

If we look at the major scale, there are two pairs of semitones, positioned asymmetrically between the 3rd and 4th degree (Mi-Fa), and the 7th and 1st degree (Ti-Do).

<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:0%5C12_2%5C12_%0A4%5C12_5%5C12_%23_semitone_(3-4)%0A7%5C12_9%5C12_%0A11%5C12_12%5C12_%23_semitone_(7-1)" title="Xenpaper" frameborder="0"></iframe>

From the perspective of deductive reasoning, the asymmetric positioning of the semitones means that two pairs of semitones (positioned correctly) can uniquely identify a major scale. Each semitone is either the Mi-Fa (3-4) or Ti-Do (7-1). If we are only given one semitone, we can narrow down the root key of the major scale to one of two options.

This is one of the motivating factors, alongside what I call the [semitone cancellation theory](#Pitch-memory-amp-semitone-cancellation-theory), to isolate two of these clausulae for the modern major tonality.

#### Perfect semitone movement (Ti-Do)

7th scale degree resolving to the 1st. Formally known as the _cantizans_ in _clausula vera_, but in modern times it makes more sense to think of it as belonging to the perfect cadence (V-I).

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(bpm%3A60)(1)%0A11%5C12_12%5C12_.%0A%5B11%5C12_7%5C12%5D_%5B12%5C12_0%5C12%5D" title="Xenpaper" frameborder="0"></iframe>

> :pencil: The use of solfege (Do Re Mi Fa So La Ti) in this article is relative to the current context's root/key center.

#### Plagal semitone movement (Fa-Mi)

4th scale degree resolving to the 3rd. Formally known as _tenorizans molle_ (soft modes, as opposed to _durum_/strong modes). In modern harmony it implies IV-I.

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(bpm%3A60)(1)%0A5%5C12_4%5C12_.%0A%5B5%5C12_%609%5C12_0%5C12%5D_%5B4%5C12_0%5C12_%607%5C12%5D" title="Xenpaper" frameborder="0"></iframe>

> :warning: The term plagal here has no correlation to the original 4 main types of cadences in the 16th century: Authentic, Plagal, Tenor, and Canto, each comprising their own unique set of clausula for each voice part. I'm using this terminology because in current times, plagal is understood as IV-I, but that was not the original use of this word. For the full picture of clausula, refer to videos by Early Music Sources: [Cadences in the 16th and 17th centuries](https://www.youtube.com/watch?v=jaCRUdxTRSM), [Durum and Molle](https://www.youtube.com/watch?v=Fdj-WPCgD8c), and [Modes in the 16th and 17th centuries](https://www.youtube.com/watch?v=lyq48eybjZw)

### How to use clausula?

If the musical context gives a key (root note) and a tonality system (e.g., Gregorian modes, Ragas, Maqamat, major, minor, blues etc...), then a clausula, in essence, is like a lever that you pull to switch train tracks to a different [side](#Tonality-as-a-state-diagram) of the tonality. In some tonalities and musical cultures (especially maqam and ECT), there are even clausula that bridges between different tonalities or keys (transposition/universal pitch offset of a tonality).

> :pencil: _Clausula_ means **little close/conclusion** in Latin.

E.g., if we assume 12edo, the key and tonality of C major, then this gives us the two default clausula that resolves in the direction of [Ti-Do](#Perfect-semitone-movement-Ti-Do) and [Fa-Mi](#Plagal-semitone-movement-Fa-Mi). As long as we don't introduce any other pairs of semitones (i.e., we stick to the 7 note major scale), the tonality won't change, and we will always have the same two clausula available.

#### Example 1: The ii-V-I

We start with a basic example. This progression is said to be the cornerstone of jazz standards, but if we remove all the extensions and strip it down to its core, it also demonstrates of the most primordial use of clausula in the major tonality and the three [primary classifications](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Diatonic-Pluralism) of [**sides**](#Tonality-as-a-state-diagram) of the major tonality: Tonic, Subdominant and Dominant. This may seem like just a beginner's exercise in voice leading for "jazz", but remember that we are trying to make the notion of "voice leading" rigorous.

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br220Hz%7D(1%2F2)%0A%5B0_4_12%5D_%5B2_5_12%5D_%5B2_5_11%5D_%5B0_4_12%5D.%0A%5B4_12%5D_%5B5_12%5D_%5B5_11%5D_%5B4_12%5D" title="Xenpaper" frameborder="0"></iframe>

In the above example, we assume the context begins in A major. We start with Mi `C#` (`4\12`) and Do `A` (`0\12` & `12\12`).

In the second chord `B D A`, we have the clausula $\texttt{C#} \to \texttt{D}$. The Mi `C#` goes to Fa `D`, which is the reversed plagal clausula of Fa-Mi. I call this the anti plagal clausula. This implies we are going from the Tonic out to some subdominant side of the tonality.

:::warning
:warning: &nbsp; **DISCLAIMER**
In the original theories of clausula in the Renaissance is not theoretically correct to understand a reversed clausula as the same clausula "going out" instead of resolving, but I have named them **anti clausulas**, just so that I may refer to them succinctly.
:::

In the third chord `B D G#`, we have the clausula $\texttt{A} \to \texttt{G#}$. The Do `A` (`12\12`) goes to Ti `G#` (`11\12`). We are now on some dominant side of the major tonality.

In the last chord `A C# A`, we have both clausulae $\texttt{G#} \to \texttt{A}$ and $\texttt{D} \to \texttt{C#}$. We return to the tonic side of the major tonality with both the perfect Ti-Do and plagal Fa-Mi clausula occurring simultaneously.

Other than cultural conventions, there is no other reason for why any of the particular configurations of clausula movements are attributed to any particular side of the tonality

#### Example 2: Prelude No. 1 (C major)

For a more complete example, let's analyze Prelude No. 1 in C major of Bach's Well Tempered Clavier 1.

<iframe width="560" height="350" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br440hz%7D%0A%23_press_the_%3E_symbol_on_the_left_to_start_playing_from_that_line.%0A%0A%23_CEGCE%0A%603_%607_%6010_3_7_%6010_3_7_%7C_%603_%607_%6010_3_7_%6010_3_7%0A%23_CDADF_Mi-Fa_anti_plagal%0A%603_%605_0_5_8_0_5_8_%7C_%603_%605_0_5_8_0_5_8%0A%23_BDGDF_Do-Ti_anti_perfect%0A%602_%605_%6010_5_8_%6010_5_8_%7C_%602_%605_%6010_5_8_%6010_5_8%0A%23_CEGCE_Ti-Do_perfect_%26_Fa-Mi_plagal%0A%603_%607_%6010_3_7_%6010_3_7_%7C_%603_%607_%6010_3_7_%6010_3_7%0A%0A%23_CEACE_(anti_tenorizans_G-A%2Fplagal_cadence)%0A%603_%607_0_7_12_0_7_12_%7C_%603_%607_0_7_12_0_7_12%0A%23_CDF%23AD_G-F%23_Do-Ti_anti_perfect_(G_in_memory)%0A%603_%605_%609_0_5_%609_0_5_%7C_%603_%605_%609_0_5_%609_0_5%0A%23_BDGDG_C-B_(Fa-Mi)_F%23-G_(Ti-Do)%0A%602_%605_%6010_5_10_%6010_5_10_%7C_%602_%605_%6010_5_10_%6010_5_10%0A%23_BCEGC_B-C_%22simultaneously%22%0A%602_%603_%607_%6010_3_%607_%6010_3_%7C_%602_%603_%607_%6010_3_%607_%6010_3%0A%23_ACEGC_B-A_tenorizans_plagal_4th_species%0A%600_%603_%607_%6010_3_%607_%6010_3_%7C_%600_%603_%607_%6010_3_%607_%6010_3" title="Xenpaper" frameborder="0"></iframe>

> :musical_note: `{12edo}{r220hz}` means numbers are interpreted as steps of 12edo from A3 = 220 Hz. The backtick &#96; shifts down by an octave.

In terms of functional harmony, there is not much to analyze about the first chord `C E G C E` except that it is C major, and that cultural entrainment will increase the probability that the major tonality over the root of C will be perceived. All analysis of function is **horizontal** (i.e., over time, over notes changing), the only purely vertical harmonic analysis is concordance.

> :pencil: **Concordance** is a word used to refer to the physiological perception of consonance, that different cultures may not have preference, or even inverse preferences for, but can still distinguish between stimuli. E.g., Gamelan (de)tunes instruments in pairs that are intentionally slightly out of tune. In ECT, that is bad and has negative connotation, and is thus considered dissonant. However, this is neither dissonant nor consonant in the original culture. It just is (for various supposed reasons, e.g., mimicking nature sounds). What is important is that the spectrum of concordance-discordance is **universally perceivable**, and that universality is what is objective.

It's not until the second chord `C D A D F` where we are 99% certain the tonality and key. However, we need to tap into vertical analysis (analyzing a single instance in time) as a confluence that can justify this. The second chord is more discordant, which implies that, instead of resolving Ti-Do, the upward semitone from E to F is a Mi-Fa that is going **out away from home** (which I call _anti plagal_). In the cultural entrainment of ECT, we usually expect resolved sounds to be more concordant (though it is [not always the case](#Tonality-is-fundamentally-cultural)). Hence, based on E-F being understood as Mi-Fa and the initial chord being major, we are now on the plagal [side](#Tonality-as-a-state-diagram) of C major tonality. However, in other contexts, it is entirely possible to perceive the exact same 2 bars of music as another key:

<iframe width="560" height="260" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br440hz%7D%0A%603_0_3_8_15_3_8_15_%7C_%603_0_3_8_15_3_8_15%0A%603_%608_1_5_12_1_5_12_%7C_%601_%608_1_5_10_1_5_10%0A%603_%607_%6010_3_7_%6010_3_7_%7C_%603_%607_%6010_3_7_%6010_3_7%0A%603_%605_0_5_8_0_5_8_%7C_%603_%605_0_5_8_0_5_8" title="Xenpaper" frameborder="0"></iframe>

> :pencil: There are different ways to measure discordance based on different psychoacoustic phenomena, such as
> * [_Gradus suavitatis_](https://mathoverflow.net/questions/136572/number-theory-underlying-eulers-theory-of-music)
> * Critical band roughness ([Sethares' Algorithm](https://sethares.engr.wisc.edu/comprog.html) on Plomp & Levelt)
> * [Harmonic Entropy](https://en.xen.wiki/w/Harmonic_entropy)
> * My unnamed [graph-based dissonance measure](https://github.com/euwbah/dissonance-wasm) that is influenced by all of the above plus (https://www.musikinformatik.uni-mainz.de/schriftenreihe/nr45/scale.pdf)

On the third chord `B D G D F`, we have the anti perfect clausula that goes out from Do to Ti. At this point, we have the two pairs of semitones, E-F and B-C, so we can be certain of the key and tonality from a purely deductive perspective.

On the fourth chord `C E G C E`, we resolve both the plagal (F-E) and the perfect (B-C) clausulae. In "modern" terms of the Viennese and neo-Riemannian system, this can be understood as the perfect cadence.

On the fifth chord `C E A C E`, there is no semitone movement. However, if we want, we can borrow from the tenorizans durum (Re-Do) and say that G-A is an anti-tenorizans which implies the "key" of G, and that we are going out from G (Do) to A (Re). However, the concept of key (the Do) in clausula is not the same as the key (universal pitch offset) of the tonality. We will tackle this in the next section about tonalities. Later on, we also learn about how the chords `Am` and `Cmaj` have the [same function and are plurally equivalent](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Diatonic-Pluralism).

### Pitch memory & semitone cancellation theory

On the sixth chord `C D F# A D`, there appears to be no semitone movement on the surface relative to the previous chord `C E A C E`. However, there's a harmonic principle that I have been withholding that we will need to make sense of this bar of music &mdash; there has been no chords at all! Yes, in the original score Bach writes for the lowest two voices to be held down, but in the xenpaper demo audio, none of the notes were held, and you've been listening to only one note at a time.

Music is believed to have started out as monophonic (one note at a time), but european culture's monophonic or iso drone music (systema teleion, byzantine music, plainchant) evolved to show a preference concordant intervals (octaves, fifths, fourths, thirds) between notes in the melody. This is unexpected: if you only have one note at a time, there is no possible physical sensation of discordance between two notes. There's many hypotheses on why this is so, the most popular being natural reverb in natural and constructed spaces (caves, cathedrals).

However, I also believe the short term memory of recently experienced pitch stimuli has a role to play (also, there has been no reverb or sustained notes on the xenpaper demos so far). Diana Deutsch's summary of her work on [short term memory for tones](https://deutsch.ucsd.edu/psychology/pages.php?i=209) captures the essence of this in empirical studies (though, replicating these studies at larger samples sizes and wider demographics would be much needed).

Here is a summary of the key findings that changed my approach to music:

* The main experiment: listen to a sequence of notes. Is the starting and ending note the same?
* If answered correctly, that implies the starting pitch was kept in memory. Otherwise, it was 'forgotten'.
* Higher probability of forgetting the note if, in between the first and last notes, there was a note a small interval away from the starting note (see [figure 4](https://deutsch.ucsd.edu/m/DD_F.jpg) of the above article). This was called a _critical intervening tone_ by Deutsch. Intervening tones that were 2/3rds tone (133.3 cents) away from the starting note had the highest error rate, followed by the semitone. It would be nice if this experiment can be replicated with higher granularity of pitch.
* Higher probability of forgetting the note if the sequence of notes was angular (full of leaps, jumpy wide intervals) and atonal.

There's many principles to be derived from these results alone, but the one that applies to bar 6 of Prelude No. 1 is what I call **semitone cancellation theory**.

:::warning
**Semitone cancellation theory**

Every note in 12edo will stay in the listener's memory and interact with future stimuli by acting as a note that viscerally exists (even if not physically sounded). It stays in memory until an intervening note that is a semitone (or approx 60¢-160¢) away from the note in memory is sounded, then the previous note will be "forgotten".
:::

> :warning: Of course, this is a oversimplification, as it is possible for two notes a semitone apart to be in memory at the same time (e.g., bar 8 of Prelude No. 1), or for notes to be forgotten because a new section of the song is expected in the moment. Discretion is required, considering current musical cultural context. Also, this study was only done on participants exposed to music of the european classical tradition (ECT), so it cannot be seen as a universal fact.

With this theory in mind, we can understand that the A of the fifth chord `C E A E A` probably did not erase G from the prior chord `C E G C E`, so we have the semitone clausula G-F#. This can be interpreted two ways, as a plagal Fa-Mi resolving to D major, or as an anti perfect Do-Ti going away from G major. Context (C is still in the short term memory and currently being sounded) and the shorter harmonic distance (G is one fifth away from C, D is two) suggests that anti perfect (Do-Ti) will be perceived with higher probability, so bar six is exploring the dominant side of G major.

We continue the analysis. Bar seven `B D G D G` has both the perfect (F#-G) and plagal (C-B) clausula that resolves to the tonic side of G major.

Bar 8 `B C E G C` is interesting. We have the semitone movement B-C repeated (and held clashing in the original score). Is this B-C an anti plagal Mi-Fa in the context of G major, or a perfect Ti-Do resolution back to C? If we analyze retrospectively from the future (which happens all the time in music), bar 9's resolution of the B-C clash hints at bar 8 being a soft (indirect) resolution to C major. In baroque terms, we can understand B as a prepared dissonance from the previous bar, or a suspension.

:::warning
As an edge case to the semitone cancellation theory and our modern clausula, if we have both semitones present at the same time, we assume the semitone to be an octave displaced major 7th, which is the most concordant interpretation, and thus the root to be the higher of the two semitones. If we have both Do and Ti, the Do takes precedence and we are on the side of Do (tonic, _finalis_). If we have both Mi and Fa, the Fa takes precedence and we are on the subdominant/plagal side of the tonality. If we have all 4 (rare in functional music), then one could interpret it as a `maj7#11` or a `7(13)add4`, at that point these are just names, the sound should take precedence.
:::

In Bar 9 `A C E G C`, we can borrow from the tenorizans durum to note the B-A Re-Do resolution. Even though I chose to focus only on semitone movements, whole tone, and fourth/fifth movements also still have an important role to play. We'll get to that soon.

#### What semitone cancellation theory & pitch memory implies for 31edo

Recall [figure 4](https://deutsch.ucsd.edu/m/DD_F.jpg) of the article [Short Term Memory for Tones](https://deutsch.ucsd.edu/psychology/pages.php?i=209) by Deutsch. This lists the error rates of incorrectly identifying whether the first and last notes of a sequence of notes are the same based on how close of an interval a in-between note got to either the first or last note.

The size of a diesis in 31edo, which is the smallest interval of 1 edostep, is 38.7 cents, which is very close to 1/6th of a tone (33.3 cents). The error rate being low suggests that there is a high probability that even if a new intervening note was played a diesis away from a note in memory, the old note in memory would still have persisted. That intervening diesis would create strong cognitive dissonance against the pitch memory of the old note, and that old note would persist much longer in memory than if that intervening tone were to be wider (66-166 cents) instead.

Hence, **dieses are not effective as functional clausula or voice movements**. Though it is possible to write a song that uses brute repetition to entrain a new musical culture or a new tonality as the song progresses (next section's topic), it will be much harder to do so if the clausulae of the tonality included dieses. No matter how hard I tried, I haven't had great success in entraining the diesis to have as much functional importance as the other intervals.

There is much microtonal music that use exceptionally small intervals (diesis or even smaller) as clausula-like intervals. It could be my own biases of ear training, but I couldn't get over the cognitive dissonance making it sound out of tune, even in my own writing.

Here is an example from Vincentino (1511-1576) himself, the earliest proponents of a tuning very similar to 31edo.

<iframe width="560" height="315" src="https://www.youtube.com/embed/_6zNO5Fieog?si=_-4oMHrXvay49Mk8" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

The piece is divided into three sections, each using more _xenharmonic_ intervals that the previous. The last section I can never understand, and it only could make sense to me in the absence of time and state.

It would be nice if this wasn't the case &mdash; 31edo has a cool feature where by moving all three voices by a diesis, you can change a major triad to a minor triad:

<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br220hz%7D%0A%5B0_10_18%5D-----_%5B1_9_19%5D-----" title="Xenpaper" frameborder="0"></iframe>

But on its own, I can't see how this can be used for any functional harmony that builds upon the existing tradition. The most I could do was to use these as internal structures of larger chords rooted in just intonation/harmonic series, e.g.,

<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br330hz%7D%0A%5B0_10_18%5D-----_%5B1_9_19%5D-----%0A%0A%5B%600_0_10_18%5D-----_%5B%60%6025_%6020_1_9_19%5D-----%0A%5B%60%6025_%6018_5_14_18%5D----_%5B%60%6023_%6013_%6018_0_10%5D-----" title="Xenpaper" frameborder="0"></iframe>

But the other notes are doing the heavy lifting of giving functional clausulae, and the diesis shifts go unnoticed.

Entraining the diesis as a clausula by consistent repetition is something I have tried in the past, but I wouldn't consider it a success:

<iframe width="100%" height="300" scrolling="no" frameborder="no" allow="autoplay" src="https://w.soundcloud.com/player/?url=https%3A//api.soundcloud.com/tracks/626046813&color=%23ff5500&auto_play=false&hide_related=false&show_comments=true&show_user=true&show_reposts=false&show_teaser=true&visual=true"></iframe><div style="font-size: 10px; color: #cccccc;line-break: anywhere;word-break: normal;overflow: hidden;white-space: nowrap;text-overflow: ellipsis; font-family: Interstate,Lucida Grande,Lucida Sans Unicode,Lucida Sans,Garuda,Verdana,Tahoma,sans-serif;font-weight: 100;"><a href="https://soundcloud.com/euwbah" title="euwbah" target="_blank" style="color: #cccccc; text-decoration: none;">euwbah</a> · <a href="https://soundcloud.com/euwbah/mood-parallax" title="Mood - Parallax" target="_blank" style="color: #cccccc; text-decoration: none;">Mood - Parallax</a></div>

This music was just an experiment I made long ago when I was still figuring out music, and not reflective of my current ideals.

:::info
:writing_hand: Can you find a way to make the diesis feel like a clausula?
:::

## Tonality

I have been using the phrases **tonality** and **side of a tonality** a lot in the previous section. This section addresses what these words mean in this article.

### Tonality is fundamentally cultural

Tonality is rooted purely in cultural entrainment, and there is no math or science that can be an etiological argument for why tonalities evolved the way it did (besides very general statements about the limits of human perception). We can only find correlations but not the causes, and oftentimes the correlations break.

For example, one popular correlation hypothesizes that the major tonality evolved because the major sonorities and chords are concordant &mdash; both of these facts are independently true, but not related. For example, the minor tonality is also commonplace in modern harmony. If we were to rank concordance of minor triads by increasing complexity/discordance of ratios:

<iframe width="560" height="260" src="https://luphoria.com/xenpaper/#embed:%7Br220Hz%7D(1%2F2)%0A4%3A5%3A6-._%23_5-limit_major_(5_odd_limit)%0A6%3A7%3A9-._%23_7-limit_subminor_(9_odd_limit)%0A10%3A12%3A15-_%23_5-limit_minor_(15_odd_limit)%0A16%3A19%3A24_%23_19-limit_minor_(19_odd_limit)" title="Xenpaper" frameborder="0"></iframe>

> :musical_note: The notation `x:y:z` means to play a chord such that the ratios of the frequencies of the notes is `x` is to `y` is to `z`. The first note of `x` units is set to the tuning frequency of 220 Hz, and the other notes will be relative to the first note. More info in the section about just intonation.

My ears find that the third minor chord `10:12:15` sounds _out_ compared to the second `6:7:9` one. Yet, in all of European classical music's conception of pitches from Euler's Tonnetz onwards, I could not find historical composers and theorists intentionally utilizing the 7-limit minor triad as a minor tonality, even though by the [odd limit metric](https://en.xen.wiki/w/Odd_limit) (i.e., the highest harmonic used in the harmonic series ruling out octave equivalences), and by sound, it is more concordant. Even through Ars Nova and the Renaissance periods when people like Zarlino and Vincentino experimented with alternative tunings and building instruments capable of approximating these ratios, they never expounded on the possibility of these triads being the building blocks of tonalities.

Because of this, some believe that [prime limit](https://en.xen.wiki/w/Harmonic_limit) (the highest prime factor used in an interval ratio) is a better measure of the concordance-discordance spectrum than the odd limit. I think that lower prime-limit correlates to ubiquity because using less primes reduces the conceptual complexity of tunings by restricting it to a [lower-dimensional space (more in Part II)](https://hackmd.io/@euwbah/extending-harmonic-principles-2?stext=28410%3A289%3A0%3A1736994971%3AAXR92R).

However, amongst all the minor triads, the last 19-limit (and 19 odd limit) one should be the most familiar to everyone exposed to modern 12edo music, because it is the closest in tuning to 12edo. Considering this, it's truly hard to make any argument for the necessity of concordance in tonalities.

### But culture can always make its own rules around physical phenomena

Even though we cannot prove causation from objective measures to the cultural construction of tonality, we can find many theorists and composers understanding the notion of a perceivable spectrum of concordance and forming culture around that.

A famous example is _mi contra fa_, i.e, the tritone being understood as an objectively discordant interval and therefore more difficult to sing (even though many modern musicians can more or less sing a tritone accurately in these days because of the cultural entrainment of the Simpsons or whatever).

> :pencil: Fun etymology fact. The famous tritone = devil rule goes _Mi contra fa diabolus est in musica_ (mi against fa is the devil in music) but the _mi_ and _fa_ are not of the same solfege that we use today, otherwise that would be a semitone, not a tritone. Instead of modern solfege, the solmization as per Guido of Arezzo (11th century) used as a teaching tool for music involved only 6 notes: Ut Re Mi Fa Sol La, between each note: Tone, Tone, Semitone, Tone, Tone. However, these 6 notes are started at different points. Relative to the C major scale, the _durum_ (hard) hexachord starts on G, the natural hexachord starts on C, and the _molle_ (soft) hexachord starts on F. The _mi_ here refers to the _mi_ of the durum or natural hexachord, and the _fa_ refers to the _fa_ of the natural or molle hexachord respectively, which is transposed up a fourth relatively, hence the tritone being a fourth plus one semitone.
>
> In this system, there were two possible ways to obtain B, the mi of the durum hexachord would be the modern B natural, and the fa of the molle hexachord would be the modern B flat. The B molle (soft B) would be denoted with a rounded "b" which became &flat;, and the B durum (hard B) would be written with a squarey 'b', which became &natural;.

Some measurable correlations are:

* More concordant = higher probability of being perceived as tonic. Some exceptions:
* 7#9 "Hendrix chord" as tonic of blues
* minor tonality exists, but not subminor tonality
* [Tonicity measure](https://github.com/euwbah/dissonance-wasm/blob/master/src/polyadic.rs). Something I came up with to model the continuous probability distribution of perceived [otonality](https://en.wikipedia.org/wiki/Otonality_and_utonality) (as opposed to actual otonality) of a note. Using some dyadic discordance measure function, the discordance of an interval is measured against the discordance of the octave inverted interval. E.g., a major third `3\12` becomes a minor sixth `9\12`. A fifth `7\12` becomes a fourth `5\12`. A minor tenth `15\12` becomes a major thirteenth `21\12`. This is done for all frequencies and interpolated in between data points. Then in the case of more than two notes, a graph-theoretic model is used to iterate through all possible ways the notes may be perceived, accounting for the context of the short term memory. This measure models the preference for otonal structures. A fourth `4/3 5\12` is rather perceived as an octave displaced fifth. A minor sixth `8/5 8\12` will rather be perceived as an octave displaced major third `5/4 4\12`. Exceptions:
* Minor tonality/triad is not otonal (unless considering the 19-limit minor `16:19:24` that is closer to that of 12edo)
* [Primodal tonalities](#Subminor-amp-orwellprimodal-6-tonalities)

### Tonality as a state diagram

This section is an exposition of my concept of tonality that models how theories of tonality wants to be rigorous, yet open to subjectivity. Music theory usually categorizes tonality as major, minor, and sometimes blues, where each tonality is a world, each set up with different expectations and clausulae. The first misconception I would like to clear up is that tonality is a scale or mode in the sense of being a set/collection of notes. I claim the opposite, that scales and modes are byproducts of tonalities.

My understanding of tonality builds off clausulae, just like the understanding of tonality in the etiology of music of the ECT, but not in the restricted original sense in which there are 8 or 12 _durum_ and _molle_ Gregorian modes and 4 cadences and 1, 2 or 4-step clausula, but in the free yet rigid sense that tonality can be expressed as a [state diagram (computer science)](https://en.wikipedia.org/wiki/State_diagram), where we use clausulae as bridges between different sides of the tonality (or even between tonalities).

#### Door state diagram


Consider a universe where there is a door with only two states, opened or closed. The above state diagram illustrates that if the door is open, the only thing you can _expect_ to do is close it. Closing the door changes the state to a closed door, and now you can only _expect_ to open it. A state is not only about where you are currently at, but it is also about what you can do once you get there, and what you could have done to get there.

### Constructing the major tonality

We could begin by saying that the major tonality has three sides/states, based on the three functions in neo-riemannian analysis, the Tonic, Dominant and Subdominant. The clausulae now become actions that you take to go between states:


The above tonality is very primitive. Starting from the Tonic side, after a Mi-Fa plagal clausula which takes you to the subdominant side of the major tonality, the tonality given by the diagram says that we cannot expect to have the anti perfect clausula (Do-Ti) while on the subdominant state. From cultural entrainment, we know that we can still have Do-Ti while on the subdominant, which still takes us to some dominant-functioning side of the tonality (a dominant 7th chord). Also, Ti-Do can take us back to subdominant from dominant if we still have Fa in the chord. Here is an attempt at representing that:


Here we run into a problem. If we are on the dominant side, and we apply the action of a Ti-Do perfect clausula, how do we know if that takes us to the subdominant or tonic side?

This is a **wrong** state/tonality diagram. The same action of the perfect clausula (Ti-Do) takes us to different places depending on the state of whether Fa is present in the chord in the dominant side, so the diagram fails to capture the state of whether Fa is in the dominant chord. We attempt to fix this by adding a state that captures whether Fa is present. We call this state dom7 after the chord that it forms:


This still isn't right. By the above diagram, this should be a progression from Tonic to Dominant:

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br3%7D(1%2F3)%0A%5B0_4_7%5D_%23_CEG%0A%5B%6011_4_7%5D_%23_BEG" title="Xenpaper" frameborder="0"></iframe>

But in the Viennese system (which I think is too antiquated for modern harmony), this is a progression to the mediant, and in the neo-riemannian (German) system, the `iiim` (minor triad built on the third degree, `Em` with respect to C), the mediant is also tonic.

We may add more rules, saying that a dominant chord must not have Mi, or if it does, then Fa must be present, so that the tritone between Ti and Fa is present to evoke the characteristic sound of dominant chords to form a `7(13)` chord. But what about `G6` (`V6`) in the key of C? It has Mi and no Fa so does that make it not a dominant chord?

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br3%7D(1%2F3)%0A%5B0_4_7%5D_%23_CEG%0A%5B%607_%6011_2_4%5D_%23_GBDE" title="Xenpaper" frameborder="0"></iframe>

A different voicing?

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br3%7D(1%2F3)%0A%5B0_4_7%5D_%23_CEG%0A%5B%607_2_11_'4%5D_%23_GDBE" title="Xenpaper" frameborder="0"></iframe>

What about if we add a b9, `G6b9`?

<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br3%7D(1%2F3)%0A%5B0_4_7%5D_%23_CEG%0A%5B%607_2_8_11_'4%5D_%23_GDAbBE" title="Xenpaper" frameborder="0"></iframe>

By now, it should be clear how hard it is to draw boundaries between the sides of a tonality, or to even know how many sides of a tonality there are. In neo-riemannian functional analysis, the aggregation of all the different sides of a tonality into 3 main functions (tonic, subdominant, dominant) allows for "mental gymnastics" (pluralism & symmetries) that we will delve into later, but within one of these functions, there are numerous sides. The above examples show that even octave equivalence may not be assumed.

As a rough guideline, for considering the 7 diatonic notes of major tonalities alone, and considering only configurations of the two pairs of semitone clausulae (Fa-Mi, Ti-Do), I find that there are at least 16 sides in the tonality!

| Do | Ti | Mi | Fa | Side |
| -- | -- | -- | -- | -- |
| :x: | :x: | :x: | :x: | Subdominant suspension |
| :heavy_check_mark: | :x: | :x: | :x: | Tonic suspension |
| :x: | :heavy_check_mark: | :x: | :x: | Dominant |
| :heavy_check_mark: | :heavy_check_mark: | :x: | :x: | Unresolved tonic suspension (Ti resolves to La or Do) |
| :x: | :x: | :heavy_check_mark: | :x: | Plural tonic suspension |
| :heavy_check_mark: | :x: | :heavy_check_mark: | :x: | Tonic |
| :x: | :heavy_check_mark: | :heavy_check_mark: | :x: | Plural tonic |
| :heavy_check_mark: | :heavy_check_mark: | :heavy_check_mark: | :x: | Unresolved tonic |
| :x: | :x: | :x: | :heavy_check_mark: | Subdominant suspension |
| :heavy_check_mark: | :x: | :x: | :heavy_check_mark: | Subdominant |
| :x: | :heavy_check_mark: | :x: | :heavy_check_mark: | Full dominant |
| :heavy_check_mark: | :heavy_check_mark: | :x: | :heavy_check_mark: | Dominant lydian V over IV |
| :x: | :x: | :heavy_check_mark: | :heavy_check_mark: | Plural subdominant suspension |
| :heavy_check_mark: | :x: | :heavy_check_mark: | :heavy_check_mark: | Plural subdominant |
| :x: | :heavy_check_mark: | :heavy_check_mark: | :heavy_check_mark: | Plural dominant |
| :heavy_check_mark: | :heavy_check_mark: | :heavy_check_mark: | :heavy_check_mark: | Subdominant lydian |

And some of these sides have more sub-sides within them based on chromatic alterations. E.g., for the all the dominant-type sides:

* If we have b9, b5, #9, or #5, then they are subcategorized as the altered-dominant-type sides of the tonality, usually built over the III and V of the current key (fun fact: these chromatic notes are borrowing the fifth-equals of the root note a tritone away &mdash; i.e., 1 = b5, 5 = b9, 2 = #5, 6 = #9, more about fifth-equals in the section about extensions & JI)
* Otherwise, if we have none of the above, but with #11 permissible, these become the lydian-dominant-type sides of the tonality, usually built over the II, IV, bVII of the current key (for major tonality)

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br%609%7D(1%2F3)%0A%5B%607_5_11_'3_'10%5D_%23_V_altered_dom%0A%5B0_11_'2_'4_'7%5D%0A.%0A%5B%605_3_7_11_'2%5D_%23_IV_lydian_dom%0A%5B0_4_7_11%5D" title="Xenpaper" frameborder="0"></iframe>

To add, a side does not exist in isolation. The above tabulation may make it seem that a side is merely determined by the notes it comprises, but in fact, the exact same set of notes can represent different sides, as we will see shortly. Sides are states, and going back to the example of [the door](#Door-state-diagram), states also embody the crucial information of where it could have came from, and where it could have gone. It is the agglutination of **context, notes and possibilities** that make up a side of a tonality.

My current model of the major tonality, in isolation, has 17 to 32 sides, depending on the period of the music I am going after, just imagine the above state diagram but with 32 sides and clausula connecting them. Some clausula may be longer than 2 notes, such as specific enclosures, patterns and cliches. I will save this for a future article.

In the state diagram, we also haven't included the culturally expected clausula that changes between tonalities, which means a single tonalities' state diagram does not exist in a vacuum, but they all combine to form one gigantic diagram which signifies the full entrainment of a musical culture.

A tonality does not exist in a vacuum, this is true for both ECT and maqam music, where there is an expectation that if a specific movement is played, there is a culturally entrained consistent shift of tonality. E.g., the combined clausula 1-7 and 5-#5, resulting in a state where there is no 5, can signify movement to the relative minor tonality instead:

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br%607%7D(1%2F3)%0A%5B0_7_'4%5D%0A%5B%6011_8_'4%5D%0A%5B%609_9_'4%5D%0A%5B9_7_'4%5D%0A%5B5_9_'4%5D" title="Xenpaper" frameborder="0"></iframe>

And here is a subgraph of the tonality state diagram that contains the above movements. The notation `&` means both clausula movement are required, `|` means either or both, and `+b7` means the "clausula" is not a movement between notes, but instead an addition of the b7 of the relative minor's root. Also, notice that the tonics of the major and minor chord are connected by a tonality realm I marked as "Relative". This tonality is where neo-soul minor 11th chords and "Dorian" sounds live, and has its own world with its own sides as well.


> :pencil: It is a common misconception to reduce the modes down to a scale with various starting notes. The origin of modes evolved all the way back from the _systema teleion_, influencing the species of fifths and fourths, and the treatment of choices of different notes for the _finalis_ in Renaissance music. A mode is its own tonality, with its own resolutions and unique clausula that distinguishes it.

Notice on the third chord in the above example, the music sounds like it has moved to the relative minor, but notice how there was no minor third in the chord! I could have added the major third instead, via a Fa-Mi in the key of VI:

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:%7B12edo%7D%7Br%607%7D(1%2F3)%0A%5B0_7_'4%5D%0A%5B%6011_8_'2_'4%5D%0A%5B%609_9_'1_'4%5D%0A%5B7_10_'1_'4%5D%0A%5B5_9_'2_'5%5D" title="Xenpaper" frameborder="0"></iframe>


Here I distinguish the higher plural subdominant side (e.g., `IVmaj7`) from the lower plural subdominant side (e.g., `iimin7`). The tonic of `2 MINOR` (minor tonality built from the 2nd scale degree relative to the major tonality) doubles as the the lower plural subdominant of the original major degree. Don't worry about these if you're unfamiliar, [I will explain plurals in Part III](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Diatonic-Pluralism).

It's geometrically impossible to draw the full state diagram of a musical culture and all its tonalities, because you would need to intersect tonalities at very intricate points, and a 2D canvas inhibits that.

Though, I believe there is still huge pedagogical merit in codifying one's cultural entrainment by drawing subsets of the abstract high-dimensional tonality diagram from short progressions, just like the above example. When one learns a particular harmonic rule/clausula from a musical culture, it would be good to see where that rule fits in the bigger picture by representing with a visual schematic of a state diagram. It still leaves room for subjectivity &mdash; tonalities are modular, and the interactions or clausulae between them can be changed according to your own preferences.

Of course, this is not how I learned music. This is merely a method of visualization of the abstract patterns I already recognize from the music I listen to, which I have learnt by osmosis.

### Some other 12edo tonalities

In this section, I briefly go through the other clausulae available in tonalities that I use most often, but there's many others I have left out.

#### Minor tonality

Generally, the tonality evoked by the words "harmonic & melodic minor" is what I would consider the minor tonality. Not to be confused with aeolian/natural minor or dorian etc... This tonality comprises three main semitone clausula:

* 7-1: The perfect clausula, similiar to the perfect clausula for the major tonality.
* 2-b3: The relative major perfect clausula, this is based on the Ti-Do clausula of the relative major.
* b6-5: The relative major plagal clausula, based on Fa-Mi of the relative major.

The minor tonality is characterized as being almost the same as the relative major tonality, with the addition of 7-1 in the minor key (or #5-6 in the relative major key).

An example to listen to would be the C minor Prelude No. 2 of Bach's Well-Tempered Clavier Book 1. This piece starts in the C minor tonality (baroque tuning, so it's a B in modern pitch), but also explores other regions (and G harmonic major) in the development section.

<iframe width="560" height="315" src="https://www.youtube.com/embed/PsEw2irUEug?si=NB1Bxp2UfHIlV41D&amp;start=765" 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>

#### Dorian tonality

This is my take on what I think "dorian" means in the modern interpretation. Intuitively speaking, having the _finalis_ on the second degree of the major scale means that we are treating the subdominant side of the major tonality as the tonic side of dorian. Referring back to the [ii-V-I example](#Example-1-The-ii-V-I), dorian is effectively the same as treating Do and Fa as resolution points for clausulae in the major tonality (instead of Do and Mi), and setting Re to be the new Do.

* 2-b3: Relative major perfect clausula based on Ti-Do of the relative major. This sets Do of the original major tonality as resolved.
* 6-b7: This sets Fa of the original major tonality as resolved. Also, a [fifth-coloring](https://hackmd.io/@euwbah/extending-harmonic-principles-2#iv-Extension-fifth-coloring) of the 2-b3 movement.
* b7-1: :question: Some Viennese & Schenkerian analysts call this the lowered leading tone movement. I don't think it is as strong as the above two movements, but there are situations where this movement is functional.

Unironically, this was the first example off the top of my head, and Simon Fransman absolutely killed it on the recorder here. This comprises mostly dorian and blues tonalities:

<iframe width="560" height="315" src="https://www.youtube.com/embed/6EFTCNWRCLc?si=iMpWbYehvwgnhkWM" 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>

#### Harmonic major tonality

This can almost be seen as the "other side" (Dominant) of the minor tonality, with two clausulae's direction of resolution flipped and the relative root changed to the 5th scale degree.

* 4-3: Comes from the inversion of the perfect clausula (7-1) from minor tonality. Similar treatment as Fa-Mi in major tonality
* b6-5: From the inversion of the relative major perfect clausula (2-b3) from minor tonality. Similar treatment as b6-5 in minor tonality.
* 7-1: Comes from the perfect Ti-Do clausula.

Backdoor `ii-V`s and minor ii-Vs resolving to a major I is the characteristic progression of this tonality.

An example of this tonality used effectively can be found in Ravel's Scarbo from Gaspard de la nuit.

<iframe width="560" height="315" src="https://www.youtube.com/embed/TQSyRXRuk6Y?si=ea1IGBJ_KngEsvMH&amp;start=765" 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>

#### Blues

I would argue that the blues can be thought of as a tonality, but it is unique because many of these clausulae/movements are not semitones. When performed by a free pitch instrument/voice or instrument with portamento/bends/slides, these clausulae seem to target just intonation ratios consistently. Even though blues is mainly played and conceived as "12edo with pitch bends", I find it useful to analyze it with higher-limit JI, because the blues is itself a masterclass on how a tonality can develop with higher-resolution intervals. I believe that the blues is evidence that a microtonal tonality is not only possible, but already widespread in modern musical culture thanks to the influences of West African, European and Cuban music on modern Black American Music.

* b3-1: Resolving from the subdominant sides, b3 is usually bent to a septimal subminor third. Can go both ways.

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:%7Br220Hz%7D(1%2F2)%0A7%2F3_2%2F1%0A%5B4%2F3_5%2F3_7%2F3%5D_%5B1%2F1_7%2F4_2%2F1%5D" title="Xenpaper" frameborder="0"></iframe>

> :musical_note: The notation `x/y` (forward slash `/`, as opposed to backslash `\`) means multiply the root tuning frequency (220 hz in the above example) by that ratio. E.g., the first note is $7/3 \times 220 \text{ Hz} \approx 513.33 \text{ Hz}$

* 6-b7: resolving from subdominant sides to tonic sides. Goes both ways.

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:%7Br220Hz%7D(1%2F2)%0A5%2F3_7%2F4%0A%5B5%2F3_4%2F3_1%2F1%5D_%5B1%2F1_5%2F4_7%2F4%5D" title="Xenpaper" frameborder="0"></iframe>

* b3-3: either resolving from subdominant to tonic if Mi is tuned to `5/4` (5-limit major third), or subdominant of subdominant if tuned to `11/9` (11-limit greater neutral third) (e.g., similar to "Jazz is a four letter word" example below). `11/9` is the 11th harmonic _semi-consonance_ of bVII. The former is not reversible, usually the direction is b3 &rarr; 3, but the latter can be reversed. I added more chords for the latter example for context.

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:%7Br220Hz%7D(1%2F2)%0A7%2F6_5%2F4%0A%5B2%2F3_5%2F6_7%2F6%5D_%5B1%2F2_7%2F8_5%2F4%5D.%0A%0A%7Br330Hz%7D%0A1%2F1_7%2F6_11%2F9_1%2F1%0A%5B1%2F2_7%2F8_1%2F1%5D_%5B2%2F3_5%2F6_7%2F6%5D_%5B4%2F9_5%2F9_7%2F9_9%2F9_11%2F9%5D_%5B1%2F2_7%2F8_1%2F1%5D" title="Xenpaper" frameborder="0"></iframe>

* 1-3, but 3 is bent flat: Here 3 is tuned somewhere between `6/5` and `11/9`. The pay attention to the sung lyric "word" that goes up in pitch (not the one that goes down).

<iframe width="560" height="315" src="https://www.youtube.com/embed/cCtQ5f4vKZw?si=SH9XUpjR6vVMx47L" 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>

* #4-4: There are too many diferent types for this one, which all temper out to the same semitone movement in 12edo. I find that #4-4 is mostly one-way: the outward motion 4-#4 can be played, but must be promptly resolved via #4-4, or by ending the phrase soon.

<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:%7Br330Hz%7D(1%2F2)%0A%23_b5_%3D_7%2F5_(greater_septimal_tritone)_treated_as_7th_harmonic_of_b6.%0A1%2F1_7%2F5_4%2F3%0A%5B1%2F2_7%2F8_1%2F1%5D_%5B4%2F5_5%2F5_7%2F5%5D_%5B2%2F3_5%2F6_7%2F6_4%2F3%5D.%0A%23_%234_%3D_11%2F8_treated_as_11th_harmonic_of_1%0A1%2F1_11%2F8_4%2F3%0A%5B1%2F2_7%2F8_1%2F1%5D_%5B1%2F2_7%2F8_9%2F8_11%2F8%5D_%5B2%2F3_5%2F6_7%2F6_4%2F3%5D.%0A%23_%234_%3D_45%2F32%2C_5th_harmonic_(maj3)_of_2%0A1%2F1_45%2F32_4%2F3%0A%5B1%2F2_7%2F8_1%2F1%5D_%5B9%2F16_27%2F32_63%2F64_45%2F32%5D_%5B2%2F3_5%2F6_7%2F6_4%2F3%5D.%0A%0A%23_and_all_the_above_examples_where_4_%3D_21%2F16_(7th_harmonic_of_5)_instead%0A1%2F1_7%2F5_21%2F16%0A%5B1%2F2_7%2F8_1%2F1%5D_%5B4%2F5_5%2F5_7%2F5%5D_%5B3%2F4_15%2F16_21%2F16%5D.%0A%0A1%2F1_11%2F8_21%2F16%0A%5B1%2F2_7%2F8_1%2F1%5D_%5B1%2F2_7%2F8_9%2F8_11%2F8%5D_%5B3%2F4_15%2F16_21%2F16%5D.%0A%0A1%2F1_45%2F32_21%2F16%0A%5B1%2F2_7%2F8_1%2F1%5D_%5B9%2F16_27%2F32_63%2F64_45%2F32%5D_%5B3%2F4_15%2F16_21%2F16%5D." title="Xenpaper" frameborder="0"></iframe>

:::info
:writing_hand: I've left out many examples in the interest of time. After completing the section about just intonation, go back to this and write out as many as you can.
:::

There are many more movements from various eras and periods of the blues, but I only have time for these few. Unlike the major and minor tonalities, which have few clausulae within one tonality, but many clausulae bridging other overlapping tonalities, the blues tonality is a monolith that stands strongly on its own with many pathways contained within itself, but lesser bridges connecting it to the other 12edo-focused "classical tonalities". This makes it feel like the blues is a tonality in a way that is more similar to the way one would consider ragas or maqamat a tonality. The tonalities of the ECT (based on the concept of function and cadence) appears to be an outlier, relative to the musical cultures the rest of the world developed.

### Making your own tonalities

Distilling the main points from the above examples, here are some findings of mine about what makes a good constructed tonality.

1. **You can create your own tonality by being consistent**. Consistency lets you create your own musical culture. This is done through a consistent repetition of expectations created, resolved, or subverted by consistently following your custom tonality/music culture model, which can entrain your audience as they listen. In fact, every individual exists in their own music bubble and perceives music through a unique filter constructed from conscious and subconscious accumulation of experiences.
1. **But you have to respect tradition. It is nigh impossible to erase or override existing cultural entrainment**. E.g., I find it impossible to override the standard Fa-Mi lemma clausula (by itself, no other clausula/signifiers) to be the bridge between the major tonality and a different one. On the subdominant sides of a major tonality, Fa-Mi on its own with no other clausulae is **always** a plagal clausula to the tonic.
1. **Each state/side of a tonality must have unique outgoing clausula/movement**. If the model of tonality is probabilistic (i.e., the same Fa-Mi movement has a 50% chance of going to X side of A tonality and 50% chance of going to Y side of B tonality), you're probably missing some state that wasn't kept track of (e.g., what notes are in the memory? which beat the previous movement was made on? how long was spent on the current state?). Consistency is what allows the entrainment and acceptance of a music, and a probabilistic model goes against that. Even the subversion of expectations is deterministic, where the realm of possible "suprises" are still eventually limited to some larger model of tonality.
1. **Besides two note clausulas, you can also consider adding notes, or a set motif/riff/lick to signify the change of state/side**, or changing tonalities entirely. This is especially prevalent in ragas (or bebop) where there are set phrases to play for the tonality sometimes signifies a certain state change. This is what the words "licks", "cliches" and "vocab" refers to. Prominent examples are: the [cool blues lick](https://www.youtube.com/watch?v=5pVxWdnInWY), the [honeysuckle rose lick](https://www.youtube.com/watch?v=uBi4BOC2nLU), ["the lick"](https://www.youtube.com/watch?v=hlY5fh3auR4), the [mario kart lick](https://www.youtube.com/watch?v=3rD-tdEFKlg). These are things that could be in your model of tonality if you wish. You can create your own clausulas/motifs/language by consistent repetition that applies a consistent cause & effect to the tonality state. As an example of this concept taken to its most extreme, the band Kneebody has developed a lexicon of musical words that have actual semantic meaning used to communicate musical direction:

<iframe width="560" height="315" src="https://www.youtube.com/embed/9EGZrck0rNU?si=7XLbQNxUiOW4CeRc" 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>

5. **Removing a note** is almost never a movement between states/sides of a tonality, unless the note removed is somewhere between 40-180 cents away from an existing note or an existing note in pitch memory. Even then, it is not a strong movement.

### 31edo tonalities

31edo is a [meantone](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Meantone) temperament. We will learn about that in section 3, but what that means for now is that, as long as you spell everything enharmonically correctly (e.g., the major third of E is G# not Ab, and the major third of G# is B# not C), **then every tonality from 12edo will carry over as per normal** (especially if it doesn't rely on [octave symmetry](https://hackmd.io/@euwbah/extending-harmonic-principles-3a#Symmetries-and-edos)). 12edo is also a meantone temperament, and most of the tonality examples we have covered are enabled by meantone temperament.

The big idea that we'll get to at the end of this series is that a tuning may offer (infinite) temperaments, and each temperament unlocks a harmonic language we can use in the tonality that is unique to that temperament, because of coincidences it creates. We'll go into this at depth in Part III, but for now, we just focus on applying the above principles of tonality construction to an example of a set of tonalities I use very often in 31edo.

The tonality state diagrams below notate both individual notes and clausula movements in edosteps from a fixed root `0\31`, since I haven't introduced notation convention for 31edo yet.

:::warning
:warning: &nbsp; **Disclaimer**

These are just small subsets of the tonalities I have developed in my personal cultural filter, there's **a lot** more to this but expositing it in full deserves its own article. The goal here is to give an example of how you could get started, and how to apply the principles for [making your own tonalities](#Making-your-own-tonalities).

This section is written with full-blown JI & temperament terminology. If you wish to obtain prerequisite knowledge, it is recommended to head to [Part II](https://hackmd.io/@euwbah/extending-harmonic-principles-2) first, and come back here after reading the rest of the series, which covers all temperaments and symmetries referenced in this section.
:::

#### Subminor & orwell/primodal-6 tonalities

Back in the section [Tonality is fundamentally cultural](#Tonality-is-fundamentally-cultural), I showed how the 7-limit subminor triad can be perceived as a consonant base to call the tonic. This is one possible realization of that concept in 31 edo, where I juxtapose traditional Ti-Do clausula, with the wider `4\31` sesquitone (i.e., an interval approximately 1.5 semitones wide) to function as a kind of "Ti-Do" instead of the usual `3\31` limma (i.e., the semitone interval between adjacent white keys E-F and B-C). The option to choose `4\31` as Ti-Do bridges to the [orwell](https://en.xen.wiki/w/Chords_of_orwell)/primodal-6 tonality that has 11:14:18 (the [mothwellsmic](https://en.xen.wiki/w/Mothwellsmic_chords) triad) as a the key "dominant" sound, which admits a non-octave symmetry of supermajor thirds `11\31` which can generate more options for dominant chords, just as how the symmetries of diminished & augmented chords in 12edo does.

> :pencil: The "official" name for the temperament (and chords offered by the temperament) is **orwell**, but I like to think of it as a tonality with my own custom clausulae. I use **Primodal-6** to refer to the subjective constructed tonality I made around this temperament, with this name referencing Amelia Huff's [Theory of Primodality](https://www.youtube.com/watch?v=KKxXdD-lkwI). The relation between primodality is that the notes are "under 6", `x/6` in just intonation. The main 'scale' of this tonality is the harmonic series from the 6th harmonic to the 12th, so we can write the scale in just intonation as `6/6 7/6 8/6 9/6 10/6 11/6 12/6`. Occasionally I would throw in `13/12` (octave reduced `13/6`) or `9/8` in there too, depending the side of tonality.

To simplify, in place of the usual `3\31` limma semitone for both Ti-Do and Fa-Mi (which map to `1\12` as the just interval `16/15`), I add the option of `2\31` for Ti-Do and Fa-Mi, or `4\31` for Ti-Do. Recall the section on [pitch memory & semitone cancellation theory](#Pitch-memory-amp-semitone-cancellation-theory), we posited (from empirical data) that of intervals between 66--166 cents are effective at creating movements between sides/states of a tonality. Notice that `2\31` &approx; 77.4&cent;, `3\31` &approx; 116.1&cent;, `4\31` &approx; 154.8&cent;

<iframe width="560" height="450" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br330Hz%7D(1%2F3)%0A%23_A%0A%5B0_7_18%5D_%5B0_13_20%5D_%5B%6028_13_20%5D_%5B0_7_18%5D.%0A%23_B%0A%5B0_7_18%5D_%5B0_13_20%5D_%5B%6028_5_13_20%5D_%5B%6026_8_18%5D.%0A%23_C%0A%5B0_7_18%5D_%5B0_13_20%5D_%5B%6027_4_12_20%5D_%5B%6025_7_17%5D.%0A%0A%23_Primodal-6_%22Dominant%22_to_Tonic.%0A%23_Fa-Do_(13-0)_plagal_bassizans%0A%23_Sesquitone_Ti-Do_(27-31)%0A%5B%6013_7_18_27%5D_%5B%600_0_7_18_31%5D.%0A%0A%23_T_S_D_T_equivalent_in_Primodal-6_tonality.%0A%5B0_7_18%5D_%5B0_13_23%5D_%5B%6027_13_23%5D_%5B0_7_18%5D.%0A%0A%23_Subminor_7th_with_narrow_Mi-Fa_(25-27)_for_7%2F4_to_go%0A%23_out_to_11%2F6_of_primodal-6%0A%23_Mothwell_clausula_(18-16)_from_mothwell_triad_(7-18-27)%0A%23_with_bassisans_superthird_non-octave_symmetry_(13-2)%0A%23_Sesquitone_Ti-Do_(27-31)%0A%23_Narrow_Fa-Mi_(2-0)_and_(20-18)_(borrowing_from_double_harmonic_major)%0A%23_Narrow_Ti-Do_(16-18)_(plural)%0A%5B0_7_18_25%5D_%5B%6013_0_7_18_27%5D_%5B%602_%6020_7_16_27%5D_%5B%600_0_7_18_31%5D.%0A%0A%23_Narrow_Ti-Do_(18-20)_from_primodal-6_to_%235_submin_side_dominant%0A%5B0_7_18_25%5D_%5B%6013_0_7_18_27%5D_%5B%6020_7_20_27%5D_%5B0_7_18_25%5D.%0A%23_Plagal_resolution_to_relative_supermajor%0A%5B0_7_18_25%5D_%5B%6013_0_7_18_27%5D_%5B%6020_7_20_27%5D_%5B%607_7_18_25%5D.%0A%23_Plagal_resolution_to_relative_major_standard_Fa-Mi_(20-17)%0A%5B0_7_18_25%5D_%5B%6013_0_7_18_27%5D_%5B%6020_7_20_27%5D_%5B%607_7_17_25%5D.%0A%23_%235_submin_as_ii%0A%5B0_7_18_25%5D_%5B%6013_0_7_18_27%5D_%5B%6020_7_27%5D_%5B%6015_2_25%5D" title="Xenpaper" frameborder="0"></iframe>


:::info
**LEGEND**

Individual circles/ovals/rectangles are sides/states of a tonality.

The `Ton` and `Plu1` (tonic) circles with a bright yellow outline are starting sides/states. All but one of the above progressions start from either of these sides/states.

Rectangular areas denotes tonalities. Overlaps mean that sides contained in the intersection of tonalities can belong in multiple tonalities.

Also, the `REL- MAJOR +7\31` tonality is split on the left & right sides due to limitations of a 2D diagram, but they're meant to be connected & the same tonality.

There are sides which are intentionally connected/overlapping. This means that the same (or very similar) set of notes can have different sides depending on the current tonality of the context. These multi-sides are effective at bridging between tonalities.

Arrows denote the direction of which one side can move to another, with the text denoting the clausulae/signifiers/movements. Some arrows are bidirectional, meaning the movements can be reversed (or undoing added notes) to return to the prior side.

As for the text, the numbers are steps of 31edo. `0` is fixed absolutely as the root of the starting point tonic (yellow outline).

`x-y` denotes a clausula/movement from one note to another.

`+x` denotes adding a note.

`[x-y]` square brackets mean that the contained movements or additions are optional.

`()` parentheses denote order of operations.

`x-y & a-b` means both `x-y` and `a-b` movements need to be executed.

`x-y | a-b` means either `x-y` or `a-b`, or both.
:::

##### Subminor progression A

* **0 7 18** Shared tonic of subminor & primodal-6 tonalities. `6:7:9` in JI representation.
* **0 13 20** Shared subdominant function of subminor & primodal-6. Subminor triad `6:7:9` built on the perfect fourth `4/3 = 13\31`. The `18-20` movement borrows from anti plagal Mi-Fa clausula, but reduced semitone size from limma (`3\31`) to apotome (`2\31`) to target `20\31` (subminor `7/6` of fourth `4/3 = 13\31`) instead of `21\31` (minor `6/5` of `4/3`).
* **&#96;28 13 20** is one of the altered dominant analogues of the subminor tonality with `20\31` db9 (sesquiflat 9, equivalent to #8) instead of b9. db9 is a suspension/common tone from septimal subminor `7/6` of fourth `4/3`. This is reached via the standard anti perfect clausula Do-Ti `31-28` which is a limma interval. Adding the note `5\31` (fifth of the fifth) is optional.
* **0 7 18** Returns back to the starting tonic via the apotome Fa-Mi `20-18` and limma Ti-Do `28-31`. Borrows from the `V7-im` cadence of the minor tonality of 12edo ECT music. Refer back to the [12edo minor tonality example](#Minor-tonality). If the optional `5\31` was present in pitch memory, there is also the apotome Ti-Do movement `5-7` which is plural from the relative supermajor, just like in 12edo's minor.

##### Subminor progression B

First two chords same as [A](#Subminor-progression-A).

* **&#96;28 5 13 20** same as third chord of A, but with added `7-5` movement, which on based on the 2-b3 resolving relative major plural Ti-Do of 12edo's minor, but reversed, so it is going out.
* **&#96;26 8 18** resolves by pluralism to the tonic of the higher relative major (`+8\31`), the [relative transformation](https://en.wikipedia.org/wiki/Neo-Riemannian_theory#:~:text=The%20R%20transformation%20exchanges%20a%20triad%20for%20its%20Relative) that anchors the 5-limit minor third `6/5 = 8\31`. Movements are: parallel subminor plural Fa-Mi `28-26` (analogue to b6-5 of 12edo); the usual clausula perfect Ti-Do `5-8` limma; and the apotome Fa-Mi 20-18, which is narrowed because of db9 `20\31` being used instead of b9 `21\31` in the prior altered dominant chord.

##### Subminor progression C

First two chords same as [A](#Subminor-progression-A).

* **&#96;27 4 12 20** dominant (dimished/altered type) of the lower relative major (`+7\31`) where we perform the [relative transformation](https://en.wikipedia.org/wiki/Neo-Riemannian_theory#:~:text=The%20R%20transformation%20exchanges%20a%20triad%20for%20its%20Relative) that anchors the subminor third `7/6 = 7\31`. The db9 of the prior chord, `20\31`, which was the subminor third `7/6 = 7\31` of the prior root, is set as an fixed anchor point. From that note, a diminished 7th chord is constructed downwards where the 5-limit minor third `6/5 = 8\31` interval is repeatedly stacked downwards.

This yields two sesquitone movements which are `4\31 = 11/10 = 12/11` (equal because the [bayatisma comma](https://en.xen.wiki/w/121/120) is temepered out). The first sesquitone movement `31-27` is the characteristic Do-Ti (anti perfect clausula) analogue of the Primodal-6 tonality, which is wider by one diesis `1\31 = 128/125` as compared to the usual limma `3\31 = 256/243 = 16/15` Ti-Do or Fa-Mi movements.

The second sesquitone movement is `0-4`, but it is optional and can instead be interpreted as adding the note `4\31` (written `+4` on the arrow text). The `0-4` movement is multi-semantic. The first way it can be interpreted is as a narrower major second movement, i.e. the anti tenorizans durum clausula Do-Re, which is common in [porcupine temperaments](https://en.xen.wiki/w/Porcupine) like 22 or 15 edo. Though, 31edo is not a porcupine temperament, the entrainment of `11/10` as a "major second"-functioning interval can still happen with enough repetition. The second way `0-4` can be interpreted would be the same way `31-27` is interpreted here, as a wider `Ti-Do`. It doesn't make intuitive sense to interpret the note `4\31` as "Do", but `4\31` is a fifth-coloring of the root of the dominant 7b9 chord (**&#96;17 &#96;27 4 12 20**) that is implied by this diminished 7th chord, which is `17\31`, so we can see it as a different shade of the "Do" of the current unplayed root `17\31` of a rootless dominant 7b9. The third way `0-4` can be interpreted would be as the anti Re-Do (b2-1) of 12edo, which evokes the bass movement of tritone substituted V-Is or the b2-1 movement of double harmonic major or phrygian tonalities. Here, `4\31` will be felt as a wide sharper "b2" (notated d2 or v2 in 31edo) the same way `27\31` is felt as a wide flatter "7" (notated d7 or v7 in 31edo). The fourth way it can be interpreted is through maqam bayati and the [harmonic language offered by the biyatismic temperment](https://hackmd.io/@euwbah/extending-harmonic-principles-3a#Biyatismic-121120).

Between this and the previous chord, there is a diesis shift `13-12` (which I did not mark as a clausula). I already mentioned [why dieses do not make good clausula, and should be avoided in general](#What-semitone-cancellation-theory-amp-pitch-memory-implies-for-31edo), and that diesis shifts create much cognitive dissonance between the short term memory of pitches and the the current external stimuli because the old pitch `13\31` is not easily forgotten when `12\31` is sounded. However, the strength of the above sesquitone movements, discordance of the gestalt of the chord, and concordance of the individual note `12\31` relative to its neighbors &mdash; these factors all contribute to the "forgetting" or "ignoring" of the note `13\31`. This is a good case study of this broken principle that shows up often in 12edo as well.

* **&#96;25 7 17** is the tonic of the lower relative major. Essentially the same clausulae as the last chord of [progression B](#Subminor-progression-B), but `20-17` is now a limma Fa-Mi instead of the apotome `20-18`. The plural Fa-Mi `27-25` and relative major limma Ti-Do `4-7` is transposed down one diesis compared to the last chord of [B](#Subminor-progression-B), but have the exact same functions.

##### The symmetric Mothwell transformation

The only movement that is not yet explained is the mothwell transformation, which is made possible with a non-octave symmetry due to the [mothwellsma `99/98 = 0\31`](https://en.xen.wiki/w/99/98) being tempered out.

:::info
:writing_hand: The rest of the progressions use side/state movements that are already discussed above, or clausulae that already exist in familiar 12edo tonalities. Explain the rest of the progressions by relating back to the clausulae which we are familiar with in 12edo. Follow the tonality state diagram and use comments in the xenpaper score as hints. Were there any other places where principles were broken? Can you justify them?
:::

We'll discuss this more thoroughly in [Part III](https://hackmd.io/@euwbah/extending-harmonic-principles-3a#Mothwellsmic-9998), but for now, understand that in 31edo, the septimal supermajor third `9/7` &approx; 435&cent; and the undecimal major third `14/11` &approx; 417.5&cent; (the interval between the 11th and 14th harmonics) both map to the same interval in 31edo, which is `11\31`.

<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:%7Br330Hz%7D(1%2F3)%0A%5B1%2F1_9%2F7%5D%0A%7Br9%2F7%7D%5B1%2F1_14%2F11%5D%0A%7Br7%2F9%7D11%3A14%3A18" title="Xenpaper" frameborder="0"></iframe>

This means that the chord `11:14:18`, formed by a `14/11` followed by a `9/7` is a stack of `11\31` in 31edo, which is like an augmented triad (1-3-#5 in 12edo), but instead of major thirds, it is formed with supermajor thirds.

<iframe width="560" height="180" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br330Hz%7D(1%2F3)%0A%5B0_11%5D_%5B11_22%5D_%5B0_11_22%5D" title="Xenpaper" frameborder="0"></iframe>

This chord I call a mothwell triad, not to be confused with the [mothwellsmic triad](https://en.xen.wiki/w/Mothwellsmic_chords) which uses the [mothwellsma `99/98`](https://en.xen.wiki/w/99/98) to add intervals to equal an octave. Instead, here I am merely considering that `14/11 = 9/7` in 31edo. In 12edo, the augmented triad divides the octave into 3 edo such that different inversions of the augmented triad are isomorphic under octave-equivalence to different augmented triads (e.g., the first inversion of `Caug` is `Eaug`, whose first inversion is `Abaug`). Unlike 12edo, the mothwell triad doesn't divide any octave, but can be seen as equally dividng the interval `18/11 = 22\31` into two equal parts, or `33\31` into three equal parts.

<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br330Hz%7D(1%2F3)%0A%5B0_11_22%5D_%5B11_22_33%5D_%5B0_11_22_33%5D" title="Xenpaper" frameborder="0"></iframe>

It is not consonant sounding nor familiar on its own. However, I find two very importaant uses of this triad as an upper structure. One built off the harmonic series from the fundamental `1:11:14:18`, octave equivalent to `8:11:14:18`, and one built off the 6th harmonic to form the primodal-6 plural tonic `6:7:9:11 = 0\31 7\31 18\31 27\31` (I used the first inversion of the mothwell triad to keep the voicing closed).

<iframe width="560" height="330" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br330Hz%7D(1%2F3)%0A%5B0_10_18%5D%0A%5B0_14_25_36%5D._%23_mothwell_built_on_14%5C31_8%3A11%3A14%3A18%0A%5B0_7_18%5D%0A%5B0_7_18_27%5D_%23_mothwell_(1st_inversion)_built_on_7%5C31_6%3A7%3A9%3A11" title="Xenpaper" frameborder="0"></iframe>

What this means is that transformations of the mothwell triad by supermajor thirds, which I call "mothwell rotations" or the "mothwell symmetry transformations", will naturally result in one voice moving by `2\31` each time, using octave-equivalence:

<iframe width="560" height="400" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br440Hz%7D(1%2F2)%0A%5B0_10_18%5D-%0A%5B0_14_25_36%5D_%5B14_25_36%5D%0A%5B14_25_34%5D_%5B%6020_14_25_34%5D%0A%5B%6020_%6030_7_14_25_34%5D-%0A%5B14_25_34%5D%0A%5B14_23_34%5D_%5B%609_14_23_34%5D%0A%5B%609_%6019_%6027_14_23_34%5D-%0A%0A%23_otonal_8%3A11%3A14%3A18%0A%5B0_14_25_36%5D_%5B%6020_14_25_34%5D_%5B%609_14_23_34%5D_%0A%5B%6029_12_23_34%5D_%5B%6018_12_23_32%5D_%5B%607_12_21_32%5D.%0A%0A%23_primodal-6_tonic_6%3A7%3A9%3A11%0A%5B%6018_14_25_36%5D_%5B7_14_25_34%5D_%5B%6027_14_23_34%5D%0A%5B%6016_12_23_34%5D_%5B5_12_23_32%5D_%5B%6025_12_21_32%5D" title="Xenpaper" frameborder="0"></iframe>

Because the movement happened to be `2\31`, which is a good interval to use as a movement that causes a change in the short term memory of pitches, I have added this harmonic language of the clausula of moving the third note of a mothwell triad (the 18th harmonic in `11:14:18`) downward by `2/31` to signify this mothwell symmetry that transforms downward by `11\31`, denoted **Moth Sym Low** in the tonality state diagram. For the inverse operation, one may raise the first note of a mothwell triad (the 11th harmonic in `11:14:18`) up by `2\31` to transform upwards by a supermajor third `11\31`.

This is the only clausula movement that I have added that is not related to any existing familiar clausula in 12edo.

##### Examples of the subminor and primodal-6 tonalities

<iframe width="560" height="315" src="https://www.youtube.com/embed/Lq9-6NnXPVg?si=D0tbRt2uPm4UL_5I&amp;start=128" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

The composer Fabio Costa has written in the description his conception of the section of the madrigal starting at 2:08, which is based on the `10:12:15` triad, and moving the minor third `12:15` in parallel by `4\31` to make use of the (13-limit:exclamation:) comma that makes `13/11 = 12/10` in 31edo, so that the notes used can be seen as part of the harmonic series `10:11:12:13:14`.

However, if we reduce the dimensionality down to the 11-limit, we can also understand this section as the primodal-6 tonality.

:::info
:writing_hand: Analyze this piece in terms the just intonation concepts by the composer, and create a tonality state diagram that can capture the essence of the tonalities in this piece.
:::

Another example would be a semi-improvised piece of mine, written with these tonalities in mind.

<iframe width="560" height="315" src="https://www.youtube.com/embed/oR8KHLdCCwA?si=brw8o7f8-4Drz8-b&amp;start=2" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

##### Learning points from the subminor & primodal-6 tonalities

To sum up the analysis of these tonalities, notice how most of the **movements are related back to the traditional Fa-Mi and Ti-Do clausulae**. One may think that abiding by the tradition of tonalities of ECT's cultural entrainment will lead to a limited use of 31edo, but this small example should give you a glimpse of how much freedom there is even within this framework if you understand the resources made available by the temperament, and the principles of symmetries and pluralism that we will cover in the coming sections. Structured freedom is a relatable and communicatable art.

#### Bayatismic tonality

> :construction: WORK IN PROGRESS.
>
> Will include this section depending on readership & demand.

### Tonalities in other tunings

11edo is possibly one of the most alien tunings relative to 12edo, because the step sizes are almost the same but all the steps are off, and all the familiar consonant just intonation structures like the perfect fifth `3/2` or major third `5/4` are out of tune and unusable as consonances.

But check out this example:

<iframe width="100%" height="300" scrolling="no" frameborder="no" allow="autoplay" src="https://w.soundcloud.com/player/?url=https%3A//api.soundcloud.com/tracks/30780344&color=%23ff5500&auto_play=false&hide_related=false&show_comments=true&show_user=true&show_reposts=false&show_teaser=true&visual=true"></iframe><div style="font-size: 10px; color: #cccccc;line-break: anywhere;word-break: normal;overflow: hidden;white-space: nowrap;text-overflow: ellipsis; font-family: Interstate,Lucida Grande,Lucida Sans Unicode,Lucida Sans,Garuda,Verdana,Tahoma,sans-serif;font-weight: 100;"><a href="https://soundcloud.com/mikebattagliaexperiments" title="Mike Battaglia" target="_blank" style="color: #cccccc; text-decoration: none;">Mike Battaglia</a> · <a href="https://soundcloud.com/mikebattagliaexperiments/tonality-patterns-in-11-edo" title="Tonality, Patterns in 11-EDO" target="_blank" style="color: #cccccc; text-decoration: none;">Tonality, Patterns in 11-EDO</a></div>

By using consistent repetition of clausula movements to form a cadence, and by acknowledging that 11edo offers a consonance in the form of the `4:7:9` triad or `4:7:9:11` tetrad, Mike Battaglia creates his own system of tonality that can result in very functional, and musical, harmony that utilizes what the tuning system has to offer while maintaining familiarity to music of the ECT.

# 2. Just Intonation (JI)

I have reached near the maximum character count for hackmd.io, later sections will be continued in Parts II and III. See [Extending harmonic principles in 12edo to 31edo. Part II: JI & Temperaments](https://hackmd.io/@euwbah/extending-harmonic-principles-2)

Thank you for hearing me!

## Support me

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

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

## Glossary

*[ECT]: European classical tradition
*[edo]: Equal division of the octave
*[edos]: Tunings which are based on equal divisions of the octave
*[12edo]: 12 equal divisions of the octave
*[31edo]: 31 equal divisions of the octave
*[otonal]: contained the overtone/harmonic series, as opposed to utonal
*[utonal]: contained in the reciprocal of the harmonic series
*[sesquitone]: An interval approximately 1.5 semitones wide
*[limma]: The semitone interval that is between two white keys. In JI, this is 256/243
*[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
*[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
*[JI]:Just Intonation

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

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

JI
: Just Intonation

limma
: The semitone interval that is between two white keys. In JI, this is 256/243

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