Seven-Limit Just Intonation
In the chapter "Five-Limit Just Intonation", I argued at length that western diatonic music is best represented with just frequency ratios having factors of 2, 3, and 5. If we include one more prime harmonic in our ratios, we get 7-limit just intonation. The adjective for fractions including factors of 7 is "septimal", and I may occasionally refer to rank-4 intervals as being septimal in so far as they are justly tuned to septimal frequency ratios. Septimal music has sounds that are quite unfamiliar to western diatonic music. You can still compose beautiful music in 7-limit, but it will often have an alien character, and one that many people will find unsettling, unapproachable, or even abrasive at first.
One of the first microtonal pieces that I like to share with people is Ben Johnstons's "String Quartet No. 4, Amazing Grace". It's in 7-limit just intonation and it's one of the best and most impressive microtonal pieces ever composed. It's also a fairly long, vast, and imposing orchestral work. If I had to select a short piece with limited instrumentation that showed more simply the beauty and importance of 7-limit JI, it would probably be "Winter Septimalia" by Mannfishh. I've occasionally been frustrated when I've tried to figure out how to compose in 7-limit and only come up with dissonance, but pieces like these two inspire me to keep honing the craft.
I once listened to a few hundred chords with septimal intervals, and rated the chords for consonance. Unfortunately, I'm so accustomed to five-limit intonation, that I basically just found septimal approximations to five-limit chords, instead of finding good chords with a beautiful new and somewhat-unsettling alien character. For example, I liked the sound of these chords:
(P1, AcM3, SpAcAA4)
(P1, SbAcd4, P5)
(P1, SpGrM3, SpGr5)
but they're effectively all just the Pythagorean major chord, (P1, AcM3, P5) with some very small septimal commas added on. Or this septimal chord I liked
(P1, SpA2, SpAcAA4),
but it basically just sounds like a just minor triad, (P1, m3, P5). A lot of the septimal intervals in chords that I liked were only different from familiar 5-limit intervals by tiny commas like the septimal super augmented zeroth interval, SpA0, which is justly tuned to 225/224. This has a hardly perceivable size of 7 cents.
If I can't trust my ear, how do I figure out how to compose beautiful music in seven-limit just intonation? We're going to try a few tricks. 1) We're going to look at ancient Greek tunings, like the septimal scales and tetrachords of Archytas and Ptolemy. 2) We're going to look at a little bit at the tonality lattices of modern music theorist Paul Erlich. 3) We're going to keep listening to septimal chords and judging their consonance, because you can't compose music without using your ears. 4) We're going to look at Mannfishh's score for "Winter Septimalia" and see what we can learn from it.
I'm excited. I hope you are too.
I will caution that septimal music sounds better on some instruments than others. When I compose with software synthesizers, things often sound better with a harp or a tuba or a clarinet than with a piano or a guitar. Or they might sound fine on your preferred instrument, but only if you play an octave higher up than you're used to playing. It can be a finnicky business, but the point is that if you're frustrated in your investigations of septimal music, you should try to experiment with a different instrument or register before giving up.
For easy reference, I'm putting a table of simple septimal intervals at the end of this chapter, with coordinates in both the rank-4 prime harmonic basis and the rank-4 Lilley-Johnston comma basis, which will both be familiar from the chapter "Prime Harmonic Bases and Comma Bases". As a refresher, the rank-4 prime harmonic basis has intervals (P8, P12, M17, Sbm21), which are justly tuned to (2/1, 3/1, 5/1, 7/1), while the rank-4 Lilley-Johnston comma basis has intervals (Ac1, A1, d2, Sp1), which are justly tuned to (81/80, 25/24, 128/125, 36/35).
: The Septimal Music of Ancient Greece
The ancient Greeks had tons of scales and tetrachords, and many of them had fairly high prime limits. Even among just the 7-limit ones, most of them sound pretty bad to me. But here are two of my favorite tetrachords that are pretty decent. I'll show them first as tetrachords spanning a perfect fourth, but the extension to a full scale is obvious: start with the tetrachord, then add an acute major second to bring us up to P5, and add on another copy of the tetrachord. My names for the tetrachords come from ex-tempore.org who seems to have done more thorough scholarship on the subject than anyone else online.
Archytas's Diatonic Toniaion tetrachord has these ratios in some order:
[8/7, 9/8, 28/27]
I think there's supposed to me a definite order, but I'm going to ignore that. These intervals are roughly sized like just or Pythagorean [M2, M2, m2], so a scale made from the tetrachord in this order will sound major, and if you swap the first two ratios, that will also sound like a major scale. You can do other arrangements of the intervals and get decent scales, but those two orders are my favorite.
Ptolemy's Diatonic Malakon tetrachord has these intervals in some order:
[8/7, 10/9, 21/20]
They're also roughly sized like [M2, M2, m2], so a scale in this order, or with the first two ratios swapped, will also sound like major scale. Other orders also make decent scales.
Those two orders of those two tetrachords are my favorites. Next I'll show the full scales and the triads formed from each scale degree:
Archytas' Diatonic Toniaion 1:
[1/1, 8/7, 9/7, 4/3, 3/2, 12/7, 27/14, 2/1] :: [0, 231, 435, 498, 702, 933, 1137, 1200] cents
I: [1/1, 9/7, 3/2]
II: [1/1, 7/6, 3/2]
III: [1/1, 7/6, 3/2]
IV: [1/1, 9/7, 3/2]
V: [1/1, 9/7, 32/21]
IV: [1/1, 7/6, 3/2]
VII: [1/1, 32/27, 112/81]
Archytas' Diatonic Toniaion 2:
[1/1, 9/8, 9/7, 4/3, 3/2, 27/16, 27/14, 2/1] :: [0, 204, 435, 498, 702, 906, 1137, 1200] cents
I: [1/1, 9/7, 3/2]
II: [1/1, 32/27, 3/2]
III: [1/1, 7/6, 3/2]
IV: [1/1, 81/64, 3/2]
V: [1/1, 9/7, 3/2]
VI: [1/1, 32/27, 32/21]
VII: [1/1, 7/6, 112/81]
Ptolemy's Diatonic Malakon 1:
[1/1, 8/7, 80/63, 4/3, 3/2, 12/7, 40/21, 2/1] :: [0, 231, 414, 498, 702, 933, 1116, 1200] cents
I: [1/1, 80/63, 3/2]
II: [1/1, 7/6, 3/2]
III: [1/1, 189/160, 3/2]
IV: [1/1, 9/7, 3/2]
V: [1/1, 80/63, 32/21]
VI: [1/1, 7/6, 40/27]
VII: [1/1, 6/5, 7/5]
Ptolemy's Diatonic Malakon 2:
[1/1, 10/9, 80/63, 4/3, 3/2, 5/3, 40/21, 2/1] :: [0, 182, 414, 498, 702, 884, 1116, 1200] cents
I: [1/1, 80/63, 3/2]
II: [1/1, 6/5, 3/2]
III: [1/1, 189/160, 3/2]
IV: [1/1, 5/4, 3/2]
V: [1/1, 80/63, 40/27]
VI: [1/1, 6/5, 32/21]
VII: [1/1, 7/6, 7/5]
Most of these are decent chords to my ear. These two show up four times each:
[P1, Sbm3, P5] # [1/1, 7/6, 3/2]
[P1, SpM3, P5] # [1/1, 9/7, 3/2]
And these make a lot of sense to me. They're the most obvious septimal triads could you ask for. So I'm glad that the scales that I like contain them repeatedly. I bet sub-minor and super-major, "Sbm" and "SpM", will keep coming up as consonant interval qualities in our investigations.
These two chords show up twice each in the scales:
[P1, SbAcm3, P5] # [1/1, 189/160, 3/2]
[P1, SpGrM3, P5] # [1/1, 80/63, 3/2]
Those look... weird and bad? Maybe they're okay. Let's also keep a look out for SbAcm and SpGrM qualities in our investigations, I guess.
We've got at least one good septimal major triad and at least one good septimal minor triad now! And that's a lot of what you need to compose music. So we've done good work here. We have some decent scales now too. They're basically just randomly mistuned major scales, but they don't sound terrible, and people know how to compose on major scales. I won't pretend that this is the divine music of the spheres, but if you want to make some septimal music, you can get a lot done in this framework.
Let's look at the triads that appear on scale degree VII in all the scales to see what a septimal diminished chord might look like.Â
[1/1, 32/27, 112/81]
[1/1, 6/5, 7/5]
[1/1, 7/6, 112/81]
[1/1, 7/6, 7/5]
For the thirds of the chords on degree VII, we have, going down the line, a Pythagorean minor third (Grm3), a just minor third (m3), and two sub-minor thirds (Sbm3). For the fifths of the chords, going down the line, we have SbGrd5, justly tuned to 112/81, and a sub-diminished fifth (Sbd5), justly tuned to 7/5, and then those two repeat again. I'm okay with calling those things diminished chords. They're not too crazy. I'm not sure if any of them has a claim to be the canonical, best, or simplest septimal diminished chord, but they're a fine start.
Let's also look at the septimal seventh chords / tetrads of those four scales. Since each scale is made of two copies of a tetrachord separated by a perfect fifth, there are going to be lots of cases where the seventh of a chord is a perfect fifth higher than the third. This
[P1, Sbm3, P5, Sbm7] # [1/1, 7/6, 3/2, 7/4]
is by far the most common extension to the sub-minor chord, with the Pythagorean Grm7 (justly tuned to 16/9) also making an appearance. Similarly for the super-major chord, this
[P1, SpM3, P5, SpM7] # [1/1, 9/7, 3/2, 27/14]
is most common seventh extension, with the 7th and 3rd both being "super major". The Grm7 (tuned to 16/9) and SpGrM7 (tuned to 40/21) also make appearances.
Here are the diatonic septimal diminished seventh chords:Â
 [1/1, 32/27, 112/81, 16/9]
[1/1, 6/5, 7/5, 9/5]
[1/1, 7/6, 112/81, 7/4]
[1/1, 7/6, 7/5, 7/4]
The first chord pairs up Grm3 with Grm7. The second one pairs up a just m3 with a just m7. And the last two chords pair up a Sbm3 with a Sbm7. There's no reason why we couldn't mix and match tetrachords either and then we'd get more thirds and sevenths of different qualities.
: Chromatic Extensions Of The Diatonic Greek Tetrachords and Related 2D Temperaments
The scales in the previous section are good tools. Anything that helps you to write good music is a good tool. But they're a little lacking in theory. "Here are four mistunings of the major scale, pick which ever one you want." We can do better.
Let's review the tetrachords.
Archytas's Diatonic Toniaion:
[8/7, 9/8, 28/27]
Ptolemy's Diatonic Malakon:
[8/7, 10/9, 21/20]
They both have a SpM2 tuned to 8/7. Then Archytas has a Pythagorean major second while Ptolemy has a just major second. Finally we have whatever altered minor second is needed to reach the perfect fourth. For Archytas, the altered minor second only has factors of 2, 3, 7, while Ptolemy's altered minor second has a factor of 5 again, like its just major second.
We might say that Archytas's Diatonic Toniaion is a septimal extension to Pythagorean tuning and Ptolemy's Diatonic Malakon is a septimal extension to five-limit just intonation. Let's try to develop minor or chromatic scales around each tetrachord that maintain these properties. Lets also try to find a nice rank-2 temperament over rank-4 interval space that shows both the Archytasian intervals and the Ptolemaic intervals at once. That will be useful for keyboard design. I've got a good feeling that one of the commas will be the schisma.
Here are the simplest intervals in the 2.3.7 just intonation subgroup, like those in the Archytas tetrachord:
P1 : (0, 0, 0, 0) :: 1/1
SpGr1 : (-1, 0, 0, 1) :: 64/63
Sbm2 : (0, 1, 1, -1) :: 28/27
Grm2 : (-1, 1, 1, 0) :: 256/243
AcM2 : (1, 2, 1, 0) :: 9/8
SpM2 : (0, 2, 1, 1) :: 8/7
Sbm3 : (1, 3, 2, -1) :: 7/6
Grm3 : (0, 3, 2, 0) :: 32/27
AcM3 : (2, 4, 2, 0) :: 81/64
SpM3 : (1, 4, 2, 1) :: 9/7
SbAc4 : (2, 5, 3, -1) :: 21/16
P4 : (1, 5, 3, 0) :: 4/3
SpGr4 : (0, 5, 3, 1) :: 256/189
SbAc5 : (3, 7, 4, -1) :: 189/128
P5 : (2, 7, 4, 0) :: 3/2
SpGr5 : (1, 7, 4, 1) :: 32/21
Sbm6 : (2, 8, 5, -1) :: 14/9
Grm6 : (1, 8, 5, 0) :: 128/81
AcM6 : (3, 9, 5, 0) :: 27/16
SpM6 : (2, 9, 5, 1) :: 12/7
Sbm7 : (3, 10, 6, -1) :: 7/4
Grm7 : (2, 10, 6, 0) :: 16/9
AcM7 : (4, 11, 6, 0) :: 243/128
SpM7 : (3, 11, 6, 1) :: 27/14
SbAc8 : (4, 12, 7, -1) :: 63/32
P8 : (3, 12, 7, 0) :: 2/1
The interval coordinates are in the rank-4 Lilley-Johnston comma basis. But on second thought we probably don't need most of those. Since 3-limit, 5-limit, and 7-limit just intonation all respect octave complementation, we should be able to look at the octave complements of our major scales to get a bunch of minor intervals.
If we look at the two Archytas major scales as giving us options for each scale degree where they differ, then under octave complementation we get this almost chromatic scale (just missing an element between P4 and P5):
Chromatic scale from Archytas's Toniaion:
P1 :: 1/1
Sbm2 :: 28/27
(AcM2, SpM2) :: (9/8, 8/7)
(Grm3, Sbm3) :: (32/27, 7/6)
SpM3 :: 9/7
P4 :: 4/3
P5 :: 3/2
Sbm6 :: 14/9
(AcM6, SpM6) :: (27/16, 12/7)
(Grm7, Sbm7) :: (16/9, 7/4)
SpM7 :: 27/14Â
P8: 2/1Â
If we treat the two Ptolemy major scales as similarly giving us options on each scale degree when they differ, and then we take octave complements, then we get this almost chromatic scale (just missing an elemnt between P4 and P5):
Chromatic scale from Ptolemy's Malakon:
P1 :: 1/1
SbAcm2 :: 21/20
(M2, SpM2) :: (10/9, 8/7)Â
(m3, Sbm3) :: (6/5, 7/6)
SpGrM3 :: 80/63
P4 :: 4/3
P5 :: 3/2
SbAcm6 :: 63/40
(M6, SpM6) :: (5/3, 12/7)Â
(m7, Sbm7) :: (9/5, 7/4)
SpGrM7 :: 40/21
P8 :: 2/1
These two chromatic scales, derived from tetrachords of Archytas and Ptolemy, have the same options for septimal alterations on the major second, minor third, major sixth, and minor seventh intervals, namely (SpM2, Sbm3, SpM6, and Sbm7) tuned to (8/7, 7/6, 12/7, and 7/4). Cool.
The two chromatic scales differ on the septimal options for minor second, major third, minor sixth, and major seventh. In the chromatic Archytas, those septimal intervals look much simpler: (Sbm2, SpM3, Sbm6, and SpM7) tuned to (28/27, 9/7, 14/9, and 27/14). In the chromatic Ptolemy scale, we've got versions of all of those that are altered by a syntonic comma, namely (SbAcm2, SpGrM3, SbAcm6, and SpGrM7) tuned to (21/20, 80/63, 63/40, 40/21).
I still really like using one tetrachord twice to make major scales. There's an audible structure there that an audience can follow. Another thing I appreciate about those scales is their simplicity and the challenge they offer to the composer to commit to doing just one thing - namely to optimize their music within the constraint of a fixed scale. And even though I have combined Archytas-derived major scales with each other, and combined Ptolemy-derived major scales with each other, to get a better picture of two septimal interval spaces, you could take the union of a single septimal major scale with its octave complement to get a single chromatic scale (with no options for alterations) if you wanted that same structure and constraint as the original major scales had.
Like here's Like here's the chromatic/octave-mirror version of Ptolemy's Diatonic Malakon 1:
P1 :: 1/1
SbAcm2 :: 21/20
SpM2 :: 8/7
Sbm3 :: 7/6
SpGrM3 :: 80/63
P4 :: 4/3
P5 :: 3/2
SbAcm6 :: 63/40
SpM6 :: 12/7Â
Sbm7 :: 7/4
SpGrM7 :: 40/21
P8 :: 2/1
It's a pretty cool scale. And this might not be fixed by the tetrachord structure, but if you wanted to fill out the gap between P4 and P5, you could do a lot worse than using Sbd5 (justly tuned to 7/5) and SpA4 (justly tuned to 10/7): these have very simple frequency ratios, and like the other intervals in the scale, they're septimal alterations of 5-limit intervals.
For the Archytas chromatic scales, if you're looking for septimal tritones to fill out the gap between P4 an P5, a decent option is to alter those two Ptolemaic tritones by syntonic commas to get version with out factors of 5, giving us SbGrd5 (justly tuned to 112/81) and SpAcA4 (justly tuned to 81/56).
: Septimal Temperaments
In the chapter "Unimodular Matrices, Isomorphic Keyboards, and Unequal Temperaments" we talked a lot about how to use interval bases to define temperaments and keyboard layouts, which are both great for playing high dimensional music.Â
Unimodular bases are great for making isomorphic keyboards, so we'll try to find some unimodular rank-4 bases to make nice isomorphic keyboards for playing septimal music.Â
For temperament tuning systems, we made a distinction between systems that temper out small unnoticeable commas, having little effect on main chromatic intervals we use most in music, and systems that temper out larger commas, which we usually do in order to simply how we think about the interval space, like how tempering out Ac1 collapses the auditory distinction between Pythagorean and just intervals, or how tempering out A1 collapses the auditory distinction between major and minor intervals, or how tempering out d2 collapses the auditory distinction between pairs of pitches with separations like that between G# and Ab, allowing for very free modulation with a small set of distinct diatonic frequencies per octave. Likewise in this chapter, we'll look into septimal temperaments that barely mistune the interval space and we'll look at septimal temperaments that significantly mistune the interval space in ways that allow us to think less.
Now it's entirely possible to mess up the interval space in weird ways by tempering out an imperceptible comma. Finding the right commas to temper out, either to not mess up the interval space or to mess it up in a desirable way, takes some work or creativity. And I'm not that creative. So let's get to work.
 Here's the first temperament that I've found that I like as a keyboard layout:
The basis is [SpGr1, Sbm2, AcAcA0, SpGrGr1], which is unimodular in the rank-4 prime harmonic basis, and then I temper out the last two intervals to get 2D grid coordinates.Â
The second to last interval in the basis, AcAcA0, is our old friend the schisma, justly tuned to 32805/32768. Similarly to how the schisma is the difference between the Pythagorean comma AcAcAcA0 and the just acute unison Ac1, the last interval in this basis, SpGrGr1, is the difference between one septimal super unison and two acute unisons. It's justly tuned to
(36/35) / (81/80)^2 = 5120/5103
which is
1200 * log_2(5120/5103)
about 6 cents.
I had correctly guessed that the simpler septimal intervals would fall just off of the chromatic diagonal in a schismatic temperament, but I was a little surprised to see their placement above. I had guessed that the unusual 5-limit intervals GrM2 and Acm2 (in contrast to the diatonic Grm2 and AcM2) would be tempered together with simple septimal intervals. And they might be in another temperament of course. But I like this one quite a lot and won't be looking too hard for other septimal schismatic temperaments.
I've said that the last two basis vectors are are tempered out, i.e. tuned to a a frequency ratio of 1/1, but what about the first two basis vectors, SpGr1 and Sbm2? We could try tuning them justly, but this will give us impure harmonics. Instead, let's figure out what tuning of the grid vectors will get us a pure octave and a pure 7th harmonic.
The easiest way I know to do this is to define a temperament that has 1) the same tempered commas as the grid layout and 2) prime harmonics for its other two basis elements:
t(P8, Sbm7, SpGrGr1, AcAcA0) = (2/1, 7/4, 1/1, 1/1)
I always support pure octaves and it seems logical to use a pure seventh harmonic (or octave reduced seventh harmonic) for septimal music. This basis isn't unimodular in the primes, so it doesn't make a nice keyboard layout, but it sounds nice and we can use it to get tunings for the grid vectors of the layout temperament. Here's the basis in Lilley comma coordinates:
[3, 12, 7, 0],
[3, 10, 6, -1],
[-2, 0, 0, 1],
[2, 0, -1, 0]
Which has this matrix for an inverse:
[5/7, -6/7, -6/7, -1/7],
[-13/14, 17/14, 17/14, 11/14],
[10/7, -12/7, -12/7, -9/7],
[10/7, -12/7, -5/7, -2/7]
This inverse basis matrix will convert interval coordinates in the Lilley-Johnston comma basis into intervals in the (P8, Sbm7, SpGrGr1, AcAcA0) basis. We simply multiply an interval's comma coordinates by the conversion matrix to get temperament coordinates out. Here are the grid coordinates:
SpGr1 = (-1, 0, 0, 1) # 64/63 -> (5/7, -6/7, 1/7, -1/7)
Sbm2 = (0, 1, 1, -1) # 28/27 -> (-13/14, 17/14, 3/14, -3/14)
The first set of coordinates are equivalent to the fact that
SpGr1Â = 5/7 * P8 + -6/7 * Sbm7 + 1/7 * SpGrGr1 + -1/7 * AcAcA0
which is an additive relation that we could express multiplicatively in frequency space as
t(SpGr1) = t(P8)^(5/7) * t(Sbm7)^(-6/7) * t(SpGrGr1)^(1/7) * t(AcAcA0)^(-1/7)
Once we tune our basis elements, t(P8, Sbm7, SpGrGr1, AcAcA0) = (2/, 7/4, 1/1, 1/1), then we can tune our target intervals:
t(SpGr1 ) = t(P8)^(5/7) * t(Sbm7)^(-6/7) =Â (2)^(5/7) * (7/4)^(-6/7) ~ 1.0155541799543253Â
t(Sbm2 ) = t(P8)^(-13/14) * t(Sbm7)^(17/14) = (2)^(-13/14) * (7/4)^(17/14) ~ 1.036548857845813
And now our keyboard layout can both look and sound good.
One thing that I particularly like about the keyboard basis can be seen on the horizontal rows around the impure intervals. For example, from left to right we have on one row the intervals [Sbm3, Grm3, m3, Acm3, Spm3], in the same order as their just tunings. This makes septimal intervals easy to find, even if your keyboard isn't labeled.
I'm pretty happy with that as a schismatic temperament which doesn't distort the most useful intervals in Pythagorean + Just + Septimal chromatic space. One thing that I don't like about the keyboard basis is that it doesn't distinguish e.g. the SbAcm2 that we saw in Ptolemy's Malakon from the Pythagorean Grm2. The best unimodular basis I've found for a 2d keyboard layout that does distinguish these has the basis:
[Ac1, Sp1, SbSbSbSbdddd4, SbSbGrGrdd3]
with the last two intervals tempered out. This temperament is exactly as ridiculous as it looks, and it's so fined grained in the intervals it can represent that an image of it wouldn't really fit on this page. So that needs some work. I know that, among equal temperaments, 60 and 65-EDO both distinguish among (P1, Sbm2, SbAcm2, Grm2, m2), so maybe I can generalize from there.
Yeah, that worked pretty well. Using the same grid intervals and two simpler commas tempered out by 60-EDO, I got this basis:
[Sp1, Ac1, SpA0, SbSbm2]Â
which looks like this:
This one has some drawbacks relative to the previous grid layout and it also has some advantages.
What if instead of wanting a finer grained view of septimal intervals, you instead wanted to to simplify the interval space? There are lots of possibilities. We might try tempering out Sp1 or SpGr1 or SpA0, or maybe Ac1 and SpGrGr1. Here's an interesting temperament I found. I wanted a temperament that showed off the just and septimal intervals while downplaying Pythagorean ones. So I thought, let's temper out the difference between the Pythagorean minor second, (Grm2 @ 256/243) and the subminor second (Sbm2 @ 28/27), and then we'll aslo include Sbm2 in the non-tempered intervals, so that we have the option to tune it purely. After a little experimentation, I found that I liked the layout when we use Sp1 as the other grid dimension and the schisma, AcAcA0, as the other tempered interval. It looks like this:
Its basis is [Sbm2, Sp1, SpGr1, AcAcA0], with the last two intervals tempered. There are many more layouts like these to find. If you find an interesting one, please let me know.
: Erlich's Septimal Tonality Lattices, The Harmonic Major Seventh Chord, and The Minor Super Thirteenth Chord
In a 2001 paper, "The Forms Of Tonality", microtonal music theorist Paul Erlich used a psychoacoustic model of consonance called "octave-equivalence harmonic entropy" to give a ranking of the harmonic strength of different just intervals. In his model, the strongest 5-limit harmonies are found at [6/5, 5/4, 4/3, 3/2, 8/5, and 5/3] and the strongest 7-limit harmonies are found at [8/7, 7/6, 7/5, 10/7, 12/7, and 7/4].
Based on these consonances, he selects two 7-limit four note chords with strong harmony between all the notes. I will call the first on the harmonic dominant 7th chord:
[4:5:6:7] . [1/1, 5/4, 3/2, 7/4] . [P1, M3, P5, Sbm7]
and I'll call the second one the minor super sixth chord:
[70:84:105:120] . [1/1, 6/5, 3/2, 12/7] . [P1, m3, P5, SpM6]
Like with most septimal chords, these can sound unpleasant on some instruments and registers while on others they sound a little unusual but inoffensive and perhaps even beautiful.
Erlich draws 3d graphs to show these chords and the intervals between all the notes, and he links the graphs up into an irregular lattice. I don't really know how to use his lattice for composition, but I think we should explore these two chords in greater detail.
A major septimal chord and a minor septimal chord are probably enough to compose quite a bit of septimal music. Let's look at subsets and inversions of these two chords to fill out our pallet and then try composing a chord progression.
Here are the harmonic dominant seventh chord and its inversions, i.e. its cyclic permutations:
[P1, M3, P5, Sbm7] # [1/1, 5/4, 3/2, 7/4]
[P1, m3, Sbd5, m6] # [1/1, 6/5, 7/5, 8/5]
[P1, Sbm3, P4, M6] # [1/1, 7/6, 4/3, 5/3]
[P1, SpM2, SpA4, SpM6] # [1/1, 8/7, 10/7, 12/7]
And here are the minor super sixth chord and its inversions:
[P1, m3, P5, SpM6] # [1/1, 6/5, 3/2, 12/7]
[P1, M3, SpA4, M6] # [1/1, 5/4, 10/7, 5/3]
[P1, SpM2, P4, m6] # [1/1, 8/7, 4/3, 8/5]
[P1, Sbm3, Sbd5, Sbm7] # [1/1, 7/6, 7/5, 7/4]
It's easy to see the triads that we get by knocking off the last interval of these. And a nice thing about seventh chords is that if we were to remove the first interval and rebase the chord on the third, we'd get that same set of triads. Let's write them out.
[1/1, 5/4, 3/2]
[1/1, 6/5, 7/5]
[1/1, 7/6, 4/3]
[1/1, 8/7, 10/7]
[1/1, 6/5, 3/2]
[1/1, 5/4, 10/7]
[1/1, 8/7, 4/3]
[1/1, 7/6, 7/5]
Much like the tetrachordal Greek scales and their chromatic extensions, I think that working within the constraint of this chord space could provide an interesting and unified sound for a piece of music.
Here's what I propose. In a major key, only the fifth scale degree gets a dominant seventh chord, so that's where we'll put the harmonic diatonic seventh chord, Â [P1, M3, P5, Sbm7]. And the only diatonic minor chord (out of scale degrees ^2, ^3, and ^6) that has a major sixth over it is ^2, so that's where we'll put the minor super sixth chord, Â [P1, m3, P5, SpM6]. Scale degrees ^1, ^3, ^4, and ^6 just get just triads. And then for scale degree ^7, we'd like to have a diminished chord, and we saw two options in our table of inversions, namely [P1, m3, Sbd5, m6] and [P1, Sbm3, Sbd5, Sbm7]. The chord tones in the first of these, over M7, have already shown within the tones of the other diatonic chords, so I think that's the more conservative addition.
All together we get a pretty basic justly tuned major with the addition of AcM2, SbAc4 ( justly tuned to 21/16), and SpGrM7 (justly tuned to 40/21). So it seems like we're not going to get very septimal music out of this, but maybe you want something conservative for your first exploration with septimality. If so, I've got just the chord space for you. On the other hand, if you have septimal chords on ^2, ^5, and ^7, you can push that really far. Like, in jazz, no one will bat an eye if you do a long sequence of 2-5-1 chord progressions based on a randomly modulating tonic. You can totally make a song that is 2/3rds septimal using just this chord space.
: Other Consonant Septimal Chords
I tried to really dial in my auditory investigation of septimal chords to just the most logical triads as tetrads - the ones that my mental model said should sound good, so if they don't then I need to update my model. Happily, between playing the tetrachord-derived scales and listening to these a lot, I've started developing an ear for what septimal chords should and should not sound like. Here's what I found out and what I'm still investigating.
Principle 1). No just with septimal thirds. The just minor seventh, m7, tuned to 9/5, sounds bad in septimal chords. I didn't guess that this would be the case, but I find it to be true quite reliably. It doesn't sound good against the septimal thirds, Sbm3 and SpM3, and it doesn't sound good against a septimal sub-diminished fifth, tuned to (7/5).
Principle 2). No M7 with septimal thirds: If you have a perfect fifth, P5, instead of Sbd5, then you can use any of the obvious thirds: [Sbm3, m3, M3, SpM3] to form a triad. Both septimal sevenths, Sbm7 and SpM7, work well as extensions to any of those triads. The Sbm7 sounds a little better, but they're both good and functional. I ruled out m7 as an extension if the third is septimal in principle 1. The just M7, as an extension to septimal chords, doesn't sound quite as bad as that, but it's still pretty bad. Definitely a step below SpM7, and I think we can do without it.
Principle 3). No SpM3 with Sbd5. The  septimal sub-diminished fifth is surprisingly beautiful. It sound good in a triad with [Sbm3, m3, or M3], respectively tuned to (7/6, 6/5, and 5/4). The Sbm3 triad sounds a little better and the the just third triads sound a little worse, but they're all good and functional. All of these triads can also be extended with (Sbm7, SpM7, or M7), tuned to (7/4, 27/14, and 15/8) respectively. The Sbd5 does not work well in a triad with SpM3, tuned to 9/7. The Sbd5 and SpM3 are separated by an interval of a SbSbd3, tuned to 49/45, which is apparently dissonant. Now we know.
4). I'm going to look into whether [Sbm6, m6, M6, SpM6, Sbd7] sound good as extensions to all the various triads and post it right here.
5) I'm going to look into whether various intervals with SbAcm and SpGrM qualities work in triads and tetrads. And I'll post that right here.
...
: Lessons From Mannfishh
In the introduction of this chapter, I talked about the piece "Winter Septimalia" by composer Mannfishh. I'm going to write about the principles and tricks behind the piece as I figure them out.
There are a few spots in Winter Septimalia where a note is basically carried over from one chord to the next, but there's a difference of a small septimal comma between the successive notes. The SpA0, justly tuned to 225/224, shows up a lot in particular, which was pleasing to me because I'd found it a lot when I was initially trying and failing to find consonant septimal chords that different in sonority from the five-limit.
I know a composer who works in just intonation and always thinks about what commas are compounding in their notes away from the base key where they started, and how they will take the commas off the stack to get back where they started. When you have barely perceptible commas and imperceptible commas, you don't have to think much about how to apply them or remove them to get somewhere in pitch space: you can apply them or remove them freely. Pretty cool, right?
I don't love reading sheet music to begin with, and Mannfishh uses Helmholtz-Ellis Just Intonation notation, in which natural intervals are Pythagorean, whereas I like to use Ben Johnston notation, in which natural intervals are 5-limit, so ... who knows if I'll figure out any more tricks from the score. But you're welcome to do so. And we'll already found a lot of good tricks for composing septimal music. Well done, us.
:: The Septimal Lydian Scale
...
:: Simple rank-4 intervals:
P1 : [0, 0, 0, 0] :: (0, 0, 0, 0) # 1/1
SpA0 : [-5, 2, 2, -1] :: (0, 0, -1, 1) # 225/224
Grd2 : [11, -4, -2, 0] :: (-1, 0, 1, 0) # 2048/2025
Ac1 : [-4, 4, -1, 0] :: (1, 0, 0, 0) # 81/80
SbA1 : [-5, -3, 3, 1] :: (0, 1, 0, -1) # 875/864
SpGr1 : [6, -2, 0, -1] :: (-1, 0, 0, 1) # 64/63
d2 : [7, 0, -3, 0] :: (0, 0, 1, 0) # 128/125
AcAc1 : [-8, 8, -2, 0] :: (2, 0, 0, 0) # 6561/6400
Sp1 : [2, 2, -1, -1] :: (0, 0, 0, 1) # 36/35
GrA1 : [1, -5, 3, 0] :: (-1, 1, 0, 0) # 250/243
Acd2 : [3, 4, -4, 0] :: (1, 0, 1, 0) # 648/625
Sbm2 : [2, -3, 0, 1] :: (0, 1, 1, -1) # 28/27
SpAc1 : [-2, 6, -2, -1] :: (1, 0, 0, 1) # 729/700
A1 : [-3, -1, 2, 0] :: (0, 1, 0, 0) # 25/24
SbAcm2 : [-2, 1, -1, 1] :: (1, 1, 1, -1) # 21/20
Spd2 : [9, 2, -4, -1] :: (0, 0, 1, 1) # 4608/4375
Grm2 : [8, -5, 0, 0] :: (-1, 1, 1, 0) # 256/243
AcA1 : [-7, 3, 1, 0] :: (1, 1, 0, 0) # 135/128
SpSp1 : [4, 4, -2, -2] :: (0, 0, 0, 2) # 1296/1225
m2 : [4, -1, -1, 0] :: (0, 1, 1, 0) # 16/15
SpA1 : [-1, 1, 1, -1] :: (0, 1, 0, 1) # 15/14
Acm2 : [0, 3, -2, 0] :: (1, 1, 1, 0) # 27/25
SbM2 : [-1, -4, 2, 1] :: (0, 2, 1, -1) # 175/162
GrM2 : [5, -6, 2, 0] :: (-1, 2, 1, 0) # 800/729
Spm2 : [6, 1, -2, -1] :: (0, 1, 1, 1) # 192/175
M2 : [1, -2, 1, 0] :: (0, 2, 1, 0) # 10/9
Sbd3 : [2, 0, -2, 1] :: (1, 2, 2, -1) # 28/25
AcM2 : [-3, 2, 0, 0] :: (1, 2, 1, 0) # 9/8
SbA2 : [-4, -5, 4, 1] :: (0, 3, 1, -1) # 4375/3888
SpGrM2 : [7, -4, 1, -1] :: (-1, 2, 1, 1) # 640/567
Grd3 : [8, -2, -2, 0] :: (0, 2, 2, 0) # 256/225
SpM2 : [3, 0, 0, -1] :: (0, 2, 1, 1) # 8/7
GrA2 : [2, -7, 4, 0] :: (-1, 3, 1, 0) # 2500/2187
d3 : [4, 2, -3, 0] :: (1, 2, 2, 0) # 144/125
A2 : [-2, -3, 3, 0] :: (0, 3, 1, 0) # 125/108
Sbm3 : [-1, -1, 0, 1] :: (1, 3, 2, -1) # 7/6
Acd3 : [0, 6, -4, 0] :: (2, 2, 2, 0) # 729/625
AcA2 : [-6, 1, 2, 0] :: (1, 3, 1, 0) # 75/64
SbAcm3 : [-5, 3, -1, 1] :: (2, 3, 2, -1) # 189/160
Spd3 : [6, 4, -4, -1] :: (1, 2, 2, 1) # 5184/4375
Grm3 : [5, -3, 0, 0] :: (0, 3, 2, 0) # 32/27
SpA2 : [0, -1, 2, -1] :: (0, 3, 1, 1) # 25/21
m3 : [1, 1, -1, 0] :: (1, 3, 2, 0) # 6/5
Acm3 : [-3, 5, -2, 0] :: (2, 3, 2, 0) # 243/200
SbM3 : [-4, -2, 2, 1] :: (1, 4, 2, -1) # 175/144
Spm3 : [3, 3, -2, -1] :: (1, 3, 2, 1) # 216/175
GrM3 : [2, -4, 2, 0] :: (0, 4, 2, 0) # 100/81
Sbd4 : [3, -2, -1, 1] :: (1, 4, 3, -1) # 56/45
M3 : [-2, 0, 1, 0] :: (1, 4, 2, 0) # 5/4
SbSb4 : [-2, -5, 2, 2] :: (1, 5, 3, -2) # 1225/972
Grd4 : [9, -4, -1, 0] :: (0, 4, 3, 0) # 512/405
AcM3 : [-6, 4, 0, 0] :: (2, 4, 2, 0) # 81/64
SbA3 : [-7, -3, 4, 1] :: (1, 5, 2, -1) # 4375/3456
SpGrM3 : [4, -2, 1, -1] :: (0, 4, 2, 1) # 80/63
d4 : [5, 0, -2, 0] :: (1, 4, 3, 0) # 32/25
SbGr4 : [4, -7, 2, 1] :: (0, 5, 3, -1) # 2800/2187
SpM3 : [0, 2, 0, -1] :: (1, 4, 2, 1) # 9/7
GrA3 : [-1, -5, 4, 0] :: (0, 5, 2, 0) # 625/486
Acd4 : [1, 4, -3, 0] :: (2, 4, 3, 0) # 162/125
Sb4 : [0, -3, 1, 1] :: (1, 5, 3, -1) # 35/27
GrGr4 : [10, -9, 2, 0] :: (-1, 5, 3, 0) # 25600/19683
A3 : [-5, -1, 3, 0] :: (1, 5, 2, 0) # 125/96
SbAc4 : [-4, 1, 0, 1] :: (2, 5, 3, -1) # 21/16
Spd4 : [7, 2, -3, -1] :: (1, 4, 3, 1) # 1152/875
Gr4 : [6, -5, 1, 0] :: (0, 5, 3, 0) # 320/243
AcA3 : [-9, 3, 2, 0] :: (2, 5, 2, 0) # 675/512
P4 : [2, -1, 0, 0] :: (1, 5, 3, 0) # 4/3
SpA3 : [-3, 1, 2, -1] :: (1, 5, 2, 1) # 75/56
Ac4 : [-2, 3, -1, 0] :: (2, 5, 3, 0) # 27/20
SbA4 : [-3, -4, 3, 1] :: (1, 6, 3, -1) # 875/648
SpGr4 : [8, -3, 0, -1] :: (0, 5, 3, 1) # 256/189
AcAc4 : [-6, 7, -2, 0] :: (3, 5, 3, 0) # 2187/1600
Sp4 : [4, 1, -1, -1] :: (1, 5, 3, 1) # 48/35
GrA4 : [3, -6, 3, 0] :: (0, 6, 3, 0) # 1000/729
SbGrd5 : [4, -4, 0, 1] :: (1, 6, 4, -1) # 112/81
SpAc4 : [0, 5, -2, -1] :: (2, 5, 3, 1) # 243/175
A4 : [-1, -2, 2, 0] :: (1, 6, 3, 0) # 25/18
Sbd5 : [0, 0, -1, 1] :: (2, 6, 4, -1) # 7/5
AcA4 : [-5, 2, 1, 0] :: (2, 6, 3, 0) # 45/32
SpSp4 : [6, 3, -2, -2] :: (1, 5, 3, 2) # 1728/1225
SbSb5 : [-5, -3, 2, 2] :: (2, 7, 4, -2) # 1225/864
Grd5 : [6, -2, -1, 0] :: (1, 6, 4, 0) # 64/45
SpA4 : [1, 0, 1, -1] :: (1, 6, 3, 1) # 10/7
d5 : [2, 2, -2, 0] :: (2, 6, 4, 0) # 36/25
SbGr5 : [1, -5, 2, 1] :: (1, 7, 4, -1) # 350/243
SpAcA4 : [-3, 4, 0, -1] :: (2, 6, 3, 1) # 81/56
Acd5 : [-2, 6, -3, 0] :: (3, 6, 4, 0) # 729/500
Sb5 : [-3, -1, 1, 1] :: (2, 7, 4, -1) # 35/24
GrGr5 : [7, -7, 2, 0] :: (0, 7, 4, 0) # 3200/2187
SbAc5 : [-7, 3, 0, 1] :: (3, 7, 4, -1) # 189/128
Spd5 : [4, 4, -3, -1] :: (2, 6, 4, 1) # 1296/875
Gr5 : [3, -3, 1, 0] :: (1, 7, 4, 0) # 40/27
Sbd6 : [4, -1, -2, 1] :: (2, 7, 5, -1) # 112/75
P5 : [-1, 1, 0, 0] :: (2, 7, 4, 0) # 3/2
Grd6 : [10, -3, -2, 0] :: (1, 7, 5, 0) # 1024/675
Ac5 : [-5, 5, -1, 0] :: (3, 7, 4, 0) # 243/160
SbA5 : [-6, -2, 3, 1] :: (2, 8, 4, -1) # 875/576
SpGr5 : [5, -1, 0, -1] :: (1, 7, 4, 1) # 32/21
d6 : [6, 1, -3, 0] :: (2, 7, 5, 0) # 192/125
AcAc5 : [-9, 9, -2, 0] :: (4, 7, 4, 0) # 19683/12800
Sp5 : [1, 3, -1, -1] :: (2, 7, 4, 1) # 54/35
GrA5 : [0, -4, 3, 0] :: (1, 8, 4, 0) # 125/81
Sbm6 : [1, -2, 0, 1] :: (2, 8, 5, -1) # 14/9
Acd6 : [2, 5, -4, 0] :: (3, 7, 5, 0) # 972/625
SpAc5 : [-3, 7, -2, -1] :: (3, 7, 4, 1) # 2187/1400
A5 : [-4, 0, 2, 0] :: (2, 8, 4, 0) # 25/16
SbAcm6 : [-3, 2, -1, 1] :: (3, 8, 5, -1) # 63/40
Spd6 : [8, 3, -4, -1] :: (2, 7, 5, 1) # 6912/4375
Grm6 : [7, -4, 0, 0] :: (1, 8, 5, 0) # 128/81
AcA5 : [-8, 4, 1, 0] :: (3, 8, 4, 0) # 405/256
SpSp5 : [3, 5, -2, -2] :: (2, 7, 4, 2) # 1944/1225
m6 : [3, 0, -1, 0] :: (2, 8, 5, 0) # 8/5
SpA5 : [-2, 2, 1, -1] :: (2, 8, 4, 1) # 45/28
Acm6 : [-1, 4, -2, 0] :: (3, 8, 5, 0) # 81/50
SbM6 : [-2, -3, 2, 1] :: (2, 9, 5, -1) # 175/108
Spm6 : [5, 2, -2, -1] :: (2, 8, 5, 1) # 288/175
GrM6 : [4, -5, 2, 0] :: (1, 9, 5, 0) # 400/243
M6 : [0, -1, 1, 0] :: (2, 9, 5, 0) # 5/3
Sbd7 : [1, 1, -2, 1] :: (3, 9, 6, -1) # 42/25
AcM6 : [-4, 3, 0, 0] :: (3, 9, 5, 0) # 27/16
SbA6 : [-5, -4, 4, 1] :: (2, 10, 5, -1) # 4375/2592
SpGrM6 : [6, -3, 1, -1] :: (1, 9, 5, 1) # 320/189
Grd7 : [7, -1, -2, 0] :: (2, 9, 6, 0) # 128/75
SpM6 : [2, 1, 0, -1] :: (2, 9, 5, 1) # 12/7
GrA6 : [1, -6, 4, 0] :: (1, 10, 5, 0) # 1250/729
d7 : [3, 3, -3, 0] :: (3, 9, 6, 0) # 216/125
A6 : [-3, -2, 3, 0] :: (2, 10, 5, 0) # 125/72
Acd7 : [-1, 7, -4, 0] :: (4, 9, 6, 0) # 2187/1250
Sbm7 : [-2, 0, 0, 1] :: (3, 10, 6, -1) # 7/4
AcA6 : [-7, 2, 2, 0] :: (3, 10, 5, 0) # 225/128
SbAcm7 : [-6, 4, -1, 1] :: (4, 10, 6, -1) # 567/320
Spd7 : [5, 5, -4, -1] :: (3, 9, 6, 1) # 7776/4375
Grm7 : [4, -2, 0, 0] :: (2, 10, 6, 0) # 16/9
SpA6 : [-1, 0, 2, -1] :: (2, 10, 5, 1) # 25/14
m7 : [0, 2, -1, 0] :: (3, 10, 6, 0) # 9/5
Acm7 : [-4, 6, -2, 0] :: (4, 10, 6, 0) # 729/400
SbM7 : [-5, -1, 2, 1] :: (3, 11, 6, -1) # 175/96
Spm7 : [2, 4, -2, -1] :: (3, 10, 6, 1) # 324/175
GrM7 : [1, -3, 2, 0] :: (2, 11, 6, 0) # 50/27
Sbd8 : [2, -1, -1, 1] :: (3, 11, 7, -1) # 28/15
M7 : [-3, 1, 1, 0] :: (3, 11, 6, 0) # 15/8
SbSb8 : [-3, -4, 2, 2] :: (3, 12, 7, -2) # 1225/648
Grd8 : [8, -3, -1, 0] :: (2, 11, 7, 0) # 256/135
AcM7 : [-7, 5, 0, 0] :: (4, 11, 6, 0) # 243/128
SbA7 : [-8, -2, 4, 1] :: (3, 12, 6, -1) # 4375/2304
SpGrM7 : [3, -1, 1, -1] :: (2, 11, 6, 1) # 40/21
SbGr8 : [3, -6, 2, 1] :: (2, 12, 7, -1) # 1400/729
d8 : [4, 1, -2, 0] :: (3, 11, 7, 0) # 48/25
SpM7 : [-1, 3, 0, -1] :: (3, 11, 6, 1) # 27/14
GrA7 : [-2, -4, 4, 0] :: (2, 12, 6, 0) # 625/324
Acd8 : [0, 5, -3, 0] :: (4, 11, 7, 0) # 243/125
Sb8 : [-1, -2, 1, 1] :: (3, 12, 7, -1) # 35/18
GrGr8 : [9, -8, 2, 0] :: (1, 12, 7, 0) # 12800/6561
A7 : [-6, 0, 3, 0] :: (3, 12, 6, 0) # 125/64
SbAc8 : [-5, 2, 0, 1] :: (4, 12, 7, -1) # 63/32
Spd8 : [6, 3, -3, -1] :: (3, 11, 7, 1) # 1728/875
Gr8 : [5, -4, 1, 0] :: (2, 12, 7, 0) # 160/81
AcA7 : [-10, 4, 2, 0] :: (4, 12, 6, 0) # 2025/1024
Sbd9 : [6, -2, -2, 1] :: (3, 12, 8, -1) # 448/225
P8 : [1, 0, 0, 0] :: (3, 12, 7, 0) # 2/2
The interval coordinates in square brackets are in terms of prime harmonics, while the interval coordinates in parentheses are in terms of Lilley-Johnston commas.