Unimodular Matrices, Isomorphic Keyboards, and Unequal Temperaments

: Interval Bases We've Known And Loved

We've already seen lots of bases for representing intervals. Rank 2-interval space was introduced along-side Pythagorean tuning using the (P5, P8) basis. The (A1, d2) basis was useful for naming rank-2 intervals. The rank-3 prime harmonic basis (P8, P12, M17) and the rank-3 Lilley comma basis (Ac1, A1, d2) were similarly valuable for 5-limit just intonation, the one true way. Let's talk more about what interval bases are and what is special about these particular bases that makes them worth knowing and using.

An interval basis is a set of intervals used for representing a whole interval space. A {rank-N} interval basis is made of {N} intervals that can each be representing with vectors of {N} coordinates, giving in total an {N * N} matrix. The intervals of the basis are themselves represented in some basis. If we represent our intervals individually in the (A1, d2) basis, then the Pythagorean basis, (P5, P8), has coordinates ((7, 4), (12, 7)). If we represent our intervals in the Pythagorean basis, then the Pythagorean basis looks like an identity matrix, ((1, 0), (0, 1)). For rank-3 intervals, the (P8, P12, M17) basis has coordinates ((3, 12, 7), (5, 19, 11), (7, 28, 16)) when intervals are represented in the (Ac1, A1, d2) basis. Et cetera.

: Unimodularity

There is something special about these bases that distinguishes them from most random sets of intervals you could come up with: unimodularity. Matrices that have determinant {1} or {-1} are called unimodular. Unimodular matrices are invertible over the integers, meaning that they have integer entries and so do their inverses. The inverse of a unimodular matrix is also unimodular. 

There's a well known geometric interpretation of matrices as coordinate transforms in a cartesian space, and in this interpretation, the determinant of a matrix tells you how a matrix scales the areas of geometric figures that it transforms. A determinant of {1} means that the areas are unchanged (but perhaps rotated or skewed. a determinant of {-1} means the areas are unchanged but also that a transformed geometric figure will have flipped chirality (left-right handedness) within the space. This chiral flip ends up not mattering to music as we hear it, but the volume-preservation of unimodularity matters a lot.

Now, unimodularity is something we see in matrices of coordinates. So an interval basis (Ac1, A1, d2) might be unimodular if we express it in coordinates using some interval space, and not unimodular if we express it in coordinates using another interval space. Further the coordinate matrix for any full-rank basis, when the basis vector are expressed in that same basis, will be an identity matrix, which is unimodular. So there's a sense in which all full-rank interval bases are unimodular.

The thing that's special about the Pythagorean basis and the Lilley comma basis and so on isn't that they're unimodular when expressed in terms of themselves: it's that they're unimodular when expressed in terms of a prime harmonic basis. A prime harmonic basis of a certain rank exists in a one-to-one correspondence with frequency space, up to a certain prime limit, where the frequencies are a product of primes raised to integer powers.

We know that the prime harmonic basis is good - it lets us abstract a notion of underlying intervals away from the notion of frequencies produced by a tuning system. And any bases that are unimodular in the prime harmonic basis will preserve this notion of goodness, in the same way that unimodular geometric transformations preserve areas.

It also happens to be the fact that any of these nice unimodular bases are unimodular when expressed in terms of each other. Starting from a prime harmonic basis, we can induce a whole family of bases that are equivalent ways of talking about intervals, and the coordinates remain invertible over the integers, so we won't get intervals with weird coordinates like (5/4, sqrt(3)) or whatever.

Now, an interval basis doesn't strictly have to be unimodular with respect to the primes, but unimodular interval bases have several nice features. One nice feature is that if you work in a unimodular interval basis and you tune your basis vectors to rational frequencies, then the entire interval space will also be tuned to rational frequencies. That is to say, unimodular bases are the natural way to define just intonation tuning systems.

It's intuitive that if we start with a prime harmonic basis and we preserve some general notion of areas, that the new interval basis would also be good at representing justly tuned frequency ratios, but let's look a little bit at why it works in practice: First we note that the coordinates of intervals in a unimodular basis are integers.  If you tune the basis intervals to rational values, then the integer coefficients of all the intervals in the space become integer powers in the tuning system, and a product of rational numbers raised to integer powers is still rational, so all of the intervals in the interval space get associated to rational values for their frequency ratios. 

Another neat perk of unimodular bases: intervals can also be placed in regular aligned spatial grids using unimodular coordinates. These grid layouts span the interval space, and if you tune them justly, then you get a frequency grid which spans the rational frequency space. For tuned rank-2 intervals let's say, you can  go far enough out, and you'll eventually hit any weird rational frequency ratio you can imagine, including things that are justly tuned arbitrarily closely to things you've already seen. Also the grids have no overlap or gaps. They're very pretty. If you use a basis which isn't unimodular in the prime harmonic basis, instead of having a grid that looks like a playable keyboard, you'll get uneven spacing, gaps, and overlaps.

There's another perk though! Isomorphism. These grids are isomorphic, meaning that if you play a chord or melody or whatever, you can move it somewhere else on the grid and you've performed a transposition. The intervals aren't warped by translation. If you go two steps up and one step over from over here, the difference is, let's say a major third, and if you go two steps up and one step over starting from over there, the difference is still a major third. This is really cool for performers: a major chord is always the same shape. And your favorite bebop lick is always the same shape. And you can modulate with a freedom that you didn't even realize was constraining you on piano. Even if you don't have a 2D keyboard to play, you can visualize music on a 2D grid on a computer screen and get this same isomorphism.

Unimodular matrices: natural for defining just intonation, and natural for designing 2D musical keyboards and 2D music visualizations.

: Visualizations Of Rank-2 Unimodular Interval Bases

Let's look at some unimodular bases! We'll also see how they can be transformed into one another through shear mappings.

Here's the Rank-2 Lilley basis, whose basis intervals are (A1, d2):

I've highlighted a chromatic scale in blue. Having one unimodular basis, we can make more through shearing. Let's slide the horizontal rows past each other so that m2 is below P1. This gives us the (A1, m2) basis, which is also unimodular.

Things have gotten a little more compact. The overall shape is now less of a line and more of a diamond, with the natural and once-modified intervals falling on a thicker diagonal. We can also shear the interval space in the other direction. Let's slide the vertical rows past each other so that M2 is next to P1. This gives us the (M2, m2) basis. 

If we perform a horizontal shear again, we get the (M2, m3) basis, which I think is very pretty. I've called it  the "Fitzgibbon basis" for a few years. The Fitzgibbon basis is very close the the layout of buttons on chromatic accordions, although the Fitzgibbon basis is isomorphic (intervallically consistent) while chromatic accordions buttons are not quite. 

Let's talk about that a little bit. Almost all of the writing I've seen about isomorphic rank-2 keyboard layouts, including chromatic accordion button layouts, is from people who are really stuck thinking in terms of 12-EDO, and they make constant mistakes about interval arithmetic because of it. Like people will say that a 2D keyboard layout increases by m3 in one direction, when actually actually the notated pitches increase inconsistently by m3 or A2. When you see this, do you declare that the verbal definition is the actual layout or that the displayed layout is the actual one? I don't know. My solution is to ignore all of it and do better. In particular, the Fitzgibbon basis is better, since it increases regularly by m3 in one direction, and increases regularly by M2 in the other direction, and doesn't equivocate between m3 and A2 just because 12-TET tunes them to the same frequency ratio. Here's what it looks like in intervals:

And here's what it looks like in pitches, with the pitch C1 rooted on the interval P1:

And that's clearly not how accordion teachers notate the pitches of their buttons.

If we keep alternating vertical and horizontal shear mappings, then we go from the (M2, m3) basis to the (M2, P4) basis, and then to the (P5, P4) basis. All unimodular. You might wonder why I keep alternating the shearing directions between horizontal and vertical directions. I'm getting somewhere. Slide the rows past each other one more time and we end on the (P5, P8) basis, our old friend the Pythagorean basis.

Which you can see has minor intervals on one side and major intervals on the other. I don't think this is a great keyboard layout, but it is another way to view the spiral of fifths. Slide vertically one more time and we've got the rank-2 prime harmonic basis, (P12, P8). I didn't know if all the rank-2 unimodular bases could be reached from each other by shearing, so I was very pleased when I started playing with them and found this chain of shear mappings that included all my favorites. I hope you enjoyed the ride.

I'll say it again: unimodular matrices are the natural way to represent and define just intonation, and they're also natural for designing 2D musical keyboards and 2D music visualizations.

Why just 2D keyboards? Well, it's hard to build 3D grids in physical space that are playable. But next we'll talk about how temperaments give us a cool workaround so we can almost play rank-3 and higher-rank music using just 1D and 2D keyboards.

:: Temperaments

A temperament is a kind tuning system. Specifically, it's a tuning system which tempers out an interval, i.e. tunes some interval besides P1 to a frequency ratio of 1/1. We've already learned about 1D temperaments with pure octaves, the "equal" temperaments or "equal divisions of the octave" temperaments or "EDO"s. But there are many more and some of them are quite beautiful.

When we looked EDOS, we tuned rank-2 interval space, with one interval being tempered, so that we got a 1D frequency space. In perfect analogy, we can take  a higher rank interval space and temper out an interval to get a frequency space that's one dimension smaller. We can temper out {N} intervals to get a frequency space that's {N} dimensions smaller.

The most historically important non-equal temperaments are the "meantone" temperaments. These temperaments are defined on interval spaces of rank-3 and higher and they temper out the acute unison, Ac1. Here's how we define the most famous one, called "quarter comma meantone". First, QC meantone has a basis made of the octave, the major third, and the acute unison, (P8, M3, Ac1) . If we express these as coordinates in the rank-3 prime harmonic basis, we get ((1, 0, 0), (-2, 0, 1), (-4, 4, -1)), which has determinant -4. It's not unimodular, and that's okay. We're not looking at just intonation for a second, we're looking at temperaments. Some of the frequencies will be irrational, just like how the EDOs have irrational frequencies.

Let's say we want to tune P5 in quarter comma meantone. This has coordinates (-1, 1, 0) in the rank-3 prime harmonic basis. First we need to do a change of basis into the (P8, M3, Ac1) basis. 

One way we can do this is to multiply the P5 interval coordinates by the inverse of the QC meantone basis matrix:

(-1, 1, 0) * inverse of ((1, 0, 0), (-2, 0, 1), (-4, 4, -1)) = (1/2, 1/4, 1/4)

That's fine if you have software to do matrix inversion for you. If not, for low dimension, Cramer's rule gives you another way to do a change of basis, and I like it because you can write the whole thing down in a few lines of code without any conditional branching, and you know exactly how many multiplications and additions it takes to get the answer. Cramer's rule gives us a closed form solution, that works well at low dimensions, whereas matrix inversion as an algorithm feels a little implicit to me. Here's some python code for Cramer's rule:

def perform_change_of_basis(B1, B2, B3, interval):

(m, n, o) = interval

(a, b, c) = B1

(d, e, f) = B2

(g, h, i) = B3

detA = a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)

x = (m * (e * i - f * h) - n * (d * i - f * g) + o * (d * h - e * g)) / detA

y = (a * (n * i - o * h) - b * (m * i - o * g) + c * (m * h - n * g)) / detA

z = (a * (e * o - f * n) - b * (d * o - f * m) + c * (d * n - e * m)) / detA

return (x, y, z)

Same result though.

However we get there, we've now have coordinates for P5 in the quarter comma meantone basis: (1/2, 1/4, 1/4). This means that, in interval space, we can say:

P5 = 1/2 * P8 + 1/4 * M3 + 1/4 * Ac1

and this is equivalent to 

t(P5) = t(P8)^(1/2) * t(M3)^(1/4) * t(Ac1)^(1/4)

in frequency space. The first relation is additive and the second is multiplicative, but they're both facts. They're a little bit unusual as facts because we're used to dealing with integer coefficients and exponents, but they're facts. Not even specific to QC meantone.

 Next we tune the QC meantone basis intervals, which allows us to tune all the other intervals of the rank-3 interval space. The first two basis intervals are tuned purely and the last one is tempered out:

t(P8) = 2/1

t(M3) = 5/4

t(Ac1) = 1/1

And now we just substitute!

t(P5) = (2/1)^(1/2) * (5/4)^(1/4) * (1/1)^(1/4) = (5)^(1/4) ~ 1.4953

You can see that quarter comma meantone has a slightly flat tuned value for P5, relative to the pure value of 3/2 = 1.5. It happens to be flat by a factor of (81/80)^(1/4), which is like a fourth of a purely tuned acute unison. This is our first hint as to where "quarter comma" comes from in the name QC meantone. This is also the normal way of temperaments: a few intervals are pure, but most are detuned by a fraction of the interval that was tempered out.

Since we tempered out the acute unison, the Ac1 component of P5 didn't really matter in the tuning: 1 raised to any real power is still going to be 1. This also means that two intervals which differ by by an acute unison (or multiple acute unisons) will be tuned to the same frequency ratio by a meantone temperament.

For example, the acute major second, purely tuned to 9/8, and the rank-3 major second, purely tuned to 10/9, differ by an acute unison, but are tuned to the same value by QC meantone. Let's verify.

In the rank-3 prime harmonic basis, our coordinates are: 

AcM2 = (-3, 2, 0)

M2 =  (1, -2, 1)

We can convert these to the QC meantone basis using inverse matrices or a change of basis with Cramer's rule, either way giving:

AcM2 = (0, 1/2, 1/2)

M2 = (0, 1/2, -1/2)

These only differ in the last component, but the last component is tempered out in tuning, so both become

t(AcM2) = t(M2) = (2)^(0) * (5/4)^(1/2) ~ 1.11803

This frequency is the geometric mean of the pure frequencies:

sqrt((9/8) * (10/9)) ~ 1.11803

And since the Pythagorean major second and the just major second are both called "tones" or "whole tones", we've now discovered where the name "meantone" comes from. (You've also heard that the acute unison is also called the "syntonic" comma. This is because if you temper it out, you unite the two major tones.) Generally, meantone systems tune AcM2and M2 to a shared value somewhere between the pure values, but not exactly at the geometric mean. Quarter comma meantone has a claim to be the canonical meantone tuning system because of this nice geometric mean, among other nice properties.

Recap: Meantone tuning systems operate on rank-3 interval space and higher rank spaces. They temper out the acute unison. We've seen that Quarter Comma Meantone has pure octaves and major thirds, but slightly flattens the tuned perfect fifth and equates the old major seconds together. I haven't shown it in detail, but I think you'll be able to see how this gives us a discrete two dimensional frequency space: everything is a product of rational powers of t(P8) and t(M3), and rational numbers are countable. Since we temper out Ac1, the Ac1 component of intervals can be ignored in our tuning calculations.

This is awesome. Do you want to play 5-limit just intonation but you don't have a keyboard with a 3D layout? No problem. Use a 2D layout. Don't know when to use an AcM2 or M2? No problem. They're the same now. Want pure major thirds so people don't point and laugh and call you a Pythagorean? You've got them.

Quarter Comma Meantone was hugely popular in Europe before 12-EDO was discovered and took over. And even a 1D keyboard with a small set of QC meantone notes can support pretty free modulation. It's a good system.

:: Unequal Temperaments Versus The Isomorphism Of Unimodular Bases

I said you can play QC meantone on a 2D keyboard, and that's true but a little deceiving. I should give you some more detail to be less deceptive. Remember when I said that non-unimodular bases generally make poor keyboard layouts with uneven spacing, like large gaps or overlaps? If you just use (P8, M3) coordinates of intervals as locations for the corresponding keyboard keys, you're going to have a bad time. If we represent (Ac1, P8, M3)  in the prime harmonic basis, we get ((-4, 4, -1), (1, 0, 0), (-2, 0, 1)), which has a determinant of -4. So we can predict some uneven spacing. Let's see what it actually looks like.

If our meantone temperament tunes multiple intervals to the same frequency ratio, then above I show only the interval with the simplest name at a point corresponding to the tempered interval coordinates. For example, you'll see m2 but not  Grm2 above, because m2 has a shorter name.

My visualization code scales the X and Y components by the absolute value of the basis matrix determinant to avoid overlaps, and sometimes that scales things too much, but you can still see that we don't have a very nice grid. Even with better spacing, we'd have every other column being offset vertically by half a tile height. 

Instead, here are some grid layouts based on bases that include Ac1 and which are unimodular in the primes. Also they have fairly small intervals for their second two members, because that makes for nice small/chromatic steps like a piano keyboard has, which I appreciate.

I'll show you four bases: [Ac1, A1, m2], [Ac1, A1, M2], [Ac1, M2, m2], [Ac1, A1, d2]. 

This layout is based on the unimodular basis (Ac1, A1, AcAcA0) which has coordinates ((-4, 4, -1), (-3, -1, 2), (-15, 8, 1)) in the prime harmonic basis. This basis is unimodular in the primes, and while that means that we can use it for rank-3 just intonation, we can also use it here for a temperament. We just detune the basis intervals in whatever way they were detuned by our original definition of 1/8-comma schismatic.

On this grid, when two intervals occupy the same tuned space, I've shown the one with the simpler name, e.g. M6 instead of GrGrGrd7. Hopefully. This is an older image and I fixed a mistake in my code since then. Some of the spots might have names that are less than their simplest. I've also highlighted perfect and imperfect intervals in blue. They make a nice chromatic diagonal with a little bit of width. The Pythagorean chromatic notes are also on that diagonal, e.g. look at the placement of AcM2 and Grm3. All of the usual chromatic Pythagorean intervals lie on that diagonal, but there are two that you won't see named in the diagram: the Pythagorean d5 and A4. These are respectively called GrGrd5 and AcAcA4 in rank-3 interval space, and while schismatic tempering places them on the main diagonal, they don't appear on the diagram because there happen to be intervals in the same place with shorter names. But, as you'd expect, the GrGrd5 is placed two squares away from d5 in the negative Ac1 direction, and AcAcA4 is places two squares away from A4 in the positive Ac1 direction. So surrounding the main diagonal we thus have all the Pythagorean and 5-limit chromatic intervals. Here they are listed by increasing frequency ratio (just frequency or schismatic frequency, it doens't make a difference to the ordering among these):

[P1, Grm2, m2, M2, AcM2, Grm3, m3, M3, AcM3, P4, A4, GrGrd5, AcAcA4, d5, P5, Grm6, m6, M6, AcM6, Grm7, m7, M7, AcM7, P8]

Nice. To use this (Ac1, A1) layout and have it sound like 1/8-comma schismatic, we just have to define 1/8-comma schismatic in terms of (Ac1, A1, AcAcA0) instead of (P8, M3, AcAcA0). And you can totally do that. There are lots of equivalent ways to define a tuning system. It's nice to define them at first in terms of intervals that are tuned to rational values, but once you've done that, you can also specify the same tuning system in terms of basis vectors tuned to irrational values - so long as the set you choose is independent (full rank), which these one are. Here are the tunings, which I figured out in the (P8, M3, AcAcA0) basis:

Ac1 = (-4, 4, -1) # (2/1)^(1/2) * (5/4)^(-3/2) ~ 1.0119288512538815

A1 = (-3, -1, 2) # (2/1)^(-5/8) * (5/4)^(17/8) ~ 1.0418136188775968

AcAcA0 = (-15, 8, 1)  # 1/1

Just like before, if you want to tune an interval, you do a change of basis into (Ac1, A1, AcAcA0), and then used the tuned values of the basis intervals to find the tuned value for your target interval.

Let's do it with P5, which is (-1, 1, 0) in the prime harmonic basis. Our new coordinates are

(-1, 1, 0) * inverse of ((-4, 4, -1), (-3, -1, 2), (-15, 8, 1)) = (10, 7, -4)

and that already tells us the location of P5 on the grid: 10 steps from P1 in the Ac1 direction and 7 steps  in the A1 direction. Have a look to verify. We of course we disregard the tempered component, -4. 

Tuning still works also, though it's a little messy:

t(P5) = t(Ac1)^(10) * t(A1)^(7)

t(P5) = ((2/1)^(1/2) * (5/4)^(-3/2))^(10) * ((2/1)^(-5/8) * (5/4)^(17/8))^7 = 10^(7/8)/5 =  (3/2) / (32805/32768)^(1/8) 

We've just derived a justly tuned P5 flattened by the 1/8 the just value of the schisma, as we were hoping and expecting. Or I was anyway. Isn't it beautiful? We have a two dimensional grid for constructability/playability, a unimodular layout for beautiful even spacing, a tuned perfect fifth that is crazy close to pure, and also we have the option to play all of the 5-limit chromatic interval and all of the Pythagorean chromatic intervals that any student of just intonation will want under their fingers.

This temperament-grid also nicely shows of a different definition of the schisma as the difference between two stacked augmented unisons, (Ac1 + Ac1 = ) AcAc1, and a rank-3 diminished second, d2, which is a relationship we can also verify from their just tunings,

(81/80)^2 / (128/125) = (32805/32768)

That's not as cool to me as the Pythagorean bridge, but it's a fairly natural way to define the interval as a difference of commas.

The schisma is also definable as the difference between the the Pythagorean comma and the syntonic comma, although this isn't very apparent on the grid. The Pythagorean comma is justly tuned to 531441/524288 and it was called A0 in rank-2 interval space, but in rank-3 interval space, it's AcAcAcA0.If we substract an Ac1, that will knock off one of the "Ac"s, and we're left with AcAcA0, a.k.a the schisma.

:: Other Unequal Temperaments

When I look up other rank-3 commas that people have tempered for the construction of unequal temperaments, I see a lot of diminished seconds:

    Grd2 = (11, -4, -2) # 2048/2025

    d2 = (7, 0, -3) # 128/125

    Acd2 = (3, 4, -4) # 648/625

I don't know what's good about these, but they're common choices.

In a future chapter, we'll talk at length about which commas you need to temper out from e.g. rank-5 or rank-4 space to get all the way down to different rank-1 EDOs. We'll learn about lots of weird commas there. But let's talk about rank-4 commas that aren't too weird before we close.

If you want a rank-4 interval to temper out that's justly associated with a small 7-limit frequency ratio, we've got some options. Here are just tunings for some options:

4375/4374 at 0.4 cents // [-1, -7, 4, 1] = (-1, 1, 0, -1): SbGrA1 or "ragisma"

2401/2400 at 0.7 cents // [-5, -1, -2, 4] = (2, 1, 2, -4): SbSbSbSbAcdd3 or "breedsma"

5120/5103 at 5.8 cents // [10, -6, 1, -1] = (-2, 0, 0, 1): SpGrGr1 or "hemifamity"

225/224 at 7.7 cents // [-5, 2, 2, -1] = (0, 0, -1, 1): SpA0 or "marvel"

1029/1024 at 8.4 cents // [-10, 1, 0, 3] = (2, 1, 1, -3): SbSbSbAcAcm2 or "gamelisma"

126/125 at 13.8 cents // [1, 2, -3, 1] = (1, 0, 1, -1): SbAcd2 or "starling"

245/243 at 14.2 cents // [0, -5, 1, 2] = (0, 1, 1, -2): SbSbm2 or "sensamagic"

In order, on each line, I've listed a small septimal frequency ratio, its size in cents, then the coordinates for the interval justly associated with that frequency ratio, first the prime harmonic coordinates in square brackets and then  the Lilley-Johnston coordinates in parentheses, then the interval's name, and finally the cutesy name for the fraction as it is known on the Xenharmonic Wiki. The last name I've included only for the sake of people coming from the Xenharmonic community, but I don't care for their names and won't be using them again. The first option, 4375/4374, has a nice interpretation as

(4375/4374) = (25/24) / ((36/35) * (81/80))

i.e. it's the difference between a grave augmented unison, justly tuned to  (25/24) / (81/80) = 250/243, and the septimal super unison of Ben Johnston, 36/35. I like that tempering this out would tune a moderately complicated rank-3 interval, GrA1, to the same value as a slightly simpler rank-4 interval, Sp1. It's a bit like the situation with GrGrGrd7 and M6 for the schisma.

The second option, the 2401/2400 ratio, does not have a nice relation with rank-3 intervals, but the third option, 5120/5103, does; it's just

    (36/35) / (81/80)^2

I also like 225/224 at 7.7 cents for this, which can be explained as:

    (36/35) / (128/125)

The 1029/1024 does not have a tidy rank-3 to rank-4 relationship.

The 14-cent 126/125 has a more complicated version of the 225/224 relationship (differing by the presence of an acute unison):

    (36/35) / ((128/125) * (81/80))

and it's significantly more perceptible, so I'm not very impressed by that option.

The last small septimal frequency ratio above, 245/243, is kind of cool:

    (245/243) = (16/15) / (36/35)^2

It's also a little too wide for my liking, but this nicely shows how two septimal commas can produce something like a minor second.

To summarize, I'd be friends with anyone who thought that the intervals justly associated with these ratios:

    SbGrA1 : 4375/4374 at 0.4 cents # (25/24) / ((36/35) * (81/80))

    SpGrGr1 : 5120/5103 at 5.8c # (36/35) / (81/80)^2

    SpA0: 225/224 at 7.7 cents cents # (36/35) / (128/125)

were cool things to temper out to reduce rank-4 intervals by a dimension. And then maybe you on top of that you'd also like to temper out the schisma or the syntonic comma at the same time and that would let you play 7-limit music on a 2D keyboard.

It's definitely not imperceptible, but I'm also fine with tempering out the super grave unison, SpGr1, justly tuned to 64/63, in order to reduce the dimension of rank-4 interval space. It's about 27 cents, and tempering it out means that Grm7 and Sbm7 have the same frequency, among other things. The Xenharmonic people call this one the Archytas comma. But to me it's a septimal super grave unison, SpGr1. Tempering out the regular septimal unison - Sp1, justly tuned to 36/35 - might also be cool. There are lots of things to play around with.

: Septimal Commas Tempered by EDOs

I definitely prefer two dimensional temperaments over EDOs, but EDOs also have their charms and are easier to program onto a standard midi keyboard. Let's see which EDOs temper out the schisma and at least one of the three imperceptible septimal commas above. It turns out that the second two septimal commas differ by a schisma,

SpA0 / SpGrGr1 = AcAcA0

(225/224)  / (5120/5103) = (32805/32768)

so an EDO that tempers out the schisma and the SpA0 will also temper out the SpGrGr1. The question now is just whether a schismatic EDO tempers out the SpA0, the SbGrA1 (justly tuned to 4375/4374), or both. Below 100 divisions, 53- and 65-EDO are the only ones that temper out the schisma and the SbGrA1 (justly tuned to 4375/4374). Below 100 division, the SpA0 (justly tuned to 225/224) is tempered out all of by (12, 29, 41, 53, 82, 94)-EDO. And 53-EDO shows up in both lists, so 53-EDO tempers out both.

How about meantone EDOs that temper out at least one of those three imperceptible septimal commas? The only meantone temperaments that also temper out SpGrGr1 (justly tuned to 5120/5103) are (5, 7, 12)-EDO, but no one is using those for their accuracy at representing rank-4 just intervals. Above 12 divisions and below 100, these meantone EDOs temper out these imperceptible septimal commas:

19-EDO: SbGrA1, SpA0

26-EDO: SbGrA1

31-EDO: SpA0

43-EDO: SpA0

45-EDO: SbGrA1

50-EDO: SpA0

62-EDO: SpA0

74-EDO: SpA0

81-EDO: SpA0

93-EDO: SpA0

and these have no overlap with the previous schismatic septimal EDOs.

Maybe we should look at schismatic and meantone EDOs that temper out the main perceptible septimal commas, namely the super unison, Sp1 (justly tuned to 36/35), and the super grave unison, SpGr1 (justly tuned to 64/63). Just like how tempering the syntonic comma has a huge impact on your interval space - in that it tunes together fundamental intervals that you might otherwise want to distinguish - in the same way, tempering out Sp1 and SpGr1 will have a huge impact on the rank-4 soundscape.

Since Sp1 and SpGr1 differ by an acute unison, any meantone EDO that tempers out one will temper out the other. The only EDOs that do this are (5, 7, 12)-EDO. And that seems normal to me. There shouldn't be very many tuning systems compatible with tempering out multiple large commas like Ac1 and Sp1.

The schismatic EDOs surprised me though. I thought there would be more. 12-EDO is both meantone and schismatic, so of course we still have the fact that 12-EDO tempers out Sp1 and SpGr1. The only new player is 17-EDO, which tempers out the both the Schisma and the Sp1Gr1.

Pretty slim pickings. If you want to ignore basic septimal distinctions while still making cooler music than 12-TET, your options are pretty much to use 17-EDO or to use a 2D or 3d temperament.

:: 2D Septimal Temperaments

When you temper out one comma from rank-3 space, you can get a perfect fifth that's detuned by a fraction of the tempered comma. When you temper out two commas from rank-4 interval space, you can get a P5 that detuned by fractions of both tempered commas.

If you have pure octaves, and temper out SpA0 and the schisma (=AcACA0), and tune M3 purely to 5/4, then you get .... just the regular 1/8 schismatic temperament. Likewise if you do all that using the syntonic comma instead of the schisma. The way you define a temperament over rank-4 intervals that doesn't just reduce to a temperament over rank-3 intervals is to keep a septimal interval tuned purely (for example, the sub-minor seventh, t(Sbm7) = 7/4), while also tempering out a septimal interval. Let's do two examples.

We'll work in the rank-4 Lilley-Johnston comma basis this time, (Ac1, A1, d2, Sp1), which would be justly tuned to (81/80, 25/24, 128/125, 36/35). 

Here's the interval we want to tune:

P5 = (2, 7, 4, 0) # 3/2

Let's have a temperament with pure octaves and a pure seventh harmonic, and also it will temper out SpA0 and the schisma, AcAcA0. Here's our basis

AcAcA0 = (2, 0, -1, 0) # 32805/32768

SpA0 = (0, 0, -1, 1) # 225/224

Sbm7 = (3, 10, 6, -1) # 7/4

P8 = (3, 12, 7, 0) # 2/1

To find P5 in the (AcAcA0, SpA0, Sbm7, P8) basis we just use the multiply the old vector by the inverse of that basis matrix:

(2, 7, 4, 0) * inverse of ((2, 0, -1, 0), (0, 0, -1, 1), (3, 10, 6, -1), (3, 12, 7, 0)) = (1/7, -1/14, -1/14, 9/14)

If we temper out the schisma and SpA0, this then our tuned value for P5 becomes

t(P5) = (7/4)^(-1/14) * 2^(9/14)

This is really close to the pure value of 3/2. It's sharp by like 0.3 cents. But what's the numeric expression for how sharp it is? It's stupidly easy to figure out.

When we say that P5 has coordinates (1/7, -1/14, -1/14, 9/14) in the  (AcAcA0, SpA0, Sbm7, P8) basis, that means

P5 =  (1/7) * AcAcA0 + (-1/14) * SpA0 + (-1/14) * Sbm7 + (9/14) * P8

Or equivalently

t(P5) = t(AcAcA0)^(1/7) * t(SpA0)^(-1/14) * t(Sbm7)^(-1/14) * t(P8)^(9/14)

Let's write in all the just tunings

(3/2) = (32805/32768)^(1/7) * (225/224)^(-1/14) * (7/4)^(-1/14) * (2)^(9/14)

If we divide the left hand side by the first two factors on the right hand side, namely, the ones associated with the commas that we're planning to temper out,

(3/2) / (32805/32768)^(1/7) * (225/224)^(-1/14)  =  (7/4)^(-1/14) * (2)^(9/14)

Then we see that the tuned value we got for P5 in this temperament (the right hand side) can be expressed in the form "flatten P5 by 1/7 of the pure value of the schisma and -1/14 of the pure value of SpA0". So this temperament is the "1/7-comma schismatic & -1/14-comma SpA0 temperament". The exponents for flattening are just the component coordinates of P5 associated with the tempered commas.

Let's work through another example very quickly.

P5 in the (SpA0, Ac1, Sbm7, P8) basis has coordinates:

(2, 7, 4, 0)  * inverse of ((0, 0, -1, 1), (1, 0, 0, 0), (3, 10, 6, -1), (3, 12, 7, 0)) = (1/10, 1/5, 1/10, 1/2)

If we temper out the first two coordinates and keep the second two pure, then the temperament we define has a P5 tuned to

(3/2) / ((225/224)^(1/10) * (81/80)^(1/5)) = (7/4)^(1/10) * (2)^(1/2)

so we just defined the 1/5-comma meantone & 1/10-comma SpA0 temperament.

That tells us how P5 is detuned, which is like an octave reduced third harmonic. How much is M3 detuned, which is the octave reduced fifth harmonic? To calculate that, we just need M3 in the Lilley-Johnston coordinates. The M3 was (1, 4, 2) in five-limit just intonation, so it'll be (1, 4, 2, 0) in the (Ac1, A1, d2, Sp1) basis. No factors of 7 in 5/4. Now we do a change of basis into (SpA0, Ac1, Sbm7, P8):

(1, 4, 2, 0)  * inverse of ((0, 0, -1, 1), (1, 0, 0, 0), (3, 10, 6, -1), (3, 12, 7, 0)) = (2/5, -1/5, 2/5, 0)

And these coordinates are a compact way of representing this fact of rank 4 interval space:

M3 = 2/5 * SpA0 + -1/5 * Ac1 + 2/5 * Sbm7 + 0 * P8

If we look at this relationship in frequency space and use just tunings, t(SpA0, Ac1, Sbm7, P8) -> (225/224, 81/80, 7/4, 2/1), it becomes the arithmetic identity:

(5/4) = (225/224)^(2/5) * (81/80)^(-1/5) * (7/4)^(2/5) * (2)^(0)

And the equivalent arithmetic identity for the tuning system that starts with the (SpA0, Ac1, Sbm7, P8) basis and tempers out the first two components while keeping the last two pure is

(5/4) / (225/224)^(2/5) * (81/80)^(-1/5) = (7/4)^(2/5) * (2)^(0)

Which is every so slightly sharp of the pure value, by

1200 * log_2((7/4)^(2/5) / (5/4)) ~ 1.2 cents

And that's how the (SpA0, Ac1) temperament alters the intervals that are purely associated with the third harmonic. Cool beans.