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 and 3-limit ratios, we have Pythagorean tones:

3-limit

(Grm2, 256/243)

(AcM2, 9/8)

(Grm3, 32/27)

(AcM3, 81/64)

(Grm6, 128/81)

(AcM6, 27/16)

(Grm7, 16/9)

(AcM7, 243/128)

For rank-3 intervals and 5-limit ratios, we have 5-limit tones:

5-limit:

(m2, 16/15)

(M2, 10/9)

(m3, 6/5)

(M3, 5/4)

(m6, 8/5)

(M6, 5/3)

(m7, 9/5)

(M7, 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:

7-limit

(Sbm2, 28/27)

(SpM2, 8/7)

(Sbm3, 7/6)

(SpM3, 9/7)

(Sbm6, 14/9)

(SpM6, 12/7)

(Sbm7, 7/4)

(SpM7, 27/14)

And also imperfect intervals that differed from those by syntonic commas:

7-limit:

(SbAcm2, 21/20)

(SpGrM2, 640/567)

(SbAcm3, 189/160)

(SpGrM3, 80/63)

(SbAcm6, 63/40)

(SpGrM6, 320/189)

(SbAcm7, 567/320)

(SpGrM7, 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.

I'm not sure which tritones go well with these two sets of septimal imperfect intervals. These tritones:

Sbd5 _ 7/5

SpA4 _ 10/7

have simple names like the first set, but the ratios have a factor of 5 on the side opposite the side with a factor of 7, like in the second set of septimal imperfect intervals. Contrarily, these interval names look more like the second set of septimal imperfect intervals:

(SpAcA4, 81/56)

(SbGrd5, 112/81)

but the ratios are more similar to the first set in that the siee which doesn't have a factor of 7 is 3-limit.

Here's a set of rank-5 imperfect intervals with 11-limit just tunings. They have {Ascendant minor} and {Descedant Major} qualities.

11-limit:

(Asm2, 11/10)

(DeM2, 320/297)

(Asm3, 99/80)

(DeM3, 40/33)

(Asm6, 33/20)

(DeM6, 160/99)

(Asm7, 297/160)

(DeM7, 20/11)

There's also a set of simple 5-limit intervals / 11-limit ratios ratios that differs from those by syntonic commas:

11-limit:

(AsGrm2, 88/81)

(DeAcM2, 12/11)

(AsGrm3, 11/9)

(DeAcM3, 27/22)

(AsGrm6, 44/27)

(DeAcM6, 18/11)

(AsGrm7, 11/6)

(DeAcM7, 81/44)

While 7-limit just intonation gives you simple ratios for flatter versions of minor intervals and for sharper versions of major intervals, in 11-limit just intonation our minor intervals and major intervals 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, 512/495)

(AsM2, 55/48)

(Dem3, 64/55)

(AsM3, 165/128)

(Dem6, 256/165)

(AsM6, 55/32)

(Dem7, 96/55)

(AsM7, 495/256)

But clearly it is the neutral interval that are simpler in their tuning, and these widened imperfect intervals are clearly less central examples of 11-limit's basic sound.

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)

There are two sets of imperfect intervals in the 13-limit that have fairly short names and simple ratios when justly tuned. They are

13-limit:

(Prm2, 13/12)

(ReM2, 128/117)

(Prm3, 39/32)

(ReM3, 16/13)

(Prm6, 13/8)

(ReM6, 64/39)

(Prm7, 117/64)

(ReM7, 24/13)

and

13-limit:

(PrSpGrm2, 208/189)

(ReSbAcM2, 14/13)

(PrSpGrm3, 26/21)

(ReSbAcM3, 63/52)

(PrSpGrm6, 104/63)

(ReSbAcM6, 21/13)

(PrSpGrm7, 13/7)

(ReSbAcM7, 189/104)

Here are two simple 13-limit tritones that are octave complements:

(ReAcA4, 18/13)

(PrGrd5, 13/9)

And two more:

(ReSbAcA4, 35/26)

(PrSpGrd5, 52/35)

And two more:

(ReA4, 160/117)

(Prd5, 117/80)

Here's a 17-limit set of imperfect intervals:

(Hbm2, 160/153)

(ExM2, 17/15)

(Hbm3, 20/17)

(ExM3, 51/40)

(Hbm6, 80/51)

(ExM6, 17/10)

(Hbm7, 30/17)

(ExM7, 153/80)

And here are two simple 17-limit tritones:

(ExA4, 17/12)

(Hbd5, 24/17)

And here are two tritones with simple interval names but complicated 17-limit tunings:

(Exd5, 918/625)

(HbA4, 625/459)

Here's a 19-limit set of imperfect intervals:

(Ltm2, 19/18)

(LwM2, 64/57)

(Ltm3, 19/16)

(LwM3, 24/19)

(Ltm6, 19/12)

(LwM6, 32/19)

(Ltm7, 57/32)

(LwM7, 36/19)

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

Here are two 19-limit tritones:

(LwA4, 80/57)

(Lfd5, 57/40)

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 the chromatic scales 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.

Let's see all the imperfect interval intonation again quickly:

3-limit

[Grm2, AcM2, Grm3, AcM3, Grm6, AcM6, Grm7, AcM7]

[256/243, 9/8, 32/27, 81/64, 128/81, 27/16, 16/9, 243/128]

5-limit

[m2, M2, m3, M3, m6, M6, m7, M7]

[16/15, 10/9, 6/5, 5/4, 8/5, 5/3, 9/5, 15/8]

7-limit A:

[Sbm2, SpM2, Sbm3, SpM3, Sbm6, SpM6, Sbm7, SpM7]

[28/27, 8/7, 7/6, 9/7, 14/9, 12/7, 7/4, 27/14]

7-limit-B:

[SbAcm2, SpGrM2, SbAcm3, SpGrM3, SbAcm6, SpGrM6, SbAcm7, SpGrM7]

[21/20, 640/567, 189/160, 80/63, 63/40, 320/189, 567/320, 40/21]

11-limit A:

[Asm2, DeM2, Asm3, DeM3, Asm6, DeM6, Asm7, DeM7]

[11/10, 320/297, 99/80, 40/33, 33/20, 160/99, 297/160, 20/11]

11-limit B:

[AsGrm2, DeAcM2, AsGrm3, DeAcM3, AsGrm6, DeAcM6, AsGrm7, DeAcM7]

[88/81, 12/11, 11/9, 27/22, 44/27, 18/11, 11/6, 81/44]

11-limit C:

[Dem2, AsM2, Dem3, AsM3, Dem6, AsM6, Dem7, AsM7]

[512/495, 55/48, 64/55, 165/128, 256/165, 55/32, 96/55, 495/256]

13-limit A:

[Prm2, ReM2, Prm3, ReM3, Prm6, ReM6, Prm7, ReM7]

[13/12, 128/117, 39/32, 16/13, 13/8, 64/39, 117/64, 24/13]

13-limit B:

[PrSpGrm2, ReSbAcM2, PrSpGrm3, ReSbAcM3, PrSpGrm6, ReSbAcM6, PrSpGrm7, ReSbAcM7]

[208/189, 14/13, 26/21, 63/52, 104/63, 21/13, 13/7, 189/104]

17-limit:

[Hbm2, ExM2, Hbm3, ExM3, Hbm6, ExM6, Hbm7, ExM7]

[160/153, 17/15, 20/17, 51/40, 80/51, 17/10, 30/17, 153/80]

19-limit:

[Lfm2, LwM2, Lfm3, LwM3, Lfm6, LwM6, Lfm7, LwM7]

[19/18, 64/57, 19/16, 24/19, 19/12, 32/19, 57/32, 36/19]