Frequency-Space Just Intonation
: Intro
: Archytas's Harmonic Means
: Primodality and NEJIs
: Otonal and Utonal Chords
: Three Scales From Ben Johnston
: Intervallic Analysis Of Erose's Otonal Chords
: Mannfishh's Spectralism
:: Intro
There are a few Just Intonation composers who do interesting work in frequency space that can be difficult to characterize in interval space. I'm going to talk about that work a little bit and try to discuss it from an intervallic perspective all the same.
:: Archytas's Harmonic Means
...
:: Primodality and NEJIs
One accomplished composer and theorist who mostly works in frequency space is Zheanna Erose. She does good work in 31-EDO, writing nice music and teaching people her own names for unusual chords and modes in 31-EDO, but also she researches high-prime limit just intonation, which is what we'll be focusing on here.
Erose finds scales of rational frequencies that share a denominator, like a scale where everything is of the form {n/11} for a whole number {n}. This is effectively looking at the harmonic series over the 11th harmonic, but she won't necessarily use all the harmonics and she won't necessarily keep them in order. Sometimes she skips notes from the harmonic series so that the scale has a logarithmically more equal spacing, and sometimes this is explicitly done by choosing scale degrees that approximate steps of an EDO, in which case she has produced a "near equal just intonation scale" or NEJI. She's not strict about even spacing though: she uses the narrowing of intervals at the high end of her scales to good musical effect, allowing for fine-grained melodic ornaments, and will also uses modal rotations of her scales to get varying sonorities. Erose claims that different denominators have their own characteristic sounds, and that you might not hear the characteristic sound in any given frequency ratio of a scale, but you hear the sonority in the overall relations between them. She calls this idea - that scales with fixed denominators have characteristic sounds tied to their denominator - "prime modality", and then misspells it "primodality" for fun. I think she'll also talk about a collection of ratios with a common denominator as being "a primodality".
Erose uses octave-equivalence productively, i.e. she would say that the ratio 25/22 is in the same prime mode as 13/11, since 25/11 is in that mode and 25/22 is one of its octave equivalents - thus a scale over a denominator of 22 could have the same sonority as a scale over 11. On the other hand, she rejects the use of octave complementation - i.e. 22/13 can not be a member of an scale over a denominator of 11.
This overtone scale approach is a very simple way to get very high prime-limit frequency ratios into your music. Like, between numerators of 11 and 22, i.e. in the first octave of harmonics over a denominator of 11, we have primes 13, 17, and 19. You might not know how to use factors of 19 in a regular intervallic way, but with Erose's methods, you don't have to: you can use 19 when it shows up in 19/11 and not use it again anywhere else. This is a very liberating aspect of Erose's philosophy of composition: it is not any harder to use larger primes.
Erose seems to have taken a lot of inspiration from the just intonation composer Harry Partch, who wrote at length about the overtone series (which describes the component vibrations inside harmonic instruments) and the undertone series (which doesn't describe anything in nature, but some people still like to compose with it). For an example of inspiration, instead of calling "11" the denominator in the {n/11} series, they would both call {11} the "numerary nexus". I have sometimes been offput by terminology like this, and since Partch didn't make good music, that was never a problem for me, but Erose makes beautiful music, so we're devoting most of chapter to her work, if not using her terminology very much.
: Otonal and Utonal Chords
Harry Partch wrote about harmonies made of overtones and "undertones", and we're going to learn about his framework a little bit so that we can better analyze some high-prime limit overtone chords and scales that Erose uses.
First up: we'll call chords made of frequency ratios "tuned chords". They differ from intervallic chords (made of intervals), like [P1, M3, P5].
Here are the just frequency ratios for a 5-limit major chord: [1/1, 5/4, 3/2]. We can multiply all the ratios by the least common multiple of the denominators to get a sequence of integers:
LCM(1, 4, 2) = 4
[1/1, 5/4, 3/2] * 4 = [4, 5, 6].
This is the overtone representation of the tuned chord. To get back to the ratios, we divide everything through by the first element of the otonal list.
[4, 5, 6] / 4 = [1/1, 5/4, 3/2]
In the language of Partch, the overtone representations of harmony is called "otonality" and the undertone representation of harmony is called "utonality", which we'll get to shortly. It's customary to write these representations with the tones separated by a colon, e.g. the tuned just major chord is written
[4:5:6]
And we'll do that, but I think it's easier to see that these are overtones above a common denominator if we write it as
[4/4, 5/4, 6/4]
Tuned chords can be transformed in two important ways: we can rotate their elements, which would be called "inversion" if we were working with regular intervallic chords, and we call also convert to between otonal and utonal representations.
Here's how to find cyclic inversions of an otonal chord. This only works well if your chord is less than octave in span, but it's still an important operation:
1) Move the first ratio up an octave, i.e. pop the first element off of the list, multiply it by 2, and stick it at the end of the list.
2) Divide all the list elements by the new first element. Multiply through by the least common multiple of the denominators if you want that chordal inversion to be represented otonally again.
For example, the otonal just major chord, [4:5:6], has cyclic inversions of [5:6:8] and [3:4:5].
The other important operation on tuned chords, at least for Partch, is to convert between otonality and utonality. If a tuned just major chord is written briefly as
[4:5:6] -> [4/4, 5/4, 6/4] = [1/1, 5/4, 3/2]
then its utonal inverse could be written
[6:5:4] -> [6/6, 6/5, 6/4] = [1/1, 6/5, 3/2]
which is a tuned just minor chord. A chord that we found as a utonal inverse can also be represented otonally: like with any set of frequency ratios, we multiply through by the least common multiple of the denominators:
[1/1, 6/5, 3/2] * LCM(1, 5, 2) = [10, 12, 15]
So this is an otonal representation of the just minor chord. Since undertones aren't a physical phenomenon, or at the very least least since vibrations of harmonic instruments aren't described by a harmonic undertone series, I mostly just work with otonal representations. Here's the procedure for finding otonal reciprocals, i.e. for going from [4:5:6] to [10:12:15]:
1) For each element {e} in a list, replace that element with its reciprocal {1 / e} .
2) Reverse the list (which will sort the numbers of new the list by size) and then multiply through by the least common multiple of denominators.
If you take the reciprocal again, then you get back to the original otonal chord.
Here are the three inversions of the otonal just major chord, each one paired up with its otonal reciprocal after a double colon:
[4, 5, 6] :: [10, 12, 15]
[5, 6, 8] :: [15, 20, 24]
[3, 4, 5] :: [12, 15, 20]
The major/minor reciprocity that I showed makes the otonality/utonality distinction look like a very natural way of representing chords, but things don't stay so well behaved when you add on more chord tones.
For example, if we add a harmonic 7th to a just major chord:
[4:5:6:7]
then the reciprocal becomes
[7/7, 7/6, 7/5, 7/4] * LCM(7, 6, 5, 4) = [7/7, 7/6, 7/5, 7/4] * 420 = [420:490:588:735]
In intervals, we would call this
[P1, Sbm3, Sbd5, Sbm7]
While you might have thought that the otonal/utonal reciprocity would always pair major chords up with minor chords, we can see here that extending a major chord with a tone can both hugely change the reciprocal chord's sonority and can make the reciprocal chord's otonal representation contain very large integers.
Now, I think Partch would have said that a chord is otonal in character if it has a small otonal representation and it is utonal in character if it has a small utonal representation; thus, for Partch, the fact that this septimal diminished 7th chord above has a 735 in its otonal representation just means that we're incorrectly representing a utonal chord. Even if we accept this, I think the fact that a tuned chord's reciprocal can change sonority so drastically when we add on a tone shows that chord reciprocity isn't as fundamental of relationship as the major/minor example would at first suggest.
:: Three Scales From Ben Johnston
Ben Johnston was a student of Harry Partch with perfect pitch who went on to become perhaps the finest microtonal composer ever. I wish I understood a fraction of what Ben Johnston does and how he composes. We are at least going to look at three of his scales, one of which will vastly simplify our analysis of chords and scales from Zheanna Erose, when we get to those.
Johnston sometimes used a chromatic scale made of octave reduced harmonics. Let's try to construct it. First we take the harmonics from 9 to 27 and reduce them to by octaves to be <= 2/1:
[9/8, 10/8, 11/8, 12/8, 13/8, 14/8, 15/8, 16/8, 17/16, 18/16, 19/16, 21/16, 22/16, 23/16, 24/16, 25/16, 26/16, 27/16]
When we put ratios is lowest terms we see that we have two copies each of [9/8, 3/2, 11/8, and 13/8], so we toss out those redundancies. If we sort the remaining ratios numerically, we get something very close to a chromatic scale. The scale gives us two options for an interval around 800 cents, namely a 5-limit augmented fifth, with ratio 25/16 at 773 cents, and the tridecimal prominent minor sixth, with ratio 13/8 at 840 cents. Johnston ultimately discards the A5, which I think is the clear and correct choice if you want to make music that sounds like it was plucked from the harmonic series. The scale also offers two options for a tritone interval around 600 cents, namely a noble acute augmented fourth, NbAcA4, with ratio 23/16 at 628 cents, and the ascendant fourth, As4, with ratio 11/8 at 551 cents. The first of these two tritones is indistinguishable from a just diminished fifth of 36/25 at 631 cents, but Johnston doesn't want to sound like 5-limit just intonation, he wants a harmonic series, so again he uses the lower prime option and 11/8 becomes his tritone of choice.
All together now, his chromatic overtone scale is:
[1/1, 17/16, 9/8, 19/16, 5/4, 21/16, 11/8, 3/2, 13/8, 27/16, 7/4, 15/8, 2/1] # [P1, ExA1, AcM2, Frm3, M3, SbAc4, As4, P5, Prm6, AcM6, Sbm7, M7, P8]
Johnston also used the octave complement of this scale, a chromatic undertone scale with powers of two in the numerators of the ratios:
[1/1, 16/15, 8/7, 32/27, 16/13, 4/3, 16/11, 32/21, 8/5, 32/19, 16/9, 32/17, 2/1] # [P1, m2, SpM2, Grm3, ReM3, P4, De5, SpGr5, m6, BrM6, Grm7, Hbd8, P8]
Sometimes he'd use both scales in one piece, with a utonal chord following an otonal chord. He was great.
Ben Johnston had another scale which is more easily explained in frequency space than in interval space. The guiding principle of the scale is to use super-particular ratios (those of the form {n / n - 1}) as steps between scale degrees. In particular, if you use super-particular ratios with numerators from 9 to 16,
[9/8, 10/9, 11/10, 12/11, 13/12, 14/13, 15/14, 16/15]
as relative scale degrees, but with a little rotation
[12/11, 11/10, 10/9, 9/8, 16/15, 15/14, 14/13, 13/12] : [DeAcM2, Asm2, M2, AcM2, m2, SpA1, ReSbAcM2, Prm2]
then you can hit lots of simple useful intervals:
[1/1, 12/11, 6/5, 4/3, 3/2, 8/5, 12/7, 24/13, 2/1] : [P1, DeAcM2, m3, P4, P5, m6, SpM6, ReM7, P8] :
This scale only has one (neutral) second, and one (minor) third, so it doesn't support chromaticism at the low end as well as the previous scales did, but you can still make some cool music out of it. Johnston called this scale "Eu15".
:: Intervallic Analysis Of Erose's Otonal Chords
Zheanna Erose has a huge list of original names for tuned otonal chords and scales on the Xenharmonic wiki. I think if we analyze them intervallically and compare our intervals with her names, we'll find some insight into how to use high-prime frequency ratios.
I went through all the 3, 5, and 7-limit chords and I had very little to say that wasn't critical. But I don't want to be critical, so we'll skip over those. Fortunately there were also a few interesting scales.
She has some chords and scales that are made of harmonics over powers of two, just like Ben Johnston's chromatic overtone scale. For example, her "Harmonic Lydian" scale mixes rank-2 ad rank-3 intervals to get ratios where all denominators have powers of 2:
"Harmonic Lydian": [32:36:40:45:48:54:60]
[1/1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8] # [P1, AcM2, M3, AcA4, P5, AcM6, M7]
I think that's a little clever. A Lydian scale is a major scale with an A4 in rank-2 interval space or an AcA4 in rank-3 interval space, as appears here. All of these intervals are in Johnston's chromatic overtone scale except for the AcA4, since Johnston uses As4 ~ 11/4 as his tritone.
Erose's "Harmonic Ionian" scale is simply a major scale subset of Johnston's chromatic overtone scale.
"Harmonic Ionian - LC": [16:18:20:21:24:27:30]
[1, 9/8, 5/4, 21/16, 3/2, 27/16, 15/8] # [P1, AcM2, M3, SbAc4, P5, AcM6, M7]
This is great! I'd do the same thing. The beauty of having Johnston's chromatic scale is that we don't need to comment on most possible overtone scales and chords with powers of two in the denominators, because they'll mostly be drawn in the obvious way from the foundational scale that we're already familiar with.
Erose has a scale called the "septimal Lydian heptad" (i.e. seven note scale). A normal Lydian scale is a major scale with an augmented fourth, and if I wanted to make this septimal, I'd raise all he major intervals by Sp1. Here's Erose's septimal lydian scale:
"Septimal Lydian Heptad": [14:16:18:20:21:24:27]
[1, 8/7, 9/7, 10/7, 3/2, 12/7, 27/14] # [P1, SpM2, SpM3, SpA4, P5, SpM6, SpM7]
It has the obvious choices for a septimal Lydian scale, but it's functional and inoffensive. Well done.
Erose also gives an over-7 major scale:
"Septimal Just Major 7 upperstructure with neutral lower": [21:24:27:32:40:48:60:72]
[1, 8/7, 9/7, 32/21, 40/21, 16/7, 20/7, 24/7] # [P1, SpM2, SpM3, SpGr5, SpGrM7, SpM9, SpA11, SpM13] :
I don't know why the M2 is repeated an octave up or why the 4thand 6th scale degrees are displaced an octave up. Using SpGr5, justly tuned to 32/21, as an over-7 approximation of P5 is pretty good - merely 27 cents sharp. I think SpA11 (or SpA4) was a bad choice. I'm not sure if "with neutral lower...." in the name was going to be addressing that before it got cut short, but if she'd just used a 28 between 27 and 32 in the otonal representation, that would have given a pure P4, since 28/21 = 4/3.
There's one septimal chord from Erose that I almost liked:
Septimal Phrygian Triad _ [14, 15, 21]
[P1, SpA1, P5] : [1, 15/14, 3/2]
A Phrygian scale has a m2 scale degree, so if you need to get a m2 sound from an over-7 ratio, SpA1 isn't a bad choice. But did it really need to be over 7? The unison and the fifth aren't septimal in reduced terms. Let's see if we can make a full septimal Phrygian scale. If the SpA1 trick worked once, it should work again, right?
"Over-7 Phrygian": [84:90:100:112:126:135:150]
[1/1, 15/14, 25/21, 4/3, 3/2, 45/28, 25/14] # [P1, SpA1, SpA2, P4, P5, SpA5, SpA6]
You can see that this doesn't work intervallically, and I bet she wouldn't even like it because the denominator 84 is quite far removed from 7, but it is a scale with a Phrygian sound that has multiples of 7 in the denominator.
Okay, time to look at some higher-rank ones. Let's go through her names for 11-limit, 13-limit, and 17-limit otonal structures now.
Erose uses a 17-limit interval as a minor ninth in an chord where the denominators are powers of 2:
"Dominant Ninth Minor" _ [8, 10, 12, 14, 17]
[P1, M3, P5, Sbm7, ExA8] : [1, 5/4, 3/2, 7/4, 17/8]
The exalted augmented eighth is justly tuned to 17/8, which has a size of 1305 cents, so that sounds like a 12-TET m9, which... feels a little dirty to me, since I'm all about replacing 12-TET mistunings with lower-prime intervals that reflect relationships among the lower stronger harmonics, whereas this is recreating a 12-TET sound using a fairly high and weak harmonic, but if you want a justly tuned .7b9 chord where all the denominators are powers of two, it is a good solution.
For an 11-limit chord over powers of 2, Erose uses 11/4 as as a tuned 11th interval:
"Harmonic Eleventh" _ [4, 5, 6, 7, 9, 11]
[P1, M3, P5, Sbm7, AcM9, As11] : [1, 5/4, 3/2, 7/4, 9/4, 11/4]
And I'm fine with that. Her harmonic 13th chord is basically the same and adds on a 13-limit ratio:
"Harmonic 13th" _ [8, 10, 11, 12, 13, 14]
[P1, M3, As4, P5, Prm6, Sbm7] : [1, 5/4, 11/8, 3/2, 13/8, 7/4]
I would have displaced the 4th and 6th by octaves to be 11th and13th intervals, because I like tertian spelling of chords, and I also would have included a AcM9 at 9/4. How does the harmonic 13 (different from the 13th harmonic) sound? Its tuning is 13/4 and it has a size of 2041 cents, so it sounds more like a neutral 13th than a minor 13th.
In another chord, Erose uses a 17-limit over-2 interval like it were a m6:
"Harmonic Aeolian Triad - R 2 b6" _ [32, 36, 51]
[P1, AcM2, ExA5] : [1, 9/8, 51/32]
So now you have some idea of how to use an ExA5 interval, if you were wondering which intervals with 17-limit just tunings had any uses.
She uses a different 17-limti interval for its resemblance to a diminished seventh:
"Harmonic Diminished Seventh" _ [10, 12, 14, 17]
[P1, m3, Sbd5, ExM6] : [1, 6/5, 7/5, 17/10]
At this rate we'll soon have a chromatic 17-limit scale, although all the interval ordinals will be wrong.
Here's an 11-limit over-11 chord:
Undecimal Sub5Neutral6add9 _ [22, 27, 36, 48, 32] ^ [P1, DeAcM3, DeAcM6, DeAcM9, De5] : [1, 27/22, 18/11, 24/11, 16/11]
Her otonal representation was out of order, so my intervals and ratios are also out of order, but it's fine. The third of this chord, 27/22, is a famous neutral third, reportedly introduced by lutenists Mansur Zalzal, that has been used in middle eastern music for like 1200 years. Not much middle eastern music has rich harmony, so it's nice to see a chord with a famous middle eastern ratio and other 11-limit ratios for harmonic context. The Zalzalian neutral third is often associated with two other over-11 ratios: a 12/11 neutral second, and an 81/44 neutral 7th. The most harmonic context I've ever seen for 27/22 was in a lute scale due to Yalcin Tura which closely approximates 24-EDO using 17-limit ratios, and through a little permutation, contains all of these over-11 ratios: [12/11, 27/22, 128/99, 16/11, 18/11, 81/44, 64/33, 24/11]. This contains all the tones in Erose's chord and more, but Tura didn't talk about how to make chords, and Erose gives us a suggestion. I haven't listened to it - it might sound horrible, but it's nice to hear that someone has harmonic suggestions.
And she's not even done. Erose uses 25/22, the justly tuned DeAcA2, like a major second:
Undecimal Sus2 Triad _ [22, 25, 33]
[P1, DeAcA2, P5] : [1, 25/22, 3/2]
And since it has a size of 221 cents, that's appropriate. That second had the interval quality "DeAcA", and the "DeAcA" interval quality shows up again in another chord:
"Undecimal Super4 Triad" _ [22, 30, 33]
[P1, DeAcA4, P5] : [1, 15/11, 3/2]
I don't know why you'd want a chord with that sharp fourth next to a perfect fifth; there's a very narrow ratio of 11/10 between those, which I would expect to sound quite grating, but maybe she likes it, or maybe she's just naming stuff because she can.
...
:: Mannfishh's Spectralism
We talked about composer Mannfishh a little bit in the chapter on seven-limit just intonation. When it comes to composing just intonation music in frequency space, Mannfishh has as interesting approach which he categorizes as part of the Spectralism movement in classical music. He will take a ratio, like 7/3, and split it up into lots of pieces, and compose short phrases with the pieces, and then split the interval up another way and repeat. These short compositions are more like portraits of a ratio than they are full musical pieces, but they're very interesting and he occasionally will develop a ratio portrait into a long-form piece with melodies and polyrhythms and other structure.
Here's how Mannfishh paints a portrait. First off, he treats 7/3 as a relationship between the third harmonic and the 7th harmonic over a fundamental. Every rational frequency thus has an implied fundamental, and he'll play this for you. It's like if we asked him about the chord [4:5:6], he'd play you [1:4:5:6]. The fundamental might be played very quietly, but he'll make you aware of it. Sometimes he'll play a full harmonic series like [1:2:3:4:5:6:7] so that you can hear how 7/3 exists within it.
Another thing Mannfishh will do to portray a ratio is to elaborate it through arithmetic division. For example to split 7/3 in half, we take the difference, 7 - 3 = 4, and split that over a mid point, i.e.
[3:5:7]
A handy way to split a ratio {m/n} into {k} divisions is to multiply the numerator and denominator by {k}, and then add the difference {m - n} to {kn} repeatedly till you reach {km}. For example, to split 7/3 into 3 divisions, we add 4 repeatedly to 3*3 until we reach 3*7:
[9:13:17:21]
Mannfishh will play lots of arithmetic divisions of a ratio like this. I think part of the appeal of arithmetic division is that the harmonic series is also an arithmetic series, so we're generalizing it. Also it's just something to do. Mannfishh will also sometimes use the geometric mean, which doesn't generally produce rational frequencies, to divide up a ratio. Sometimes you just want to play with a thing.
Mannfishh occasionally writes about "difference tones" in the marginal remarks on his pieces when he elaborates a ratio. I'm not sure I know what difference tones are; psychoacoustic literature talks about difference tones, combination tones, Tartini tones, heterodyne tones, and resultant tones, and I'm confused by the whole mess. I'm not sure which of those phenomena is involved, but supposedly, when two tones are played loudly, the sum and/or difference of their frequencies can also be heard, possibly in addition to sums and differences of integer multiples of the two frequencies. I'll just focus on the sum and difference of the unscaled frequencies, and call them sum and difference tones. Depending on the situation, these extra tones might just be a result of nonlinear response characteristics in your ear, or they might correspond to vibrations in your environment when your instruments have their own non-linear response characteristics. In so far as these tones exist, let's investigate them by considering two frequencies at 880hz * 1/1 and 880hz * 7/5.
The sum of these frequencies will be
880hz * (1 + 7/5) = 880hz * 12/5
and of the two differences that we could form, the one which is greater than 0hz will be
880hz * (7/5 - 1) = 880hz * 2/5
Clearly the base frequency factors out and doesn't need to be considered. If we combine the ratios of the two tones with their sum and difference tones, then in order of size we get:
[2/5, 1/1, 7/5, 12/5]
and we can multiply this through by 5 to get an otonal chord:
[2:5:7:12]
If we had just started with numerator {n} and denominator {d} in their otonal form,
[d:n]
then we could have gone straight to the answer:
[n - d : d : n : d + n]
I'm not positive, but I think Mannfishh will sometimes elaborate a ratio like 7/5 by playing [2:5:7:12]. And I'm sure he has other tricks up his sleeve. Every piece he makes is a little different. But the point is that he introduces higher primes using frequency based manipulations, like how dividing 7/3 into three got produced factors of 13 and 17:
[9:13:17:21]
or how using sum and difference tones could introduce factors of 3 and 11 into the characterization of the ratio 7/4:
[3:4:7:11]
And these introduced ratios are somewhat interpretable: they're audible sum and difference tones, or they have an arithmetic structure like the harmonic series.
Here's a question: Can we give an intervallic interpretation to Mannfishh's procedures of arithmetic ratio division, or to the notion of sum and difference tones? My guess is not, since these introduce new primes, or since they're additive in frequency space, whereas the analog of interval arithmetic is multiplicative in frequency space, but I'm open to hearing suggestions.
:: NEJI Scales
We talked a lot about how just interpretatons of Erose style primodal scales. Next we're going to talk about irrational interpreatations. In particularly, we're going to look more concretely at Erose-style NEJI scales, i.e. "near equal just intonation scales" that are designed specifically to sound like an EDO. We'll start by picking a number of divisions for an EDO and a "nexus", i.e. a denominator to use for our fractions. Then for each step {i} of from [0, edo-divisions], we find the numerator {y} such that {y / nexus} is cloest to 2^( i / divisions) for each step. If a numerator is repeated for multiple steps when you do this prodcedure, you could try a numerator of less good fit, but I just throw the whole thing out: this nexus didn't work to make a NEJI scale for this EDO; let's try again with another. When you're done, you can judge the goodness of fit of your NEJI scales in two ways: you can look at the largest error in cents between an EDO step and your rational approximation, and you can also look at the sum of all of the errors. If both are small, then well done, you've found a fairly good NEJI scale.
Here are two good NEJIs for 12-EDO:
12-EDO # [22:23:25:26:28:29:31:33:35:37:39:42]
12-EDO # [23:24:26:27:29:31:33:34:37:39:41:43]
Here are two good NEJI scales for 19-EDO
19-EDO # [33:34:35:37:38:40:41:43:44:46:48:49:51:53:55:57:59:61:64]
19-EDO # [36:37:39:40:42:43:45:46:48:50:52:54:56:58:60:62:65:67:69]
Here are three good ones for 24-EDO:
24-EDO # [42:43:44:46:47:49:50:51:53:54:56:58:59:61:63:65:67:69:71:73:75:77:79:82]
24 EDO # [43:44:46:47:48:50:51:53:54:56:57:59:61:63:64:66:68:70:72:74:77:79:81:84]
24-EDO # [46:47:49:50:52:53:55:56:58:60:61:63:65:67:69:71:73:75:77:80:82:84:87:89]
As you get to higher EDOs, it's easier to approximate steps with just ratios. For 31-EDO, all of these nexus denominators:
[43, 46, 51, 52, 53, 55, 57, 58, 60, 61]
provide total error < 20 cents and largest absolute error < 15 cents. Are any of these scales good? I don't know. I think Erose would accept a worse fit for a NEJI scale to its EDO if she got a simpler nexus out of it, so she might use different NEJI scales than these. I think maybe she'd like the over-52 and and over-60 version of 31-EDO on that basis, since 52 is just a few factors of 2 away from 13 and 60 is just a few factors of two away from 15.
If we're looking for denominators that are factors of 2 larger than [2, 3, 5, 7, 11, 13, 15], then the denominator 64 produces NEJI scales with very good approximations to EDOs [32, 35, 40, 45, 46, and 47]. The denominator 60 gives good approximations to 35-EDO and 41-EDO.
For 53-EDO, the denominators of 104 (an over-13 scale) and 120 (an over-15 scale) do quite well.
Anyway, pick a scale, play around, see if you like the sound of it, see what you can do with it. One thing you can do is feel smug that you kind of know how to use very high prime limit frequency ratios. Also you can write beautiful music with just frequency ratios structured in an interesting way.