SKULO interval names

SKULO names are the successor to SHEFKHED interval names. SKULO is an extended-diatonic system, wherein prefixes are added for sub-chroma alterations from standard Pythagorean diatonic intervals.

S signals the septimal intervals, altering a Pythagorean interval by the septimal or Archytas comma, 64/63.

K signals the classic (klassisch) intervals, and other intervals altered from Pythagorean by the syntonic comma (komma), 81/80. K is used instead of C to avoid confusion with note names.

U signals the undecimal intervals, altering a Pythagorean interval by the undecimal quarter tone or undecimal comma, 33/32.

S/s, Super/sub, septimal intervals

A capital 'S' signals alternation upward by 64/63, and a lower case 's' signals alteration downward by 64/63. The alteration is typically applied upward for 'positive' Pythagorean intervals - i.e., P5, major, augmented, doubly augmented intervals etc., and downward for 'negative' Pythagorean intervals, i.e., P4, minor, diminished, and doubly diminished intervals etc.

'S' is short for 'super', and 's' is short for sub.

For example, 7/4 is a subminor seventh, sm7, 7/6 is a subminor third, sm3, and 9/7 is a supermajor third, SM3.

This mirrors common practice for naming septimal intervals. These intervals may alternatively be described as the septimal minor seventh, minor third, and major third respectively.

S/s can be used to label all the intervals of 19edo and 26edo in such a way that only diatonic Pythagorean intervals are used, and the generic interval is strictly increasing (you don't have any seconds higher than any thirds etc.)

i.e.,

19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7 SM7/s8 P8.

26edo: P1 S1 sm2 m2 M2 SM2 sm3 m3 M3 SM3 s4 P4 A4 SA4/sd5 d5 P5 S5 sm6 m6 M6 SM6 sm7 m7 M7 SM7 s8 P8.

These are meantone edos, wherein 81/80 is tempered out, so M3 ~ kM3 ~ 5/4, for example.

K/k, komma-wide/komma-narrow, klassisch intervals

Similarly, a capital 'K' signals alteration upward by 81/80, and a lower case 'k' signals alteration downward by 81/80. The alteration is typically applied downward for positive Pythagorean intervals, and upward for negative Pythagorean intervals.

5/4 is kM3, a klassisch / classic major third, and 15/8 is kM7, a klassisch / classic major seventh. 6/5 is labelled Km3, a klassisch / classic minor third, a 9/5 is labelled a klassisch / classic minor seventh, or a komma-wide / comma-wide minor seventh, considering that 10/9 is probably better labelled a komma-narrow / comma-narrow major second. 27/20 and 40/27 are KP4 and kP5 respectively, komma-wide / comma-wide perfect fourth and komma-narrow / komma-narrow perfect fifth.

We can use K/k to label all the intervals of 10edo, 15edo, and 22edo, i.e.,

10edo: P1/m2 Km2/kM2 M2/m3 Km3/kM3 M3/P4 K4/k5 P5/m6 Km6/kM6 M6/m7 Km7/kM7 M7/P8.

15edo: P1/m2 Km2 kM2 M2/m3 Km3 kM3 M3/P4 K4 k5 P5/m6 Km6 kM6 M6/m7 Km7 kM7 M7/P8.

22edo: P1 m2 Km2 kM2 M2 m3 Km3 kM3 M3 P4 K4 kA4/Kd5 k5 P5 m6 Km6 kM6 M6 m7 Km7 kM7 M7 P8.

64/63 is tempered out in these three edos, so M3 ~ SM3 ~ 9/7, for example.

U/u, Uber/unter, undecimal intervals

A capital 'U' signals alteration upward by 33/32, and a lower case 'u' signals alteration downward by 33/32. The alteration is typically applied downward for positive Pythagorean intervals, and upward for negative Pythagorean intervals.

11/8 is UP4, and Uber perfect fourth and 11/6 is Um7, and Uber minor seventh. We can also add N/n, the greater and lesser neutrals, where Um is n, and uM is N, so that 11/6 is n7, the lesser neutral 7th, 11/9 is n3, the lesser neutral third, and 27/22 is N3, the greater neutral third. In tunings where the difference between a lessor and greater neutral of the same generic interval, is. 243/242 is tempered out, we can just use 'N' for 'neutral'.

We can use U/u and N to give familiar labels to 17edo, or to label 10edo as a neutral system, i.e.,

10edo: P1/m2 N2 M2/m3 N3 M3/P4 U4/u5 P5/m6 N6 M6/m7 N7 M7/P8.

17edo: P1 m2 N2 M2 m3 N3 M3 P4 U4 u5 m6 N6 M6 m7 N7 M7 P8.

SKU

We can use a combination of U/u and N and S/s to label 24edo, 31edo, 38edo, and 45edo.

24edo: P1 S1/U1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 U4 A4/d5 u5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/u8/s8 P8.

31edo: P1 S1/U1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 U4 A4 d5 u5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8/u8 P8.

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 U4 A4 SA4/sd5 d5 u5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8.

45edo: P1 S1/U1 uA1 A1 sm2 m2 n2 N2 M2 SM2 A2/d3 sm3 m3 n3 N3 M3 SM3 d4 s4 P4 U4 A4 SA4 sd5 d5 u5 P5 S5 A5 sm6 m6 n6 N6 M6 SM6 A6/d7 sm7 m7 n7 N7 M7 SM7 d8 Ud8 s8/u8 P8.

We have to use chromatic interval names in 38edo and 45edo, like in many larger edos.

We can use a combination of K/k and S/s to label the intervals of 29edo, i.e.,

29edo: P1 K1/S1/sm2 m2 Km2 kM2 M2 SM2/sm3 m3 Km3 kM3 M3 SM3/s4 P4 K4 kA4/d5 A4/Kd5 k5 P5 S5/sm6 m6 Km6 kM6 M6 SM6/sm7 m7 Km7 kM7 M7 SM7/S8/k8 P8.

We can use a combination of all the prefixes introduced so far to label the intervals of edos which do not temper out 64/63 or 81/80, i.e.,

34edo: P1 K1/S1/sm2 m2 Km2 N2 kM2 M2 SM2/sm3 m3 Km3 N3 kM3 M3 SM3/s4 P4 K4 U4/d5 kA4/Kd5 A4/u5 k5 P5 S5/sm6 m6 Km6 N6 kM6 M6 SM6/sm7 m7 Km7 N7 kM7 M7 SM7/k8/s8 P8.

41edo: P1 K1/S1 U1/sm2 m2 Km2 N2 kM2 M2 SM2 sm3 m3 Km3 N3 kM3 M3 SM3 s4 P4 K4 U4 kA4 Kd5 u5 k5 P5 S5 sm6 m6 Km6 N6 kM6 M6 SM6 sm7 m7 Km7 N7 kM7 M7 SM7/u8 k8/s8 P8.

46edo: P1 K1/S1 U1/sm2 m2 Km2 n2 N2 kM2 M2 SM2 sm3 m3 Km3 n3 N3 kM3 M3 SM3 s4 P4 K4 U4 uA4/d5 kA4/Kd5 A4/Ud5 SA4/u5 k5 P5 S5 sm6 m6 Km6 n6 N6 kM6 M6 SM6 sm7 m7 Km7 n7 N7 kM7 M7 SM7/u8 k8/s8 P8.

An astute reader, or a fan of 27edo, may be wondering what about 27edo? 27edo can be labelled using K/k and U/u (and N), though for 27edo this means using the 27e mapping, where 11/8 is mapped to it's second best approximation. 27e is generally preferred to 27p (patent 27edo, using the best approximation of 11/8 as well as the other primes in the 11-limit).

27edo: P1 K1/m2 U1/Km2 N2 kM2 M2 m3 Km3 N3 kM3 M3 P4 K4/d5 U4/Kd5 kA4/u5 A4/k5 P5 m6 Km6 N6 kM6 M6 m7 Km7 N7 kM7/u8 M7/k8 P8.

LO

SKU can be combined to get any 11-limit interval, however, two others are additionally added so that 14/11 can be described as a type of (major) third, and 11/10 can be described as a type of major second:

L alters by the pentacircle comma, 896/891, where LM3 is 14/11.

O alters by 45/44, the undecimal 1/5th tone, where oM2 is 11/10.

L/l, Large/little

14/11 is a large major third