High Prime-Limit Just Intonation

The unison, perfect fourth, perfect fifth, an octave (P1, P4, P5, P8) are all "perfect" intervals. The imperfect intervals with ordinals between P1 and P8 are 2nds, 3rds, 6ths, and 7ths. You can mess with the tuning of these more readily than the imperfect intervals; they're more fluid. Let's look at some ways to tune the imperfect intervals at various prime limits.

For rank-2 intervals (3-limit ratios), we have Pythagorean tuning:

[Grm2, AcM2, Grm3, AcM3, Grm6, AcM6, Grm7, AcM7] : [256/243, 9/8, 32/27, 81/64, 128/81, 27/16, 16/9, 243/128]

For rank-3 intervals (5-limit ratios), we have 5-limit just intonation:

[m2, M2, m3, M3, m6, M6, m7, M7] : [16/15, 10/9, 6/5, 5/4, 8/5, 5/3, 9/5, 15/8]

In the post on 7-limit just intonation, we looked at scales that had imperfect intervals with super-major and sub-minor qualities, and also a separate set of imperfect intervals that differed from those ones by syntonic commas:

[Sbm2, SpM2, Sbm3, SpM3, Sbm6, SpM6, Sbm7, SpM7] : [28/27, 8/7, 7/6, 9/7, 14/9, 12/7, 7/4, 27/14]

[SbAcm2, SpGrM2, SbAcm3, SpGrM3, SbAcm6, SpGrM6, SbAcm7, SpGrM7] : [21/20, 640/567, 189/160, 80/63, 63/40, 320/189, 567/320, 40/21]

I'm here to tell you that if you want to get familiar with higher rank interval spaces and higher prime-limit ratios, you can just keep making scales by messing with the imperfect intervals.

For rank-5 interval space (11-limit ratios), we have {Ascendant minor} and {Descedant Major} qualities.

[Asm2, DeM2, Asm3, DeM3, Asm6, DeM6, Asm7, DeM7] : [11/10, 320/297, 99/80, 40/33, 33/20, 160/99, 297/160, 20/11]

and a related set that differs from that by syntonic commas:

[AsGrm2, DeAcM2, AsGrm3, DeAcM3, AsGrm6, DeAcM6, AsGrm7, DeAcM7] : [88/81, 12/11, 11/9, 27/22, 44/27, 18/11, 11/6, 81/44]

While the simplest 7-limit ratios give you flattened minor intervals and sharpened major intervals, in 11-limit just intonation our minor intervals and major intervals instead meet near the middle at neutral seconds, thirds, sixths, and sevenths.

Interestingly, if we switch which of (As|De) pairs with (M|m) in the first set, we still get a set of moderately simple just ratios: 

[Dem2, AsM2, Dem3, AsM3, Dem6, AsM6, Dem7, AsM7] : [512/495, 55/48, 64/55, 165/128, 256/165, 55/32, 96/55, 495/256]

But clearly it is the neutral intervals of the first two 11-limit tone sets that are simpler in their tunings, and these widened imperfect intervals are less central examples of 11-limit's core sound.

There are two sets of rank-6 imperfect intervals (with 13-limit just tunings) that have fairly short names and simple justly tuned ratios. They are

[Prm2, ReM2, Prm3, ReM3, Prm6, ReM6, Prm7, ReM7] : [13/12, 128/117, 39/32, 16/13, 13/8, 64/39, 117/64, 24/13]

[PrSpGrm2, ReSbAcM2, PrSpGrm3, ReSbAcM3, PrSpGrm6, ReSbAcM6, PrSpGrm7, ReSbAcM7] : [208/189, 14/13, 26/21, 63/52, 104/63, 21/13, 13/7, 189/104]

Here's an imperfect interval set in rank-7 interval space (and 17-limit just intonation):

[Hbm2, ExM2, Hbm3, ExM3, Hbm6, ExM6, Hbm7, ExM7] : [160/153, 17/15, 20/17, 51/40, 80/51, 17/10, 30/17, 153/80]

And here's an imperfect interval set in rank-8 interval space (and 19-limit just intonation):

[Lfm2, LwM2, Lfm3, LwM3, Lfm6, LwM6, Lfm7, LwM7] : [19/18, 64/57, 19/16, 24/19, 19/12, 32/19, 57/32, 36/19]

Though I'm still deciding on adjectives / interval qualities for 19-limit and this last list may change soon.

The 3-limit, the 7-limit and the 17-limit intonations behave similarly in that they sharpen the major intervals relative to 5-limit and they flatten the minor intervals relative to 5-limit.

3: AcM, Grm (Acute major, grave minor)

7: SpM, Sbm (Super major, sub minor)

17: ExM, Hbm (Exalted major, humbled minor)

The 11-limit, 13-limit, and 19-limit intonations are similar in that they place the sharpening accidental on the minor intervals and the flattening accidental on the major intervals:

11: DeM, Asm (Descendant major, Ascendant minor)

13: ReM, Prm (Recessed major, Prominent minor)

19: LwM, Ltm (Lowly major, Lofty minor)

To be adept with high prime-limit just intonation requires a lot more than using one of these interval spaces; I think you have to be able to use intervals from lower rank spaces simultaneously. There's more to composing well with these things than mistuning a chromatic scale. But if you want to get familiar with intervals of each rank, then chromatic scales based on these imperfect interval sets are a good starting place. We'll discuss more tricks for using high rank intervals and high prime limit ratios in the chapter on Frequency Space Just Intonation.

A chromatic scale has perfect intervals, minor and major imperfect intervals, and modified imperfect intervals at A4 and/or d5. I confess that I don't have a general procedure for pairing up tritones with the previous tone spaces so that we get full chromatic scales. I will still show you some tritones of different prime-limits.

These 7-limit tritones:

(SpA4, 10/7) _ (Sbd5, 7/5)

have simple names like the first set of imperfect intervals, but the ratios have a factor of 5 on the side opposite the side with a factor of 7, which is like the second set of septimal imperfect intervals. Contrarily, the names of these intervals names look more like those in my second 7-limit tone space:

(SpAcA4, 81/56) _ (SbGrd5, 112/81)

But the ratios are more similar to the first set in that the side which doesn't have a factor of 7 is 3-limit. These 7-limit tritones have interval names that look a little funny to me, but the ratios are fairly simple:

(SpSpA4, 72/49) _ (SbSbd5, 49/36)

Here are two tritones with simple 11-limit tunings:

(DeAcA4, 15/11), (AsGrd5, 22/15)

Other 11-limit tritones are significantly less simple in their just tunings:

(AsA4, 275/192) _ (Ded5, 384/275)

(DeA4, 400/297) _ (Asd5, 297/200)

Or significanly less simpel in their interval names (or both):

(DeDeAcAcA4, 162/121) _ (AsAsGrGrd5, 121/81)

(DeAcAcA4, 243/176) _ (AsGrGrd5, 352/243)

Although perhaps these two tones can function as tritones:

(As4, 11/8) _ (De5, 16/11)

And if so, they'd have the simplest names and ratios of the bunch.

Here are three pairs of 13-limit tritones:

(ReAcA4, 18/13) _ (PrGrd5, 13/9)

(ReSbAcA4, 35/26) _ (PrSpGrd5, 52/35)

(ReA4, 160/117) _ (Prd5, 117/80)

Here are some 17-limit tritones:

(ExA4, 17/12) _ (Hbd5, 24/17)

(Exd5, 918/625) _ (HbA4, 625/459)

Here are two pairs of 19-limit tritones:

(LwAcA4, 27/19) _ (LfGrd5, 38/27)

(LwA4, 80/57) _ (Lfd5, 57/40)

...

:: High Prime-Limit Commas

While I think there is value in mastering high prime limit just intonation, you  might be surprised to learn how little the basic intervals I showed above differ from each other.

For example, the octaved reduced 19th harmonic, 19/18, differs from the pythagorean minor third by a mere 3 cents:

(19/16) / (32/27) = 513/512 @ 3c

And this 2.3.19 comma relates all of the basic 19-limit impure intervals to the pythagorean intervals, e.g.

(19/18) / (256/243) = 513/512

(9/8) / (64/57) = 513/512

(81/64) / (24/19) = 513/512

This fact might lead you to think that 19-limit just intonation doesn't really differ from pythagorean intonation - that there is no point in it. I'm not sure if there is or not, but I'm not impressed with someone simply because they can make good music using 19-limit intonation.

It's not quite as startling, but the basic 11-limit and 13-limit frequency ratios have a very similar sound also.

For example, these important pairs differ from one another by a mere 5 cents:

(88/81) / (13/12) = 352/351

(44/27) / (13/8) = 352/351

(11/9) / (39/32) = 352/351

Now, if you want to be a great microtonalist, maybe you have to care about a difference of 5-cents and not shrug off these coincidences. I'm not good with 5-cent differences, but I still think that 13-limit intonation has value, both because it has historic use in analyzing middle eastern microtonal music and because the 13th harmonic is fairly perceptible, which means that it's not silly to include the 13th harmonic in chords - that's just reproducing some of the audible structure of the overtone series that exists in basically all resonant instruments.

What other commas might give us pause about the value of high-prime-limit just intonation? Can we say anything mean about 17-limit just intonation? I don't think so; 17-limit just intonation does a really good job of filling in the large ~26 cent gaps between 3-limit and 7-limit just intonation, and it does so very evenly, with a ~13-cent gap on either side. See for example this list of low-complexity major second intervals which increase in the size of their just tunings:

AcM2 # 9/8 _ 204c            3-limit

ExM2 # 17/15 _ 217c         17-limit

SpM2 # 8/7 _ 231c            7-limit

AsM2 # 55/48 _ 236c         11-limit

SpAcM2 # 81/70 _ 253c        7-limit

I haven't invested much effort into making music in 17-limit just intonation, but from this alone, it clearly has a unique sound for anyone who can hear 10 cents of frequency difference. So let's use it. Go forth and multiply frequency ratios.