# Extending harmonic principles in 12edo to 31edo. Part IIIa: Octave symmetries, application & summary
$$
\def\monzo#1{\left[ \begin{matrix}#1\end{matrix} \right>}
\def\cov#1{\left< \begin{matrix}#1\end{matrix} \right]}
\def\braket#1#2{\left< \begin{matrix}#1\end{matrix}\right. \mid \left. \begin{matrix}#2\end{matrix}\right>}
$$
:::warning
:warning: Read this in [dark mode](https://hackmd.io/@docs/how-to-set-dark-mode-en#Set-View-page-theme).
:::
Previous: [Part III: Temperament, vals, mapping, scales, functional harmony](https://hackmd.io/@euwbah/extending-harmonic-principles-3).
:::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]
# Temperaments & Mapping
## 31edo vs 1/4-comma meantone
Before continuing, I want to make a quick digression to how 31edo relates to 1/4-comma meantone.
Recall that [1/4-comma meantone was called 1/4-comma](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Why-it-is-called-14-comma-meantone) because the size of a `3/2` fifth (also the 3rd harmonic/3rd prime interval) was reduced by a quarter of the syntonic comma `81/80`. This resulted in 4 stacked fifths being detuned down by exactly one syntonic comma, which made it exactly equivalent to the 5th harmonic/prime interval 5.
Recall that 31edo has the default edo mapping (patent val) $\cov{31 & 49 & 72 & 87 & 107}$
If we look at the size of the fifth $3/2 \equiv \monzo{-1&1}$ in 31edo, which maps to:
$$\braket{31&49}{-1&1} = 18 \setminus 31$$
edosteps of 31edo, then its size in cents is $\frac{18}{31} \cdot 1200 \approx 696.774¢$, which is very similar to the size of the fifth we arrived at in 1/4-comma meantone: 696.578¢.
By sharpening the size of the fifth by a very tiny almost impossible to detect interval of around 0.2¢, we are able to close back the circle of fifths after 31 fifths, which is what makes 31edo an edo (rank-1), having a finite number of notes within each octave.
## Symmetries (and edos)
We are continuing where we left off in [Symmetries (and edos)](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Symmetries-and-edos)
### 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).
> :nerd_face: :point_up: We are dealing with [cyclic groups](https://en.wikipedia.org/wiki/Cyclic_group) of finite order and their [subgroups](https://en.wikipedia.org/wiki/Subgroup) (and if we are considering negative harmony, [dihedral groups](https://en.wikipedia.org/wiki/Dihedral_group))
Let's add notes to corners of the square. Let's assume we are in 12edo = $\cov{12&19&28}$ in the JI subgroup `2.3.5`, and we have these notes, where `0\12 = C4`:
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A0_3_6_9_%5B0_3_6_9%5D-" title="Xenpaper" frameborder="0"></iframe>
If we add a minor third $3\setminus 12 \equiv \monzo{1&1&-1}$ to all of the notes, this is what we get:
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A%5B0_3_6_9%5D-_%5B3_6_9_12%5D-" title="Xenpaper" frameborder="0"></iframe>
It's almost the same, but not exactly. However, if we assume octave equivalence, then we can ignore the octave numbers, and edosteps will be taken modulo 12 (that means once we reach 11\12, it loops back to 0 instead of going up to 12).
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A%5B0_3_6_9%5D-_%5B3_6_9_0%5D-" title="Xenpaper" frameborder="0"></iframe>
And what we realize is that moving all notes up by `3\12` is actually **equivalent**, up to the octave, to the exact same structure.
:::info
:writing_hand: **Exercise**
What comma is being tempered out in the above example?
:::spoiler Hint
We are equating the action of going up by 4 minor thirds $4 \cdot \monzo{1&1&-1}$ to the action of going up one octave $\monzo{1}$
:::spoiler Answer
Thus, the subtraction of the octave from 4 minor thirds must map to zero:
$$4 \cdot \monzo{1&1&-1} - \monzo{1} = \monzo{3 & 4 & -4} \equiv \frac{648}{625}$$
Hence, 648/625, the [diminished comma](https://en.xen.wiki/w/648/625), is what we're tempering.
:::
#### Octave symmetries
The above example is only a symmetry because we consider octaves to be equivalent. For that reason, we call it an **octave symmetry** (or a "symmetry up to an octave")
### Symmetries of 12edo music & its applications
#### 12edo itself
If we collect all 12 notes of 12edo, and increase every note by 1 semitone (edostep), we still end up with the same 12 notes if we assume octave equivalence.
This is a trivial fact for all edos.
> :nerd_face: :point_up: This is the trivial subgroup of the cyclic group $\mathbb{Z}_{12} \cong \mathbb{Z} / n\mathbb{Z}$ for $n$ edo.
#### Whole tone scale (aka 6edo)
We can also collect every _other_ note, e.g. `C D E F# G# A#`, or `Db Eb F G A B`, and increase every note by 2 semitones:
This structure can both be a scale and a chord (e.g., `C7aug#11`), but the application of this as a scale for melodic use on its own is difficult because it presents **no standard clausula** of the entrainment of the ECT.
However, the practice of using **only** these notes is made famous by Claude Debussy:
<iframe width="560" height="315" src="https://www.youtube.com/embed/FVV0jkZC4jI?si=KFUmCVwdvQIu2Fgr" 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>
For which a novel system of tonality must be entrained and executed within the song.
#### Diminished tetrad (dim7, 4edo)
The diminished tetrad officially has the scale degrees `1 b3 b5 bb7`, but in 12edo, or [compton temperaments](https://en.xen.wiki/w/Compton_family#Compton) tempering the Pythagorean comma `531441/524288`, these scale degrees are enharmonically equivalent to `1 b3 #4 6`. We've already gone through why this is an octave symmetry at the start.
This symmetry has huge applications for harmony in 12edo music.
To explain this, we'll need to two tools: [extended pluralism](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Chromaticism-in-extended-pluralism) and the [Fa-Mi & Ti-Do clausulae](https://hackmd.io/p28Z1haLTmWWwTooWk34sg#The-modern-clausulae-of-semitones).
Let's use 1 = C as our tonic root.
We regard `A min` as the plural of `C maj`. Then, the dominant `V` of `A min` is `E maj`.
We can treat `6 = A` as the _other_ Do (written **Do2**), because it is the root of the plural equivalent `A min`. By adding E maj, we add the note `G# = #5`, which gives us two different valid clausula `#5-6`, which is Ti2-Do2 with respect to 6, and `#5-5`, which is _tenorizans molle_, (or can be seen as Fa2-Mi2 of _yet another_ plural Eb major of C minor, which is the parallel transformation of C).
That's a lot to take in, so take your time to ponder over the diagram and the principles we've gone through — we're putting all the pieces of the puzzle together here.
From the use of the extended plural borrowing from the plural `vi m` tonic chord, we obtain two new clausulae, which we can add either, or both (in more modern musical cultures, e.g. Barry Harris' 6th-diminished) to the existing `4-3` and `7-1` that we have from the major tonality.
The presence of this clausula implies that on top of only using Ti-Do and Fa-Mi for our perfect cadence `V - I`...
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A'5_'4_%23_Fa_Mi%0A11_'0_%23_Ti_Do%0A%5B7_11_'2_'5%5D_%5B0_7_'0_'4%5D_%23_perfect_cadence" title="Xenpaper" frameborder="0"></iframe>
...we can also add, say, the tenorizans molle `#5-5`:
<iframe width="560" height="400" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A'5_'4_._%23_Fa_Mi%0A11_'0_._%23_Ti_Do%0A8_7___._%23_tenorizans_molle%0A%5B7_8_11_'2_'5%5D_%5B0_7_'0_'4%5D._%23_perfect_cadence_%2B_tenorizans_molle%0A%0A%5B8_11_'2_'5%5D_%5B0_7_'0_'4%5D_%23_remove_G_to_increase_concordance" title="Xenpaper" frameborder="0"></iframe>
Notice that G `7\12` can be removed to increase concordance — it does not contribute to any clausulae, so it is alright to remove.
Once we remove G, notice that our first chord `8 11 '2 '5` is the **octave symmetric diminished chord** `Ab dim7`. If we increase every note by `3\12`, we get `11 '2 '5 '8` `B dim7`, which is octave equivalent.
In fact, `Abdim7 = Bdim7 = Ddim7 = Fdim7` are all octave equivalent. If we understand the function of the `dim7` chords by its clausulae (Ti-Do, Fa-Mi, and tenorizans molle), we can find that the G, Bb, Db, and E dominant 7b9 chords also share the exact same clausulae (in the same key), so they can have the same function, and extending the transitivity more leniently, we can say that, with respect to C, G7 and Db7 are equivalent because they share all 3 clausulae notes (F, Ab, B), and E7 and Bb7 are similar because they share 2 of 3 clausulae (G#/Ab, B)
:::spoiler :writing_hand: Notice G# and Ab are now the same note because of 12edo. What 3 limit comma must be tempered out to allow this?
The Pythagorean comma $531441/524288 = \monzo{-19&12}$
:::
This means that any of the above chords can be resolved to C major (and also Eb, F#, and A, because of symmetry).
But how about resolving to minor chords? Thinking in `C = 1`, instead of having the note `3 = E`, we have `b3 = Eb`, which means the Fa-Mi (F-E) clausula does not apply anymore.
Recall the [minor tonality of 12edo](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Minor-tonality) we glossed over in Part I.
This tonality is characterized by the three semitone clausulae `7-1`, `2-b3`, and `b6-5`.
But we can do some magic, now that we know pluralism exists. Notice what happens when we lower all those scale degrees by a minor third, so that $\flat 3 \mapsto 1$, which is effectively changing the relative root to b3.
Now we get the clausulae `#5-6`, `7-1`, and `4-3`. Looks familiar? These are 3 out of 4 clausulae we obtained when applying extended pluralism to the principle of the Ti-Do and Fa-Mi clausulae! This is written as "plural perf. clausula" in the diagram:
This means, not only the above equivalences of dominant `7b9` and `dim7` apply when resolving to major triads, but it also applies when resolving to minor triads!
This is the principle why parallel (e.g., C major to C minor) and relative (C major to A minor) transformations share a very similar set of dominants in 12edo, and is the motivating principle behind neo-riemannian theory's PLR transformations (or modal interchange).
More generally, the principle here is that certain sets of clausulae are especially effective in certain temperaments because of **musical puns** that the temperament allows. In 12edo, having the clausulae Ti-Do and Fa-Mi in our culture of tonality is strong because of the symmetry of the tritone (enabled by compton & dimipent temperaments), and having tenorizans molle is strong because of the meantone temperament that enables pluralism between the `vi minor` and the `I` major via fifths-coloring.
> :pencil: If we already have a meantone temperament, adding either the additional constraint of dimipent, augmented, or compton temperaments to it will result in 12edo. The math of "mixing and matching" temperaments is [Regular Temperament Theory](https://en.xen.wiki/w/Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT) which uses linear algebra, or [exterior algebra](https://en.xen.wiki/w/Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_EA_for_RTT), or even [Clifford algebra](https://github.com/xenharmonic-devs/temperaments).
<iframe width="560" height="640" src="https://luphoria.com/xenpaper/#embed:(1)%7Br%603%5C12%7D%7B12edo%7D%0A%5B0_7_'4%5D-_________%23_C_maj%0A%5B2_8_11_'5%5D_______%23_Ddim7____________Dom_(sym)_of_C%0A%5B4_11_12_'7%5D______%23_Cmaj7%2FE_%3D_C%2BEmin_Ton_plu_of_C%0A%0A%23_E7b9%2FF__Dom_(sym)_of_C%2C_Dom_(plu_of_F_%3D_Am)%0A%5B5_'2_'4_'8_'11%5D--%0A%5B5_'0_'5_'9%5D______%23_F___________Ton_of_F%0A%23_Bb7%2FAb________Dom_(sym)_of_C%2C_Sub_of_F%0A%5B8_'5_'10_''2%5D-_%0A%5B10_'3_'8_''0%5D-___%23_Bb9sus______Sub_of_Ab%2FFm%0A%5B5_'0_'2_'8%5D-_____%23_Fm6_(D07%2FF)_Ton_of_Ab%2FFm%2C_Sub_of_C%0A%5B1_8_11_'2_'5%5D-___%23_Db7b9_______Dom_(sym)_of_C%0A%5B0_7_'0_'4%5D---____%23_Cmaj________Ton" title="Xenpaper" frameborder="0"></iframe>
Above is an example of the application of both pluralism and symmetry to freely change the tonic. Notice that some chords have double functions, one function in one key, but another function in the next:
* `E7b9/F` is the dominant of C via diminished (4edo) symmetry which relates it to `G7b9` (`Abdim7`), but it also is the dominant of F via the [diatonic plural](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Diatonic-Pluralism) relation `Fmaj` = `Am`, and `E7b9` resolves to `Am` via the three minor clausulae `#5-6`, `7-1`, `4-3`
* `Bb7/Ab` is a dominant of C via diminished (4edo) symmetry, but also the subdominant of F according to the Fa-Mi (Bb-A) clausula (note that `IV7` is a different side of the tonality than `IV`/`IVmaj7#11` in the present cultural entrainment. To distinguish it as a cultural consonance, 7-limit harmony is often used for `b7 = Ab = 7/4`)
Notice that in the first chord `E7b9/F`, or its simplified equivalent `Fdim7`, we said that it is also the dominant of F via the diatonic plural F (`I`) ≈ Am (`iii`). This means that a single diminshed chord (Fdim7) not only resolves 4 ways (major or minor of C, Eb, F#, A), but 8 ways (major or minor of C, D, Eb, F, F#, G#, A, B), by applying the symmetry to the diatonic plural!
<iframe width="560" height="390" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A%5B5_8_11_'2%5D_%5B4_7_'0%5D._%23_C%0A%5B5_8_11_'2%5D_%5B5_9_'0%5D._%23_F%0A%5B5_8_11_'2%5D_%5B7_10_'3%5D.%23_Eb%0A%5B5_8_11_'2%5D_%5B8_'0_'3%5D.%23_Ab%0A%5B5_8_11_'2%5D_%5B6_10_'1%5D.%23_F%23%0A%5B5_8_11_'2%5D_%5B6_11_'3%5D.%23_B%0A%5B5_8_11_'2%5D_%5B4_9_'1%5D._%23_A%0A%5B5_8_11_'2%5D_%5B6_9_'2%5D._%23_D" title="Xenpaper" frameborder="0"></iframe>
<iframe width="560" height="315" src="https://www.youtube.com/embed/euE0wyRmyZs?si=uFCIYfZxGezcJ7a9&start=38" 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>
#### Augmented triad (3edo)
The augmented triad (scale degrees `1 3 #5`) being an octave symmetry was the [motivation for constructing 12edo](https://hackmd.io/@euwbah/extending-harmonic-principles-3#edos-the-hard-way-tempering-commas) in the first place.
Just like before, we relate its applications to the clausula.
<iframe width="560" height="300" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A11_'0_._%5B2_7_11%5D_%5B4_7_'0%5D_._%23_Ti-Do_of_C%0A'3_'4_._%5B6_11_'3%5D_%5B7_11_'4%5D_._%23_Ti-Do_of_Em%0A%23_Plural_C_%3D_Em%0A%5B11_'3%5D_%5B'0_'4%5D_._%5B7_11_'3%5D_%5B0_7_'0_'4%5D_%23_Gaug_-%3E_C" title="Xenpaper" frameborder="0"></iframe>
The augmented chord supports the plural between C and its fifths-coloring (extension) Em by containing the "Ti" of both the Ti-Do of C and Em.
This configuration of clausulae can appear frequently over three different roots:
<iframe width="560" height="240" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A%5B%607_5_11_'3%5D-_%5B0_4_'0_'4%5D-_.._%23_Gaug_-%3E_C%0A%5B%605_9_11_'3_'7%5D-_%5B%604_7_'0_'4%5D-._%23_F7%2311_-%3E_C%0A%5B%608_7_11_'3%5D-_%5B0_4_'0_'4%5D-_%23_Abmin(maj7)_-%3E_C" title="Xenpaper" frameborder="0"></iframe>
The augmented triad has 3edo octave symmetry, which gives us 3 different keys to resolve to:
<iframe width="560" height="315" src="https://www.youtube.com/embed/KwIC6B_dvW4?si=gJI93uSYPZkfuSGA" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
<iframe width="560" height="315" src="https://www.youtube.com/embed/OkMLYNWP-yo?si=ZWrXagbfFRQnhQl4&start=144" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
<iframe width="560" height="315" src="https://www.youtube.com/embed/s76aFImPhzw?si=0meS6MPMbd6Xv-j0&start=38" 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>
#### Tritone dyad (2edo, tritone substitution)
This is a subset (subgroup) of [the diminished tetrad (4edo)](#Diminished-tetrad-dim7-4edo), so we have already covered its rationale and applications.
For the sake of completeness, I'll explain a common 12edo centric explanation: the Ti (deg 7, 11\12) and Fa (deg 4, 5\12) of the scale form a tritone which resolves to Do (deg 1, 0\12) and Mi (deg 3, 4\12). Since it forms a tritone which has an octave symmetry, we can swap the functions of Ti and Fa, such that Ti is now scale degree 4 (the previous Fa), and Fa is degree 7 (the previous Ti), hence, instead of $4 \to 3,\ 7 \to 1$, we have $4 \to \sharp 4,\ 7 \to \sharp 6$:
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br3%5C12%7D%7B12edo%7D%0A%5B5_11%5D_%5B4_'0%5D_._%5B%607_5_11%5D_%5B0_4_'0%5D.%0A%5B5_11%5D_%5B6_10%5D_._%5B1_5_11%5D_%5B%606_6_10%5D" title="Xenpaper" frameborder="0"></iframe>
# Putting everything together
We now combine everything we have learnt from deconstructing 12edo into JI and its musical offerings by its supported temperaments.
Even though 31edo doesn't support the same temperaments as 12edo, we can still try to generalize these principles of how temperaments have influenced harmony, to create a framework to make music that is culturally familiar, not just in 31edo, but in any tuning system.
## Principle of musical puns
There's a consistent theme in what we've gleaned from 12edo so far:
* Because of temperament, one tempered interval can assume the function of various JI intervals
* And from that, we create the idea of pluralism, where one function can comprise many different sides of a tonality
* We also have symmetries, where one chord can be seen as the equivalent of several other symmetrically equivalent chords.
* And from the combination of all the above, we find that one chord can simultaneously act as one function of some key, and another function of a new key, acting as a functional bridge between keys and functions.
I like to call this phenomenon **musical puns**: the idea that one object/structure/chord can act as **many things**, not just itself, and because that structure is **multi-purpose**, it can be used as a **bridge** between musical ideas, tonalities and cultures.
This concept underpins everything in this section, we're trying to find these musical puns to connect different harmonic places together, so that we can move around freely, yet be rooted in the existing culture of music.
## Higher-limit consonances
In 12edo, we are only familiar with cultural consonances in the 3 limit (fifths and fourths) and the 5-limit (major/minor thirds and sixths), which are culturally regarded as the consonant intervals.
However, 31edo allows us to distinguish and approximate primes effectively in the 11-limit, with intervals in the JI subgroup `2.3.5.7.11`.
In order to effectively use these new worlds of intervals, we to make use of concordance that makes it easier to entrain consonance in our constructed tonalities.
There are models of dyadic (two-note) roughness and discordance that you can find on the [Xenharmonic Wiki](https://en.xen.wiki/w/Main_Page), such as [Harmonic Entropy](https://en.xen.wiki/w/Harmonic_entropy) and [Sethares' dissmeasure algorithm](https://en.xen.wiki/w/Bill_Sethares), which work on the continuous frequency spectrum, or using [JI interval heights](https://en.xen.wiki/w/Height) when thinking in terms of JI's rational ratios.
In general I find that, on top of all the usual 3 and 5-limit consonances:
* 7/4 and 7/6 can be used as strong consonances as tonic sides/states of the tonality.
* 9/7 (septimal supermajor third) can be used as 75% consonances to temporarily resolve to relative supermajor (which is relative of subminor) in order to reduce diesis shifts, but the triad will not be stable and sounds like it needs to resolve. Unlike the duality of major/minor in the sense that both can be tonics, the 7/6 subminor tonality feels "more tonicizable" than 9/7.
* 11/8 and 11/10 can be used as 50% consonances or 50% dissonances, usually for subdominant function chords, and sometimes dominant function chords (if there are other more discordant substructures), I would rank these as more crunchy versions of the major and minor seconds.
* 11/9 (neutral third) is 25% consonance and 75% dissonance. It is useful as substructures to outline some otonal upper structure triad, but on its own the neutral third doesn't make a tonality with triads, but instead it makes tonalities with melodies (like [maqam rast](https://www.youtube.com/watch?v=M9OVTNkVPXg))
Other than using concordant intervals as consonances, we can also entrain by exposure and repetition. In Part I, we already covered why [concordance is not the only factor in perceiving consonances or building tonalities](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Tonality-is-fundamentally-cultural).
## Lessons from symmetry, application to 31edo
31 is a prime number — we cannot have any octave symmetries because we can't divide 31 into any number of equal parts (other than itself or 31).
In order to apply the lessons from the symmetries of 12, we need to deconstruct the principles of why symmetry works in the first place. A symmetry, in essence, is:
1. A set of **common tones** that
2. up to a certain **interval of equivalence**, that is preferably concordant.
3. can be **rotated**
4. so that **clausulae are rotated**, **transposing** an existing culturally entrained (or constructed) **tonality** to a new root note (new "key")
5. while the set of **common tones stay the same** with respect to the interval of equivalence.
Since 31edo doesn't have any octave symmetries, choosing the octave as the interval of equivalence will not yield any symmetries. Instead, 31edo has:
1. **Detuned octave symmetries** e.g., $125/64 \equiv \monzo{-6&0&3} \equiv 30 \setminus 31$, or $729/343 \equiv \monzo{0&6&0&-3} \equiv 33 \setminus 31$ and we assume "equivalence" of these notes even though they may not be perceived as equivalent.
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A0_10_20_30_._%5B0_10_20%5D-_%5B10_20_30%5D-.%0A0_8_16_24_'1_._%5B0_8_16_24%5D-_%5B8_16_24_'1%5D-.%0A0_11_22_33_._%5B0_11_22%5D-_%5B11_22_33%5D-" title="Xenpaper" frameborder="0"></iframe>
2. **Non-octave symmetries** such as **fifth-equivalences**. E.g., we can instead rely on the second most concordant interval as an interval of equivalence (which is also the same principle as extensions/fifths-coloring. The mapping of $3/2 = \monzo{-1&1}$ in 31edo is `18\31`, a composite number that can be equally divided into 2, 3, or 6 steps.
<iframe width="560" height="260" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A0_18_._%23_1ed_fifth%0A0_9_18_._%23_2ed_fifth%0A0_6_12_18_._%23_3ed_fifth%0A0_3_6_9_12_15_18_._%23_6ed_fifth" title="Xenpaper" frameborder="0"></iframe>
We'll cover these as temperaments in [Turning temperaments into harmonic tools](#Turning-temperaments-into-harmonic-tools), rather than symmetries, as they relate less to the 12edo symmetries we're familiar with.
The thing to note about using these non-octave symmetries is that the cultural entrainment of equivalent octaves will still always be present. Usually a clausula will form between non-common tone notes when the non-octave symmetric chord is being rotated. These clausulae can be entrained into a new constructed tonality, as we will see in the following examples.
### Transferring 12edo clausulae to 31edo: lessons from augmented and diminished substitution
#### Augmented substitutions in 31edo
Unlike 12edo, `Caug` and `Eaug` are no longer the same chord up to octave equivalence.
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A10_20_'0_%5B10_20_'0%5D-_._%23_Caug%2FE%0A10_20_30_%5B10_20_30%5D-_%23_Eaug%0A" title="Xenpaper" frameborder="0"></iframe>
According to the original 5-limit Ti-Do and Fa-Mi clausulae being mapped to the interval $16/15 = \monzo{4 & -1 & -1}$, this maps to $\braket{31&49&72}{4&-1&-1} = 3 \setminus 31$ edosteps. When we consider the Ti-Do clausula applied to the augmented triads, there is only one root each augmented chord resolves to (unlike 12edo where there are 3):
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B10_20_'0%5D-_%5B13_23_'0%5D-._%23_Caug%2FE_-%3E_F%0A%5B10_20_30%5D-_%5B10_23_'2%5D-_%23_Eaug_-%3E_A%0A" title="Xenpaper" frameborder="0"></iframe>
However, even if we resolve from the "wrong" augmented triad, such as:
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B10_20_30%5D-_%5B13_23_'0%5D-._%23_Eaug_-%3E_F%0A%5B10_20_'0%5D-_%5B10_23_'2%5D-_%23_Caug%2FE_-%3E_A%0A" title="Xenpaper" frameborder="0"></iframe>
These still sound fine!
* `Eaug -> F` has the same two clausulae because `E = 10\31` and `G# = 20\31` are both common tones of `Caug` and `Eaug`. The difference is, now there is a diesis shift from `B# = 30\31` to `C = 0\31`. We know from [Part I that diesis shifts don't make good clausulae](https://hackmd.io/@euwbah/extending-harmonic-principles-1#What-semitone-cancellation-theory-amp-pitch-memory-implies-for-31edo) because it does not mark a significant change in pitch memory. However, this diesis shift does not contribute to any clausulae, so it is actually good that it doesn't cause any change, because we didn't want the note to sound like it has changed at all! In most cases, we can hide diesis shifts by surrounding it with highly discordant, or highly concordant structures, effectively tempering it out psychologically/psychoacoustically, rather than in the tuning itself.
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A30_'0._%5B10_20_30%5D%5B13_23_'0%5D-._%23_Eaug_-%3E_F" title="Xenpaper" frameborder="0"></iframe>
When isolated, the diesis shift becomes a lot more noticeable.
* `Caug/E -> A` has the original Ti-Do clausula of `G# = 20\31` to `A = 23\31` (≈ 116.1¢), but the other clausula `C = 0\31` to `C# = 2\31` is of a different size (≈77.4¢). The entrainment of the Ti-Do and Fa-Mi clausulae applies to a range of intervals.
#### Lesson: range of clausulae's tuning
Even if the diesis `128/125` was not tempered in 31edo, the "augmented symmetry" still can be used because the size of Fa-Mi and Ti-Do clausulae has some leniency (for me, anything between 70¢-120¢ passes), and the diesis shift is hidden. In 31edo, we have the diatonic semitone of `3\31` (e.g., E-F, B-C), but we also have the apotome (chromatic semitone) of `2\31` (e.g., `E-E#`, `Fb-F`), both of which strongly evoke the Ti-Do and Fa-Mi clausulae.
<iframe width="560" height="390" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A28_'0_._%5B5_18_28%5D%5B10_18_'0%5D-._%23_diatonic_semitone_Ti-Do%0A29_'0_._%5B5_18_29%5D%5B10_18_'0%5D-._%23_chromatic_semitone_Ti-Do%0A13_10_._%5B13_18_28%5D%5B10_18_'0%5D-._%23_diatonic_semitone_Fa-Mi%0A12_10_._%5B12_18_28%5D%5B10_18_'0%5D-._%23_chromatic_semitone_Fa-Mi" title="Xenpaper" frameborder="0"></iframe>
#### Diminshed substitutions
<iframe width="560" height="550" src="https://luphoria.com/xenpaper/#embed:(1%2F2)%7Br8%5C31%7D%7B31edo%7D%0A%23_resolving_to_C_maj%0A%5B%6028_5_13_21%5D_%5B0_10_18%5D_._%23_Bdim7__(B__D__F__Ab)_usual_clausulae%0A%5B%6020_%6028_5_13%5D_%5B%6018_0_10%5D.%23_G%23dim7_(G%23_B__D__F_)_%238-8%0A%5B12_20_28_'5%5D_%5B10_18_'0%5D_.%23_E%23dim7_(E%23_G%23_B__D_)_%233-3%2C_%238-8%0A%5B5_13_21_29%5D_%5B10_18_'0%5D_._%23_Ddim7__(D__F__Ab_Cb)_b1-1%0A%0A%23_far%0A%5B13_21_29_'6%5D_%5B10_18_'0%5D_._%23_Fdim7_(F__Ab_Cb_Ebb)_b1-1" title="Xenpaper" frameborder="0"></iframe>
A minor third $6/5 \equiv \monzo{-1&-1&1}$ maps to `8\31`, so 4 of it exceeds an octave by a diesis.
However, a "rotation" of the diminished chord (by adding/subtracting `8\31`) still obtains 3 out of 4 common tones each time.
Recall the original rationale for diminished chords working in 12edo was the [presence of 3 (or 4) clausulae](#Diminished-tetrad-dim7-4edo): `b6-5` (tenorizans molle), `4-3` (Fa-Mi), `7-1` (Ti-Do), and optionally `#5-6` for borrowing Ti-Do of the relative minor via pluralism/fifths-coloring.
We can also find these clausulae in slightly altered forms if we rotate the diminished tetrad in 31edo.
The default alignment for `G7b9` would be `Bdim7`, which resolves to `C maj` via all the standard clausulae, the above example shows how different alignments shrink the size of clausulae.
Notably, the E#dim7 will invoke the harmonic 7th (augmented 6th) of G, which makes it a nice **bridge** between the 5-limit harmonies of stacked `6/5`-s and the 7-limit `7/4`.
#### Lesson: Rotating too far
In the above example, rotating till `Fdim7` introduced `Ebb` from the scale degree bb7, (enharmonically equivalent to `D^` in 31edo, mapped by the septimal supermajor second $8/7 \equiv 6 \setminus 31$), which is beyond the standard major/minor tonalities that we want to work with.
Using the septimal second will require the construction of a new tonality, so it is important to be careful of which rotation of diminished is used in 31edo.
### Duality/Multiplicity: lesson from tritone substitution (and clausulae)
:::warning
:warning: Not related to Ernst Levy's dualism.
:::
In 12edo, recall that when only given a single clausula of the size `1\12`, e.g. `B-C`, without any context, it would be impossible to determine whether B-C is Ti-Do (perfect clausula, resolving), or if it is Mi-Fa (anti plagal clausula, going "out").
More context (prior tonality sides/states) is required to determine what tonality, and in what key it is. For example, if in addition to B-C, F-E (or its reverse) was present, then we'd know that B-C was Ti-Do and the tonality was C major (or A minor, etc...), but if F#-G (or its reverse) was present, then we'd know that B-C was Mi-Fa. This was the effect used in the demonstration of the changing of tonality perception of [Bach's Prelude No. 1 in C major WTC 1 in Part I](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Example-2-Prelude-No-1-C-major).
This is one way duality/multiplicity can manifest, which happens when there are **multiple clausulae of the same interval size**. This is not unique to 12edo and can happen in most tunings:
<iframe width="560" height="300" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A28_'0_.._13_%5B0_10%5D_%5B5_28%5D_%5B10_'0%5D_.._%23_B-C_as_Ti-Do%0A%0A28_'0_.._18_15_%5B10_28%5D_%5B5_'0%5D_%5B15_23%5D_18-_%23_B-C_as_Mi-Fa" title="Xenpaper" frameborder="0"></iframe>
Later on, we'll find a harmonic tool enabled by the [Bayatismic temperament](#Biyatismic-121120) which invokes this form of duality/multiplicity.
But there's **another kind of duality** that only happens as a result of the 2edo symmetry in 12edo: even if we have **both** of the starting notes of both clausulae Ti-Do and Fa-Mi, (i.e., Ti & Fa), we still cannot guess the key & tonality. The 2edo symmetry of Ti `11\12` and Fa `5\12` means we cannot actually discern if Ti and Fa was `11\12` and `5\12` respectively, or the other way around., until more notes are played.
However, in 31edo, we can't divide 31 into 2, which means we have to make a choice about which Ti and which Fa we are using even before proceeding to resolve to Mi and Do, because the choice of Ti and Fa affect the spelling of notes!
Recall that, since 31edo is a meantone temperament, simply naming our major scales with the correct enharmonics (i.e., for F# major, we write `F# G# A# B C# D# E# F#` instead of `F# Ab A# Cb Db D# F Gb`), will yield the 5-limit major scale that we've worked hard to obtain in [Part II](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Scales).
This means that if B is Ti, then enharmonically, that implies Fa must be F, but if B is Fa, then enharmonically, Ti must be E#, not F.
<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A28_13_._%5B28_13%5D_%5B'0_10%5D_.._%23_B_as_Ti_implies_F_is_Fa%0A28_12_._%5B28_12%5D_%5B25_15%5D_.._%23_B_as_Fa_implies_E%23_is_Ti" title="Xenpaper" frameborder="0"></iframe>
But notice what happens when we intentionally flip the enharmonics:
<iframe width="560" height="360" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%23_flipped%0A%5B28_12%5D_%5B'0_10%5D_.._%23_B_E%23_-%3E_C_E%0A%5B28_13%5D_%5B25_15%5D_.._%23_B_F_-%3E_A%23_F%23%0A%0A%23_diatonic%0A%5B28_13%5D_%5B'0_10%5D_.._%23_B_F_-%3E_C_E%0A%5B28_12%5D_%5B25_15%5D_.._%23_B_E%23_-%3E_A%23_F%23" title="Xenpaper" frameborder="0"></iframe>
They still sound valid! Remember that [clausulae can have variable sizes](#Transferring-12edo-clausulae-to-31edo-lesson-from-augmented-and-diminished-substitution), and `2\31` and `3\31` can have similar clausulae functions in 31edo.
In the flipped variants, if we add the root note (scale degree 5) of the dominant chord, notice that we get the spelling of [augmented 6th chords](https://en.wikipedia.org/wiki/Augmented_sixth_chord) (as in scale degree #6, not the modern chord name `aug6` as in `#5(add6)`) in classical ECT theory.
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%23_aug_6th_chords%0A%5B%6018_12_28%5D_%5B0_10_'0%5D_.._%23_G_E%23_B%0A%5B3_13_28%5D_%5B%6015_15_25%5D_.._%23_Db_F_B" title="Xenpaper" frameborder="0"></iframe>
If we analyze the interval of $\sharp 6$ in 31edo, as in [stacking up 10 fifths](https://hackmd.io/@euwbah/extending-harmonic-principles-3?stext=32455%3A13%3A0%3A1736826939%3AYvTrIO), we obtain the mapping
$$\braket{31 & 49}{-15&10} = 25 \setminus 31$$
And if we look at the octave reduced 7th harmonic $7/4 \equiv \monzo{-2&0&0&1}$, it is mapped to:
$$\braket{31&49&72&87}{-2&0&0&1} = 25 \setminus 31$$
which is the same interval!
:::info
:pencil: **Exercise**
Which comma is tempered out in the above example?
:::spoiler Answer
$$\monzo{13&-10&0&1} = \frac{59049}{57344}$$
which is [Harrison's comma](https://en.xen.wiki/w/Harrison%27s_comma)
:::
<!--:::-->
If we entrain the interval $7/4$ as a higher-limit consonance, we can use the augmented 6th as another type of dominant chord which can also function as tonic.
However, using this variant of the dominant chord makes it tricky to use scale degree 4 `13\31`, because here our 'Fa' is mapped to `12\31`. We have to construct other tonalities that can make use of **musical puns** to navigate around this lowered 4th scale degree, or sneakily hide the diesis shift somehow:
<iframe width="560" height="480" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%23_subminor_tonality%2C_D_submin_has_Fv%0A%5B2_10_18_25%5D-%5B5_12_23%5D-%5B%6018_12_28%5D-_%5B0_10_'0%5D-_..%0A%23_sneaking_E%23_as_Aaug%2C_D_submin9_with_Ev_(neutral_3rd_of_C)%0A%5B0_18_'10_'18%5D-%5B%6023_23_'2_'12%5D-%5B5_12_23_30_'9%5D-%5B%6018_12_28_'5%5D-_%5B0_10_'0%5D-..%0A%23_D_primodal-6_tonality%2C_using_Ebdim7_to_anchor_C%5E%0A%5B0_18_28_'10%5D-%5B13_20_26_'5%5D%5B9_16_24_'1%5D%5B5_12_23_'1%5D-%5B%6018_12_28_'5%5D-%5B0_10_'0%5D-" title="Xenpaper" frameborder="0"></iframe>
:::info
:writing_hand: **Exercise**
In the last progression of the above example, the chord before `Ebdim7 [9 16 24 '1]` is `Bb!7/F [13 20 26 '5]` (harmonic dominant). Assuming [we allow diminished symmetries](#Transferring-12edo-clausulae-to-31edo-lesson-from-augmented-and-diminished-substitution) (even without the comma(s) tempered), what are some ways view the function of these chords in this context?
:::spoiler Answer
If we view `Bb!7/F` as a [7-limit consonance](#Higher-limit-consonances) (due to its concordance as an otonal chord), then it functions as both its own tonic (of Bb) as well as a variant of the diminished symmetric dominant `G7db9` (where the 7th harmonic `vb7`/`db7` of Bb is the `db9`/`#8` of G), which makes it a dominant of C via the `#5-5` and `4-3` clausulae (with respect to C).
With the `Bb!7/F` as a bridge acting as its own tonic, the next chord `Ebdim7` is the dominant of this new tonic, via another diminished symmetry that still maintains three clausulae: `4-3` (Eb-D), `b6-5` (Gb-F), and `7-1` (A-Bb) (with respect to Bb).
The "walk-down" chords from `Ebdim7` to `D smin9` are relying on another cultural entrainment of modern harmony:
<iframe width="560" height="315" src="https://www.youtube.com/embed/F4YFwtupS48?si=ISyGckWYGgwSJaT2&start=4826" 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>
Draw out the tonality state diagram that describes the above chords for a better picture.
:::
<!--:::-->
As discussed in Part I, it is hard to entrain the diesis as a functional movement, so we don't want to [make the diesis shift obvious](https://hackmd.io/@euwbah/extending-harmonic-principles-1#What-semitone-cancellation-theory-amp-pitch-memory-implies-for-31edo):
<iframe width="560" height="270" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B%6013_13_23%5D-%5B%6018_12_28%5D-_%5B0_10_'0%5D-_.._%23_diesis_shift_noticeable%2C_weird%0A%0A13-12-_%23_%3F%3F" title="Xenpaper" frameborder="0"></iframe>
## Turning temperaments into harmonic tools
### Meantone 81/80
We already have gone through this extensively in Part III. Meantone unlocks [diatonic pluralism](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Diatonic-Pluralism), and by extension, [chromatic pluralism](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Chromaticism-in-extended-pluralism).
### Biyatismic 121/120
The [Biyatismic](https://en.xen.wiki/w/Catalog_of_rank-4_temperaments#Biyatismic_.28121.2F120.29) temperament tempers the [Biyatisma](https://en.xen.wiki/w/121/120) comma `121/120`, which can be seen as the difference between `11/10` and `12/11`, two adjacent superparticular ratios in the harmonic series. The 11th harmonic splits the 5-limit minor third `6/5` into two parts, and in 31edo, this is split equally into two sesquitones (3/4 tones) because it tempers the Biyatisma `121/120`.
> :pencil: Presumably named by Gene Ward Smith, the name comes from maqam bayati, which contains the two sesquitones forming the first nucleus of the maqam.
> <iframe width="560" height="315" src="https://www.youtube.com/embed/1DrzYlc8DAM?si=dXVV8x8W6o2UUnzR" 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 (undecimal) neutral second $4 \setminus 31 \approx 154.8¢$ (e.g., `E-F^`, `Gbb-G`) can also have a similar effect to the Fa-Mi or Ti-Do clausula, in the sense that it [contributes a strong movement in the pitch memory](https://deutsch.ucsd.edu/m/DD_F.jpg) (recall the principle of [Pitch Memory](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Pitch-memory-amp-semitone-cancellation-theory)), but the interval is far away enough from either the 12edo semitone `1\12` or tone `2\12`, that it is perceived as its own class of intervals.
<iframe width="560" height="170" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A27_'0_._%5B7_18_27%5D%5B10_18_'0%5D-." title="Xenpaper" frameborder="0"></iframe>
But the beauty of the neutral second `4\31` is that it divides the minor third into two equal parts, meaning we can entrain two clausulae of the same interval size, analogous to how Ti-Do and Fa-Mi are the same size, to create [duality](#DualityMultiplicity-lesson-from-tritone-substitution-and-clausulae). However, instead of keys ending up being a tritone apart if we swap Ti and Fa, in this case, swapping the roles of the `4\31` clausula result in chords a fourth apart if we entrain them according to to the sequence of harmonics `10:11:12` like so:
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A27_'0_._%5B7_18_27%5D%5B10_18_'0%5D-.%0A27_23_._%5B7_18_27%5D%5B0_13_23%5D-.%0A%0A%5B%6023_13_'0%5D--%5B%6013_0_23%5D%5B%6013_7_18_27%5D-%5B%6017_4_17_27%5D-%0A%5B%6019_9_14_27%5D--%5B%609_%6027_9_19%5D%5B%609_3_14_23%5D-%5B%6013_0_13_23%5D-%0A%5B%6015_5_10_23%5D--%5B%605_%6023_5_15%5D%5B%605_%6030_10_19%5D-%5B%609_%6027_9_19%5D-%0A%5B%6011_1_6_19%5D---" title="Xenpaper" frameborder="0"></iframe>
We can also entrain a similar tonality to maqam bayati (built on C, it is `C Dv Eb F G Ab Bb C`, with nucleus/finalis on C Eb and F), but with the addition using `Dv = 4\31`, the middle of the minor third `C-Eb` as the 11-th harmonic semi-consonance `11/8` of `Ab`:
<iframe width="560" height="260" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B%600_4_8_18%5D-%5B%600_4_21%5D%5B%600_8_26%5D%0A%5B%6021_15_'4%5D-%5B%6021_13_'0%5D-%0A%5B%6013_0_8_26%5D-%5B%608_4_15_21%5D-%0A%5B%6014_%6029_4_18%5D-%5B%6018_%6029_8_18%5D-%5B%600_0_18%5D-" title="Xenpaper" frameborder="0"></iframe>
Here, we use the `4\31` interval as a clasula not only in between the minor third (i.e, `Dv-Eb`, and `Dv-C`), but also up to the fifth as a type of subdominant `F^-G` ($14 \to 18$ edosteps), via the chord `Dv!11/F#v` (the 7th and 11th harmonic dominant chord). The 11th harmonic of `Dv` is `G`, which allows G to anchor the cadence as a common tone, and instead of `2-5-1` as the root notes of our subdominant-dominant-tonic progression (like the minor `ii0-V-i` in jazz theory), we have `v2-5-1`, but we hide the awkward interval between `v2-5`.
Also, notice the second last chord is `Cbaug/G` (rather than Gaug $\texttt{[`18 `28 7 18]}$), because we don't want the diesis shift from $29 \to 28$, and we also want to invoke the "Ti-Do" (`Dv-Eb` $4 \to 8$) (rather than introduce a new clausula and note $4 \to 7$). This comes from the principle of scales: we want to reduce the number of unique notes and movements to ease the perception of the song, especially if we are entraining new tonalities, repeating movements and notes are always good.
#### Lesson: Hiding "bad intervals" with clausulae & octave displacement
The reason why that "root motion" `v2-5` didn't sound out of tune was because we didn't exaggerate the detuned fourth between `v2` and `5`, instead we hid it by
1. Making the clausula `^4 = F^` go up to the `5 = G`, so another voice takes the sound of the `5`, rather than having a voice going `v2-5` directly like:
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B%604_%6014_%6029_18%5D-%5B%6018_%6028_18%5D-.%0A%5B%604_4%5D-%5B%6018_18%5D-_%23%3F%3F" title="Xenpaper" frameborder="0"></iframe>
2. Not including the 5th of the G chord, which would highlight the diesis shift from `Dv` to `D`:
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B%6014_%6029_4_18%5D-%5B%6018_%6028_5_18%5D-.%0A4-5-_%23%3F%3F" title="Xenpaper" frameborder="0"></iframe>
Even though the diesis shift is not _that_ obvious, it evokes a similar cultural entrainment as the backdoor-type subdominant, where an non-standard clausula is combined with an expected clausula creating a shift that suddenly increases the prominence of the 5 of 5 (i.e., `2 = D`):
<iframe width="560" height="240" src="https://luphoria.com/xenpaper/#embed:(1%2F2)%7Br8%5C31%7D%7B31edo%7D%0A%5B%6014_%6029_4_18%5D-%5B%6018_%6028_5_18%5D-.(1)%0A%5B%6026_13_21_'5_'13%5D-%5B%6026_13_18_'0_'10%5D%5B%6026_13_21_'5_'13%5D%5B%6018_5_18_28_'5%5D---" title="Xenpaper" frameborder="0"></iframe>
This is the same principle of choosing clausulae with intention and regard for pitch memory as referenced in the (negative) example of [using pluralism without principles](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Dont-forget-the-principles) in Part III.
Another way to hide these awkward intervals or diesis shifts is to displace them by octaves, so that the ear doesn't pick up on these shifts so easily, which is actually what happened between the chords $\texttt{[`8 4 15 21]} \to \texttt{[`14 `29 4 18]}$
### Mothwellsmic 99/98
The [mothwellsma](https://en.xen.wiki/w/99/98) is the comma between the interval $9/7 \equiv \monzo{0&-2&0&1}$ (the septimal supermajor third, between the 9th and 7th harmonic, mapping to 435¢ in JI), and the interval $14/11 \equiv \monzo{1&0&0&1&-1}$ (the undecimal major third/pentacircle major third mapping to 417.5¢ in JI):
$$\frac{9}{7} \div \frac{14}{11} = \frac{99}{98}$$
In JI, expressing these ratios as higher-limit consonances between harmonics:
<iframe width="560" height="360" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%0A7%2F4_9%2F4_._%5B4%3A5%3A7%3A9%5D--.%0A%0A%7Br14%2F11%7D%0A11%2F8_14%2F8_._%7Br1%2F2%7D_%5B4%3A6%3A10%3A11%3A14%5D--%0A%7Br440hz%7D%7Br8%5C31%7D%7Br1%2F2%7D%0A%5B1%2F1_5%2F4_7%2F4_9%2F4%5D-%5B7%2F11_21%2F22_35%2F22_7%2F4_49%2F22%5D-%5B1%2F1_5%2F4_7%2F4_9%2F4%5D-%5B7%2F11_21%2F22_35%2F22_7%2F4_49%2F22%5D-" title="Xenpaper" frameborder="0"></iframe>
In 31edo, both $\frac{9}{7}$ and $\frac{14}{11}$ map to the same `11\31` (supermajor third, one diesis above the mapping of `5/4` major third `10\31` in 31edo). If we were to map those same JI ratios to 31edo using the default edo mapping $\cov{31&49&72&87&107}$:
<iframe width="560" height="200" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B0_10_25_'5%5D-%5B%6011_%6029_21_25_'5%5D-%5B0_10_25_'5%5D-%5B%6011_%6029_21_25_'5%5D-" title="Xenpaper" frameborder="0"></iframe>
We now get the [musical pun](#Principle-of-musical-puns) of the two different JI ratios being the exact same interval, but the expense of sounding less stable in 31edo, because the error of the prime interval 11 is -9.4¢, which is rather high compared to the [rest of the prime intervals' mappings](https://en.xen.wiki/w/31edo#Prime_harmonics).
But the magic isn't just that these two intervals are the same, that alone is rather trivial considering that any finite-rank tuning system (and edos are rank-1, 1 is definitely finite) will have an infinite number of unique, linearly independent commas tempered out.
The tool that we use here is that
1. the chord `11:14:18`, which I call the **mothwell (superaugmented) triad** (not to be confused with [mothwellsmic triad](https://en.xen.wiki/w/Mothwellsmic_chords)) comprises exactly these two intervals $14/11$ and $9/7$ (reduced $18/14$), and
2. this chord is a concordant otonal structure built of the harmonic series that can be entrained as a semi-consonance:
<iframe width="560" height="330" src="https://luphoria.com/xenpaper/#embed:(1)%7Br16%5C31%7D%7B31edo%7D%0A%5B0%5D___________________%23_1%2F1%0A%5B0_10%5D________________%23_4%3A5%0A%5B0_10_18%5D_____________%23_4%3A5%3A6%0A%5B0_10_18_25%5D__________%23_4%3A5%3A6%3A7%0A%5B0_10_18_25_'5%5D_______%23_4%3A5%3A6%3A7%3A9%0A%5B0_10_18_25_'5_'14%5D--_%23_4%3A5%3A6%3A7%3A9%3A11" title="Xenpaper" frameborder="0"></iframe>
I like to call this chord the "Duke Ellington chord", just because it reminds me of the dominant `7#11` voicings in the band, or the "Chelsea Bridge chord", especially when the melody is intentionally bent flat:
<iframe width="560" height="315" src="https://www.youtube.com/embed/NGBw09LUzKw?si=GW17AxKdT86P-fpM" 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>
If we also choose to consider 12edo in the 11-limit subgroup `2.3.5.7.11` with the mapping for $7/4 \approx 10 \setminus 12$ and $11/8 \approx 6 \setminus 12$, then 12edo also tempers this exact same comma in the exact same way, where it forms, not only a triad stacked from two `4\12` major thirds, but a symmetric 3edo structure (augmeted triad), since $4 \times 3 = 12$.
In 31edo, this structure is not symmetric (if it is, the interval of equivalence is `33\31`, which is a chromatic semitone above an octave, a bit of a _stretch_ to call that an octave). However, rotating this structure by `11\31` will result in 2 common tones and a chromatic semitone `2\31` clausula. If I apply both rotations and inversions to keep the 3 voices of the mothwellsmic triad
<iframe width="560" height="480" src="https://luphoria.com/xenpaper/#embed:%7B31edo%7D%7Br8%5C31%7D%0A(1%2F2)%0A36_34_._%5B0_14_25_36%5D-_%5B%6020_14_25_34%5D-.%0A%0A(1%2F3)%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>
We actually already have thoroughly covered the applications of the mothwellsmic temperament to a constructed subminor & primodal-6/orwell tonality [all the way back in Part I](https://hackmd.io/@euwbah/extending-harmonic-principles-1#The-symmetric-Mothwell-transformation), but now's a good time to revisit that section and analyze it with the new principles, and language of JI, mappings, and temperaments that we've gone through.
:::info
:pencil: **Exercise**
Revisit the [subminor & orwell/primodal-6 tonality](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Subminor-amp-orwellprimodal-6-tonalities) in Part I and convert (detemper) edosteps to JI ratios. If there are multiple interpretations for JI ratios, write all of them that you think are significant in terms of contributing [musical puns](#Principle-of-musical-puns) to the tonality.
:::
### Orwellismic 1728/1715 & nuwell 2430/2401
The comma $1728/1715 = \monzo{6&3&-1&3}$, known as the [orwellisma](https://en.xen.wiki/w/1728/1715), is formed between three stacks of $7/6 \equiv \monzo{-1&-1&0&1}$, i.e., $343/216 \equiv \monzo{-3&-3&0&3}$ and a 5-limit minor sixth (minor third plus a fourth): $8/5 \equiv \monzo{3 & 0 & -1}$.
In 31edo, both of these map to the same interval `7\31` (subminor third):
<iframe width="560" height="260" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A0_7_14_21_._%5B0_7_14_21%5D-_%5B0_21%5D-.%0A0_8_21_._%5B0_8_21%5D-..%0A%0A%5B0_7_14_21%5D-%5B0_10_18%5D-" title="Xenpaper" frameborder="0"></iframe>
and in fact, four stacks of $7/6$ will give us a major 7th, which is made possible by tempering out the [nuwell comma](https://en.xen.wiki/w/2430/2401).
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A0_7_14_21_28_._%5B0_7_14_21_28%5D-_%5B0_28%5D-.%0A0_10_18_28_._%5B0_10_18_28%5D-.%0A%5B0_7_14_21_28%5D-%5B0_10_18_23%5D-" title="Xenpaper" frameborder="0"></iframe>
Despite how wild this temperament is in theory, there's a familiar feeling to these chords here, especially in those example progressions.
Let's go back to principles and analyze the clausulae we have at hand to see why it may be familiar.
* `C` is a common tone between both chords.
* `D#-E` creates the classic `7-1` Ti-Do of Em which is a plural tonic of C (`#2-3`) (refer back to [diminished tetrad symmetries in 12edo](#Diminished-tetrad-dim7-4edo))
* `Ab-G` creates a Fa-Mi of the tenorizans molle (`b6-5`), or also can be see as Fa-Mi of the relative major if the parallel C minor was taken as the root (modal interchange/PLR transformations)
* `F^-G` is the bayatismic sesquitone `4\31` movement which can be entrained as the 11th harmonic moving up to the 5 (or down to the 3). I drew this motion with a dotted line because the Ab-G motion wins by familiarity within the context of ECT, so it's more probable that Ab-G will be perceived first than `F^-G`.
* Optionally the note `A` can be present in the resolved chord to form the chord `C6`, and we also can have the clausula `Ab-A` (which is an intonation variant of `G#-A`, the relative minor plural Ti-Do).
Indeed, this clausula are very similar to these in 12edo:
<iframe width="560" height="230" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B12edo%7D%0A%5B0_3_6_9%5D_%5B0_4_7%5D-.___%23_Cdim7_-%3E_Cmaj%0A%5B0_3_6_8%5D_%5B0_4_7_9%5D-._%23_Ab7%2FC_-%3E_C6%0A%5B0_3_6_11%5D_%5B0_4_7_9%5D-._%23Cdim(maj7)_-%3E_C6" title="Xenpaper" frameborder="0"></iframe>
E.g.:
<iframe width="560" height="315" src="https://www.youtube.com/embed/wGKlEY3UdJg?si=uCYJveM0byyIlHoH" 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>
And just as how stacking minor thirds yields an octave symmetry in the form of common tones and common clausulae in 12edo:
<iframe width="560" height="300" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B12edo%7D%0A%5B0_3_6_11%5D_%5B0_4_7_9%5D%0A%5B3_6_9_'2%5D_%5B3_7_10_'0%5D%0A%5B6_9_'0_'5%5D_%5B6_10_'1_'3%5D%0A%5B9_'0_'3_'8%5D_%5B9_'1_'4_'6%5D%0A%5B'0_'3_'6_'11%5D_%5B11_'3_'6_'9%5D_%5B4_11_'3_'8%5D--" title="Xenpaper" frameborder="0"></iframe>
We can stack (up to five!) subminor thirds to yield a non-octave symmetry, but still has common tones and common clausulae:
<iframe width="560" height="315" src="https://luphoria.com/xenpaper/#embed:(1)%7Br8%5C31%7D%7B31edo%7D%0A%5B0_7_14_21_28%5D_%5B0_10_18_23%5D%0A%5B7_14_21_28_'4%5D_%5B7_17_25_30%5D%0A%5B14_21_28_'4_'11%5D_%5B14_24_'1_'6%5D%0A%5B21_28_'4_'11_'18%5D_%5B21_'0_'8_'14%5D%0A%5B28_'4_'11_'18_'25%5D_%5B28_'7_'15_'21%5D_%5B'0_'10_'18%5D--_%23!!!" title="Xenpaper" frameborder="0"></iframe>
# Summary
Our journey began [naively choosing the closest 31edo tuning](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Why-not-just-approximate-frequencies) to the notes that we were familiar with in 12edo, but it did not work because it [inconsistently mapped the same interval in 12edo to differently sized intervals](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Whats-the-problem) in 31edo.
The point of this was to develop a theory around 31edo harmony that was culturally familiar to our conception of modern 12edo harmony, and that begun with studying [cultural entrainment](https://hackmd.io/@euwbah/extending-harmonic-principles-1#1-Cultural-entrainment).
In cultural entrainment we encountered the principle of [clausulae: the history and etiology of voice leading](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Voice-Leading), empirically backed by the psychoacoustic phenomenon of [pitch memory & semitone cancellation](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Pitch-memory-amp-semitone-cancellation-theory), which we can generalize to a [larger range of intervals](https://deutsch.ucsd.edu/m/DD_F.jpg).
We also covered how tonality is [fully a product of cultural entrainment and familiarity](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Tonality-is-fundamentally-cultural), rather than any mathematical model of perception, or the Pythagorean ideal of mathematical beauty. However, the [lineage of music from the ECT](https://hackmd.io/@euwbah/extending-harmonic-principles-2#It-matters-because-concordance-is-part-of-the-cultural-entrainment-of-western-tonality) has long included theorists who have [injected mathematics into the culture](https://hackmd.io/@euwbah/extending-harmonic-principles-1#But-culture-can-always-make-its-own-rules-around-physical-phenomena), such that we have a subjective notion of consonance and dissonance that is roughly based on physically or mathematically measurable concordance.
We covered [tonality (and the interaction/bridges between tonalities) being represented as a state diagram](https://hackmd.io/@euwbah/extending-harmonic-principles-1#Tonality-as-a-state-diagram) of particular actions (clausulae & note additions) we are culturally entrained to expect upon being on a particular state/side of a tonality, with predefined movements between states. While tonality is a probabilistic and subjective perception of the infinity, the state diagram is a finite model that approximates tonality with greater detail and freedom than other restricted models of harmony, at the [cost of verbosity](https://hackmd.io/@euwbah/extending-harmonic-principles-1?stext=40993%3A44%3A0%3A1736892151%3AUkFS4G).
The state diagrams encode information about the clausula available coming from each side/state of the tonality, and we have [briefly gone over several tonalities available to us in 12edo.](https://hackmd.io/@euwbah/extending-harmonic-principles-1?stext=40993%3A44%3A0%3A1736892151%3AUkFS4G), which we now have the gargantuan task of generalizing and expanding to 31edo, without erasing or forgetting the impact of any of the existing culturally entrained tonalities.
[We went through an example of multiple interacting 31edo tonalities](https://hackmd.io/@euwbah/extending-harmonic-principles-1#31edo-tonalities), but only covered the cultural aspects of the thought process behind choosing those specific movements. Why they were even feasible or possible in the first place, and how do we know that these tonalities can be culturally entrained as a constructed tonality as an extension of existing tonality was not yet known.
For that, we went back to the absolute basics, and covered the [origin of notes](https://hackmd.io/@euwbah/extending-harmonic-principles-2#The-harmonic-series) and [intervals](https://hackmd.io/@euwbah/extending-harmonic-principles-2#What-are-intervals-Logarithms), both as naturally occuring physical phenomenon and our [subjective, logarithmic perception](https://hackmd.io/@euwbah/extending-harmonic-principles-2#So-what-do-logarithms-have-to-do-with-pitch) of them. We used the [fundamental theorem of arithmetic](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Decomposing-intervals-into-prime-harmonics-amp-naming-notes) as a way to name these notes and intervals, by the prime intervals that decide the [octave numbering](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Prime-interval-2-octave), [nominals and pythagorean accidentals](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Prime-interval-3-fifth--octave), and [higher-limit accidentals to adjust by commas to name higher-limit notes](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Prime-interval-5-major-third--2-octaves-amp-what-does-major-third-even-mean).
Along the way, we also covered [classical interval naming conventions](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Music-theory-crash-course-interval-names-in-classical-music-theory), [scale degrees](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Scale-degree-notation), as well as [chord names](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Chords), so that we can maintain some semblance of mutual intelligibility with the existing music theory of today, while cultivating a new way of seeing these existing nomenclature without the bias of 12edo, but still culturally rooted in the ECT.
Next, we had to see [notes with JI ratios as points in space represented as monzos](https://hackmd.io/@euwbah/extending-harmonic-principles-2#Monzo), and intervals as distances between those points, because the notion of intervals living in high-dimensional space, and each prime being its own dimension, allows us to develop the [spatial intuition of temperaments](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Rank-nullity-theorem-How-many-intervals-to-temper-out-before-it-becomes-an-edo) (which we now understand as the rank-nullity theorem of linear algebra).
But before that, we were motivated to consider temperaments because of the [cultural and cognitional importance of scales](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Scales) as tools for reducing the vast infinity into a manageable few and giving it names. Yet, the [impossibilities and "strange loopyness" of the major scale in JI](https://hackmd.io/@euwbah/extending-harmonic-principles-3#The-syntonic-comma-problem) meant that we had to think of tempering the [meantone comma](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Meantone).
Once we've obtained our temperament, we had to come up with a [new way to obtain our intervals via mapping](https://hackmd.io/@euwbah/extending-harmonic-principles-3#14-comma-meantone-as-a-mapping), because they were no longer rational ratios. Using this, we could finally listen to the intervals of our temperament.
We covered the [impact of meantone on modern harmony of the ECT](https://hackmd.io/@euwbah/extending-harmonic-principles-3#What-meantone-enables-for-modern-harmony), and how it enabled pluralism, but also how it [still had infinite notes](https://hackmd.io/@euwbah/extending-harmonic-principles-3#But-a-temperament-like-meantone-is-not-always-an-edo), and we had to [temper even further](https://hackmd.io/@euwbah/extending-harmonic-principles-3#Symmetries-and-edos) to obtain an edo.
Or, we could just [start with an edo](https://hackmd.io/@euwbah/extending-harmonic-principles-3#edos-the-easy-way-Constructing-directly) and work backwards to find what commas were tempered.
We noticed that in 12edo, we could [divide it evenly across the octave in a multitude of ways, giving rise to octave symmetries](#Symmetries-of-12edo-music-amp-its-applications) that created functional equivalences that, together with pluralism, extended the possiblities of 12edo even further, and [created many bridges between tonalities](#Diminished-tetrad-dim7-4edo).
We got stumped by 31 being a prime number, which meant 31edo had no octave symmetries, but that didn't stop us from distilling the lessons we've learnt from symmetries and pluralism into principles, and these principles are the foundation of music in the ECT, showing itself in the lineage of ECT from 3000 BC till the present:
* Importance and permanence of cultural entrainment
* Ability to construct new tonalities by being consistent to the constructed finite-state model of tonality, [pandering to concordance](#Higher-limit-consonances), and not going against existing entrainments
* Pitch memory
* Old and new clausula
* [Musical puns](#Principle-of-musical-puns) to create [duality & multiplicity](#DualityMultiplicity-lesson-from-tritone-substitution-and-clausulae)
* Building bridges between tonalities, eras, and musical cultures with musical puns.
* Mathematical temperaments as portals, with [each temperament unlocking a unique harmonic tool](#Turning-temperaments-into-harmonic-tools) in the palette of tonalities.
My goal of writing this was to push the spirit of curiousity and love of music out into the world, for anyone who wants to resonate. These ideas are all opinionated and not academic, but they are a journal and summary of my musical life. I've always had questions about music that no one around me could answer, so these were the best answers I could come up with. I just feel these things should be written down somewhere, just in case something good may come from it.
------------------------
Congratulations on making it to the end, and thank you for reading this far!
I am open for queries, lessons and collaboration. Reach me [@euwbah](https://www.instagram.com/euwbah/) on instagram.
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# Glossary
*[apotome]: The semitone interval contributed by a sharp or flat. In JI, this is 2187/2048
*[cents]: 1/100ths of a 12edo semitone
*[cent]: 1/100ths of a 12edo semitone
*[comma]: Small tuning discrepancy between intervals
*[detemper]: Reinterpreting an interval from a tempered tuning system as a JI interval
*[detempering]: Reinterpreting an interval from a tempered tuning system as a JI interval
*[diatonic]: Acccording to the 7 notes of the major scale, instead of fixed intervals.
*[diesis]: One step of 31edo, or 128/125 in just intonation
*[dieses]: Plural of diesis, steps of 31edo or 128/125
*[discordance]: The part of dissonance that is objectively identifiable, but with no inherent favor or preference for concordance or discordance
*[concordance]: The opposite of discordance
*[ECT]: European classical tradition
*[edo]: Equal division of the octave
*[edos]: Tunings which are based on equal divisions of the octave
*[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.
*[octave symmetry]: If we assume octave-equivalence, then the set of notes has some kind of symmetry
*[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.
octave symmetry
: If we assume octave-equivalence, then the set of notes has some kind of symmetry
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
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