User:FloraC/Analysis on the 13-limit just intonation space: episode i

This page is a work in progress.

Preface

Just intonation is the repertoire of pitch materials that is the most concerning. In this essay, it refers to the infinite-dimensional harmonic space consisting of all rational numbers, sometimes dubbed as rational intonation.

It should take no effort to recognize the source of its significance, but making it clear will benefit. Just intonation is an elephant in the room – in that one cannot turn a deaf ear to it. Again, it matters to point it out that, analytically, inharmonicity is not the same as harmonicity, just like fractions are not the same as integers. Additions and multiplications in the harmonic space always result in something in the said space. In math, it is called a ring. The same cannot be said of arbitrary inharmonic spaces.

Of course, simply disregarding irrational intervals is not doing it just, as explained in my previous essay There Is Not a Third Side of the River, since the semioctave, for example, is a highly characteristic sound just as important as those rational intervals. These quadratic irrationals will be addressed in the later sections. It is an indispensible part of this analysis.

As for the artificial attempts where utterly irrelevant entities are forced together, like using π as frequency ratio of the sound, I have criticized in Fundamental Principles to Musical Sense and have no reason to discuss even further.

The 13-limit just intonation is of particular interest for multiple reasons. The structure is neat and cognitively accessible for homo sapiens, yet is sophisticated enough to breed rich harmonic gestures and tempering options. Each prime harmonic in this space has a relatively distinct identity, which unfolds fairly consistently, unlike higher primes.

This essay aims to be objective. Explanations on my favorite temperaments will not be present – at least not in sufficiency to single them out.

Now let us start with Pythagorean tuning aka 3-limit just intonation.

Chapter I. How Pythagoras Broke the Tuning

The Pythagorean tuning or 3-limit just intonation is the backbone of the interval space. My classification of Pythagorean intervals is exactly the same as Functional Just System (FJS), so I guess I should only cite it.

FJS technique #1: to convert from a Pythagorean ratio to an FJS representation

1. Factorize the ratio.

2. Initially ignore octaves (powers of two).

3. If the power of three is positive, move that many steps by fifths clockwise; if negative, move anticlockwise. Convert that number to an interval.

4. Adjust octaves as required.

Example: To convert 9/8 to the FJS, we factorize: 2^-3 3². We ignore the factor of two. The power of three is +2, so we move two fifths clockwise: C–G–D. We have a major second. No octave adjustment needs to be made. The answer is M2.

FJS Technique #2: to convert from an FJS representation of a Pythagorean ratio back to the ratio.

1. Initially ignore octaves.

2. Convert the interval to the number of steps by fifths, name it n.

3. Calculate red(3ⁿ)

4. Adjust octaves as required.

Example: To convert the FJS interval m3 to a Pythagorean ratio, we convert it first to -3 fifths: C–F–B♭–E♭. We now raise 3 to that power: 3^-3. This is 1/27. To bring this number between 1 (inclusive) and 2 (exclusive), we multiply by 32 to get the answer: 32/27.

and

This is pretty boring for now.

This tuning naturally gives us the 7-tone diatonic scale and the 12-tone chromatic scale. Based on the diatonic scale, each interval is assigned a diatonic degree by the mapping of 7et:

[math]\displaystyle{ \langle \begin{matrix} 7 & 11 \end{matrix} ] }[/math]

Each degree changes its size through rotation. As such, each interval is assigned major or minor based on their size. Meanwhile, the chromatic scale can be generated this way (discarding either of the tritones), and implies the mapping of 12et:

[math]\displaystyle{ \langle \begin{matrix} 12 & 19 \end{matrix} ] }[/math]

Together, as is found out by Mike Battaglia, the change of basis

[math]\displaystyle{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \rightarrow \begin{bmatrix} 7 & 11 \\ 12 & 19 \end{bmatrix} }[/math]

is helpful for Pythagorean interval classification. In particular, it is easy to identify the following tempered monzos:

[math]\displaystyle{ [ \begin{matrix} 1 & 1 \end{matrix} \rangle }[/math]

is a diatonic semitone, in this case 256/243. A movement by this would change the diatonic degree as well as the chromatic degree.

[math]\displaystyle{ [ \begin{matrix} 0 & 1 \end{matrix} \rangle }[/math]

is a chromatic semitone, in this case 2187/2048. A movement by this does not change the diatonic degree but only the chromatic degree.

[math]\displaystyle{ [ \begin{matrix} 1 & 0 \end{matrix} \rangle }[/math]

is an enharmonic diesis, in this case the Pythagorean comma (531441/524288). A movement by this changes the diatonic degree but not the chromatic degree.

It is possible to build tonality using the Pythagorean tuning, but the major and minor categories of intervals here are only to be understood in combination with diatonic degrees for the purpose of marking generator steps i.e. fifth shifts, and are not to be confused with the major and minor tonality. An interval is perfect if its number of generator steps is -1, 0, or +1, major if it is +2 to +5, and minor if it is -5 to -2.

Figure 1: diatonic generator step markers

Meantone tempers out 81/80, the syntonic comma, identifying 5/4 by 2 diatonic degrees and 4 generator steps. That is how 5/4 is called a major third.

5-limit just intonation can be analysed as the syntonic comma added to the Pythagorean tuning, or meantone with the syntonic comma recovered, represented by the following mapping:

[math]\displaystyle{ \begin{bmatrix} 7 & 11 & 16 \\ 12 & 19 & 28 \\ 0 & 0 & 1 \end{bmatrix} }[/math]

or in terms of generator steps:

[math]\displaystyle{ \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix} }[/math]

with the syntonic comma mapped to the tempered monzo of [0 0 -1⟩ in both cases. Since the first two entries are zero, this tempered monzo marks neither diatonic moves nor chromatic moves, but a commatic one, which only alters the color of the interval. A Pythagorean interval altered by the syntonic comma is dyed with the "color of 5". For example, 81/64 is M3, whereas 5/4 is M3⁵. While both are members of major thirds, 5/4 involves otonal-5.

Now these are some of the diatonic semitones:

Name	Ratio	Monzo	Size (¢)	FJS
Large limma	27/25	[0 3 -2⟩	133.2	m225
Classical limma	16/15	[4 -1 -1⟩	111.7	m25
Pythagorean limma	256/243	[8 -5⟩	90.2	m2

Each is separated by 81/80.

These are some of the chromatic semitones:

Name	Ratio	Monzo	Size (¢)	FJS
Pythagorean chroma	2187/2048	[-11 7⟩	113.7	A1
Large chroma	135/128	[-7 3 1⟩	92.2	A15
Classical chroma	25/24	[-3 -1 2⟩	70.7	A125

Each is separated by 81/80.

Chapter II. Septimal Voice Leading

81/80 translates a Pythagorean interval to a classical one. What is its septimal counterpart, which translates a Pythagorean interval to a septimal one? The answer is 64/63, the septimal comma.

Superpyth is the corresponding temperament of the septimal comma. It is the opposite of meantone in several ways. To send 81/80 to unison, meantone tunes the fifth flat. To send 64/63 to unison, superpyth tunes the fifth sharp. In septimal meantone, intervals of 5 are simpler than those of 7, whereas in septimal superpyth, intervals of 7 are simpler than those of 5, and their overall complexities are comparable. George Secor identified a few useful equal temperaments for meantone and superpyth. He noted 17, 22, and 27 to superpyth are what 12, 31, and 19 to meantone, respectively. I call those the six essential low-complexity equal temperaments.

The significance of the septimal comma is successfully recognized by notable notation systems including FJS, HEJI (Helmholtz–Ellis Just Intonation), and Sagittal. It corresponds to the following change of basis, in terms of generator steps.

[math]\displaystyle{ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 1 & 0 & 4 \\ 0 & 1 & 4 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} }[/math]

Each of most diatonic degrees comes in three flavors: a Pythagorean one, a classical one, and a septimal one. The best example for this is the minor third, they are 32/27 (m3), 6/5 (m3₅), and 7/6 (m3⁷).

Voice leading plays a significant role in traditional harmonies. It is customary to prefer the diatonic semitone to the chromatic semitone for this purpose. Consider 7-limit harmony, the class of diatonic semitones has three varieties as discussed above. Besides 256/243 (m2), there are 16/15 (m2₅), sharp by 81/80, and 28/27 (m2⁷), flat by 64/63. In 12et, the syntonic comma, the septimal comma and the Pythagorean comma are all tempered out, so all varieties of semitones are conflated as one, which is very adequate for voice leading. The classical diatonic semitone, however, is larger. Consequently, the traditional dominant chord with this semitone would be very weak. The Pythagorean variant is not ideal either, since it lacks color and concordance. The septimal version is a much stronger choice.

A basic form of dominant–tonic progression is, therefore, a septimal major triad followed by a classical major triad:

[math]\displaystyle{ 3/2–27/14–9/4 \rightarrow 1–5/4–3/2 }[/math]

where 27/14 resolves to 2/1.

21/20 (m2⁷₅), the 5/7-kleismic diatonic semitone, is another possible candidate. Compound in color, however, it is not as easy to grasp as 28/27, nor is it as strong, since it is only flat of the Pythagorean version by 5120/5103, the 5/7-kleisma aka the hemififths–amity comma. In contrast, 28/27 creates more cathartic effects for voice leading.

Actually, septimal harmony entail different chord structures from classical ones, and 21/20 has a niche from this perspective. This will be discussed in Chapter VII.

Chapter III. Avicenna's Gift

The harmonics 11 and 13 can be modeled similarly to 5 and 7. 33/32, the undecimal quartertone, lends itself to translate the Pythagorean perfect fourth to 11/8, the octave reduced 11th harmonic. 1053/1024, the tridecimal quartertone, lends itself to translate the Pythagorean minor sixth to 13/8, the octave reduced 13th harmonic.

Again, FJS successfully recognizes these, and extends the mapping to

[math]\displaystyle{ \begin{bmatrix} 1 & 1 & 0 & 4 & 4 & 6 \\ 0 & 1 & 4 & -2 & -1 & -4 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix} }[/math]

The four intervals are related through the uniquely most important comma in the 13-limit, 2080/2079, the ibnsinma, as

[math]\displaystyle{ 2080/2079 = ((64/63)(1053/1024))/((81/80)(33/32)) }[/math]

The ibnsinma fits the difference between 8 pairs of 13-limit superparticular commas, not to mention a lot more non-superparticular but useful identities. Its monzo is

[math]\displaystyle{ [ \begin{matrix} 5 & -3 & 1 & -1 & -1 & 1 \end{matrix} \rangle }[/math]

As its orders of 5, 7, 11, and 13 are unity, any no-5, no-7, no-11, or no-13 subgroup temperament can be immediately extended to the full 13-limit by adding this comma to the comma list, and it typically makes sense. The ibnsinma is a prime-limit-inclusive fully entangled comma (PIFE comma). As Scott Dakota has noted, these commas are highly portable among numerous systems. Indeed, the ibnsinmic temperament can be defined by the merge of five comma-size equal temperaments whose tuning profiles are vastly different from each other: 41, 46, 53, 58, and 72. I call those the five essential comma-size equal temperaments.

An equal temperament is a point in the tuning space where it happens to temper out certain additional commas. Appendix D shows a list of efficient 13-limit equal temperaments, gated by TE relative error < 5.5% and cut off at 494. 31 out of the 42 equal temperaments temper out the ibnsinma. Although this density drops if the cutoff is set higher, it is expected for any comma.

To temper out the ibnsinma requires a fairly accurate representation of harmonic 3, since the corresponding order is three. Meantone or superpyth would not work with it; otherwise it is very portable.

Another potential weakness of the ibnsinma is the seesaw effect of error accumulation, observed in 118et.

Figure 2: tuning profile of pure-octave 118et (118f val)

118et is famous for being the first equal temperament which clearly gives 5-limit microtempering, with errors well under half a cent. While the harmonics 3 (-0.26¢) and 5 (+0.13¢) are accurate enough, the 7 (-2.72¢) and 11 (-2.17¢) are moderately flat. These may not seem much, yet all the errors backfire on the 13 in the same direction, demonstrated below.

[math]\displaystyle{ T \vec b = 0 \\ T = J + E }[/math]

where T is the tuning map, J the JIP, E the error map, and b the monzo of the ibnsinma. The error of harmonic 13 is given by

[math]\displaystyle{ E_{\pi(13)} = 3E_{\pi(3)} - E_{\pi(5)} + E_{\pi(7)} + E_{\pi(11)} - J \vec b \\ = -3 \times 0.26 - 0.13 - 2.72 - 2.17 - 0.83 ¢ \\ = -6.63 ¢ }[/math]

With optimal octave stretch, it is less cursed, at -4.48 ¢. Although this tuning does nudge it into the range of the closest approximation, the level of stretch has destroyed the microtempering quality of the 5-limit.

The equal temperaments that do not temper out the ibnsinma typically miss an accurate harmonic 3 or the aforementioned seesaw effect is too prominent. Examples are 103, 121, and 190.

Another comma that is potentially important is 4096/4095, the schismina. It creates simple connections among 5, 7, and 13. Although not a PIFE comma by definition, it shares virtually all properties with PIFE commas such as the ibnsinma. It identifies 1053/1024, the tridecimal quartertone, by 36/35, the septimal one; it also equates 64/63, the septimal comma, with 65/64, the wilsorma.

However, relating the schismina is at best poor taste due to its restrictive nature. Since its monzo is

[math]\displaystyle{ [ \begin{matrix} 12 & -2 & -1 & -1 & 0 & -1 \end{matrix} \rangle }[/math]

the only entry with a positive index is the 2, so tempering it out removes the possibility of all-flat systems. It removes the possibility of all-sharp systems too if we neglect its size. Furthermore, the flatter the 3, 5, and 7 are tuned, the sharper the 13 is required to compensate, amplifying the differential error to twice. That is the undesirable quality I call the shearing effect of error accumulation.

It is tempting to add the schismina along with the ibnsinma due to the identities shown in Figure 3.

File:Factorizations of the ibnsinma.png

Figure 3: factorizations of the ibnsinma

Note also tempered out is their stack, the olympia, 131072/130977, equating the undecimal quartertone with a stack of two septimal commas. The tempering calls for an accurate tuning of the septimal comma for it to make sense at all, which is not so common. Hence, the portability is remarkably impaired. Plenty of efficient 13-limit equal temperaments do not follow it. Notable examples include 58 and 72.

Chapter IV. Distribution of Superparticular Commas

Superparticular commas are special in that they are more efficient in dominating the tuning. Up to the same size, they typically have a shorter path in the interval space due to their simplicity. Thus, looking into the distribution of superparticular commas may prove valuable. The phenomenon that the 13-limit optimal tuning often shows a shift from the 11-limit one could be understood as correlated to a distribution shift of superparticular commas. To understand it we need to go through a few superparticular commas first, in particular, superparticular kleismas that form a distribution peak in the 13-limit. They are 352/351 (minthma), 325/324~385/384 (marveltwin~keenanisma), 364/363~441/440 (gentle~werckisma), 540/539~729/728 (swetisma~squbema), and 351/350 (ratwolfsma). Note three of the five come in pairs separated by the ibnsinma.

352/351 (minthma)

Also known as 11/13-kleisma, this comma measures 4.93 ¢ and marks the difference between 11/9 and 39/32, and between 27/22 and 16/13.

Separated by the ibnsinma are also 5120/5103, the 5/7-kleisma aka the hemififths–amity comma, and 847/845, the cuthbert comma.

325/324~385/384 (marveltwin~keenanisma)

The marveltwin measures 5.34 ¢, and is the difference between 16/13 and a stack of two 10/9's or between 13/9 and a stack of two 6/5's.

The keenanisma measures 4.50 ¢, and is the difference between 33/32 and 36/35.

364/363~441/440 (gentle~werckisma)

The gentle comma measures 4.76 ¢, and makes fifth complements of 14/11 and 13/11.

The werckisma measures 3.93 ¢, and is the difference between 21/20 and 22/21.

540/539~729/728 (swetisma~squbema)

The swetisma measures 3.21 ¢, and is the difference between 15/11 and a stack of two 7/6's or between 33/20 and a stack of two 9/7's.

The squbema measures 2.38 ¢, and is the difference between 27/26 and 28/27.

351/350 (ratwolfsma)

The ratwolfsma measures 4.94 ¢, and is the difference between 13/10 and 35/27.

All are equipped with respective 13-odd-limit essentially tempered chords, implying their worth of study in far greater detail than this essay can cover.

Since those are five independent commas on top of the ibnsinma, it is impossible to temper out all of them. Nonetheless, each of the five essential comma-size equal temperaments does all but one. That which is not tempered out is tuned to one step. 41et does not temper out the ratwolfsma; 46et the swetisma~squbema; 53et the gentle~werckisma; 58et the marveltwin~keenanisma; 72et the minthma. Consider the step sizes and the comma sizes. When not tempered out, it is extended to a comma size, about five times the size it should be, and the consequence is clearly reflected in each tuning profile.

Notice

[math]\displaystyle{ 41 + 46 + 53 + 58 + 72 = 270 }[/math]

More specifically, it is the vals that add up:

[math]\displaystyle{ V_{41} + V_{46} + V_{53} + V_{58} + V_{72} = V_{270} }[/math]

So 270et is the equal temperament where all those kleismas are tuned to one step, which explains its exceptional strength.

Among the five equal temperaments, 46et is the weakest as the swetisma~squbema pair is too small compared to the other kleismas, yet it is special for the same reason. Observe that

[math]\displaystyle{ 41 + 53 + 58 + 72 = 224 }[/math]

So 224et is the equal temperament where the swetisma~squbema remains tempered out while the other kleismas are tuned to one step. In fact, 224et outperforms all before it and is only bettered by 270et.

224et and 270et can be seen as dual to each other. 224et tempers out the swetisma~squbema; 270et tunes it to one step like the other kleismas. The abigail temperament is the merge of them. It makes the swetisma~squbema a variable while maintaining the other identities. Their sum, 494et, achieves an optimum by tuning it to half of the other kleismas, and this temperament is even more excellent than 270et.

Chapter V. Other Side of the River

Although the FJS is a highly coherent interval space model in the 13-limit, it does not capture the harmonic nature of implicit intervals.

Specifically, quartertones separate interval classes just as semitones do. An implicit interval such as a neutral third is, after all, neither major nor minor, and two neutral thirds make up a perfect fifth. Matthew Yacavone recognized part of this facet and developed the Neutral Functional Just System (NFJS), although it only adopted 2.sqrt (3/2). The sqrt (2).sqrt (3) interval space lends itself better to an enhancement on Pythagorean tuning's role, such that quadratic irrationals are used as anchors for neutrals, interseptimals, quartertones and the tritone.

Quartertones come in different types. Aura analysed quartertones in the 2.3.11 interval space as follows. The first quartertone is the Alpharabian semiaugmented unison, also dubbed the Alpharabian parachromatic quartertone, with the ratio 33/32, measuring 53.3 ¢. Another quartertone is the Alpharabian inframinor second, also dubbed the Alpharabian paradiatonic quartertone, with the ratio 4096/3993, measuring 44.1 ¢. We shall see the whole tone comprises an Alpharabian chromatic semitone and an Alpharabian diatonic semitone. An Alpharabian chromatic semitone comprises two Alpharabian parachromatic quartertones, whereas an Alpharabian diatonic semitone comprises an Alpharabian parachromatic quartertone and an Alpharabian paradiatonic quartertone. Here is an illustration.

Figure 4: decomposition of the whole tone

The same arithmetics applies to the sqrt (2).sqrt (3) interval space. The Pythagorean parachromatic quartertone has the ratio of sqrt (2187/2048) and the sqrt (2).sqrt (3) subgroup monzo [-11 7⟩, measuring 56.8 ¢. The Pythagorean paradiatonic quartertone has the ratio of sqrt (134217728/129140163) and the sqrt (2).sqrt (3) subgroup monzo [27 -17⟩, measuring 33.4 ¢.

There is a more balanced model for this space, though. Thereby I present three distinct semitones each separated by half the Pythagorean comma.

Name	Alternative Name	Ratio	S. Monzo	Size (¢)
Chromatic semitone	Augmented unison	2187/2048	[-22 14⟩	113.7
Pseudoambitonal semitone	N/A	sqrt (9/8)	[-3 2⟩	102.0
Diatonic semitone	Minor second	256/243	[16 -10⟩	90.2

The pseudoambitonal semitone is the mean of the chromatic and diatonic. It is also the perfect fifth displaced by the semioctave. The term pseudoambitonal is explained later in this chapter.

Name	Alternative Name	Ratio	S. Monzo	Size (¢)
Metachromatic quartertone	N/A	(20 digits)	[-30 19⟩	68.6
Hemichromatic quartertone	Semiaugmented unison	sqrt (2187/2048)	[-11 7⟩	56.8
Hemidiatonic quartertone	N/A	sqrt (256/243)	[8 -5⟩	45.1
Metadiatonic quartertone	Inframinor second	(18 digits)	[27 -17⟩	33.4

Granted, none of the quartertones is rational. I avoid Aura's terms for my coherence (and not as a dismissal): the hemichromatic quartertone is the same as the parachromatic quartertone, and the metadiatonic quartertone is the same as the paradiatonic quartertone.

The hemichromatic semitone is hemichromatic because it is half the chromatic semitone; same for the hemidiatonic semitone. We shall see this hemichromatic and hemidiatonic pair is central, in contrast to the hemichromatic and metadiatonic pair shown above. Here is an illustration.

Figure 5: revised decomposition of the whole tone

With this model, undecimal intervals may take a new class. Previously, 11/8 is a perfect fourth, and two make up a minor seventh. Now, 11/8 is a semiaugmented fourth, and two make up a major seventh. That undoubtedly captures the harmonic characteristics much better. The 2.3.5.11 subgroup mapping is

[math]\displaystyle{ \begin{bmatrix} 2 & 2 & 0 & 4 \\ 0 & 2 & 8 & 5 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} }[/math]

Hence it follows that 11/8 is the Pythagorean semiaugmented fourth, sqrt (243/128), altered by half the rastma, sqrt (243/242). The half rastma overtakes the role of the undecimal quartertone in the previous model.

Harmonic 11 demonstrates the characteristic of ambitonality. It is ambitonal in the sense that otonal and utonal intervals share the same class. With respect to 5, 5/4 is otonal, and 6/5 is utonal. Like any other classical intervals, they belong to distinct classes. With respect to 11, however, 11/9 is otonal, and 27/22 is utonal. Both are neutral thirds, only separated by the rastma. It even applies to 11/10 and 12/11, separated by the biyatisma, 121/120, which is also commatic since

[math]\displaystyle{ 121/120 = (81/80)/(243/242) }[/math]

Another observation is that all the interval classes of undecimal pairs are implicit, and vice versa, all the implicit interval classes are undecimally ambitonal. That is why the sqrt (9/8) is called the pseudoambitonal semitone – it is an ambitonal class (further discussed in Chapter VIII) but takes a position of an explicit interval.