SKULO interval names

From Xenharmonic Wiki
Jump to navigation Jump to search

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'. I also like to use N/n for fourths and fifths, where 11/8 is a (lesser) neutral 4th and 16/11 is a (greater) neutral fifth, but I have used only U4, u5, uA4 and Ud5 instead of n4, N5, N4, and n5 respectively in my edo summaries.

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

Basic SKU extended diatonic intervals
Short-form Mid-form Long-form JI interval represented
P1 prf 1st perfect unison 1/1
K1 k-wde 1st komma-wide/comma-wide unison 81/80
S1 sup 1st super unison 64/63
U1 ubr 1st uber unison 33/32
sm2 sub-min 2nd subminor second 28/27
m2 min 2nd minor second 256/243
Km2 kla-min 2nd klassisch/classic minor second 16/15
Um2/n2 ubr-min 2nd / lsr-ntl 2nd uber minor second / lesser neutral second 88/81
uM2/n2 unt-maj 2nd / gtr-ntl 2nd unter major second / greater neutral second 12/11
kM2 k-nrw 2nd komma-narrow/comma narrow major second 10/9
M2 maj 2nd major second 9/8
SM2 sup-maj 2nd supermajor second 8/7
sm3 sub-min 3rd subminor third 7/6
m3 min 3rd minor third 32/27
Km3 kla-min 3rd klassisch/classic minor third 6/5
Um3/n3 ubr-min 3rd / lsr-ntl 3rd uber minor third / lesser neutral third 11/9
uM3/N3 unt-maj 3rd / gtr-ntl 3rd unter major third / greater neutral third 27/22
kM3 kla-maj 3rd klassisch/classic major third 5/4
M3 maj 3rd major third 81/64
SM3 sup-maj 3rd supermajor third 9/7
s4 sub 4th sub fourth 21/16
P4 prf 4th perfect fourth 4/3
K4 k-wde 4th komma-wide/comma-wide fourth 27/20
U4 ubr 4th (/ lsr-ntl 4th) uber fourth (/ lesser neutral fourth) 11/8
uA4 unt-aug 4th (/ gtr-ntl 4th) unter augmented fourth (/ greater neutral fourth) 243/176
kA4 kla-aug 4th klassisch/classic augmented fourth 45/32
A4 aug 4th augmented fourth 729/512
SA4 sup-aug 4th super augmented fourth 81/56
sd5 sub-dim 5th sub diminished fifth 112/81
d5 dim 5th diminished fifth 1024/729
Kd5 kla-dim 5th klassisch/classic diminished fifth 64/45
Ud5 ubr-dim 5th (/lsr-ntl 5th) uber diminished fifth (/lesser neutral fifth) 352/243
u5 unt 5th (/gtr-ntl 5th) unter fifth (/greater neutral fifth) 16/11
k5 k-nrw 5th komma-narrow/comma-narrow fifth 40/27
P5 prf 5th perfect fifth 3/2
S5 sup 5th super fifth 32/21
sm6 sub-min 6th subminor sixth 14/9
m6 min 6th minor sixth 128/81
Km6 kla-min 6th klassisch/classic minor sixth 8/5
Um6/n6 ubr-min 6th / lsr-ntl 6th uber minor sixth / lesser neutral sixth 44/27
uM6/N6 unt-maj 6th / gtr-ntl 6th unter major sixth / greater neutral sixth 18/11
kM6 kla-maj 6th klassisch/classic major sixth 5/3
M6 maj 6th major sixth 27/16
SM6 sup-maj 6th supermajor sixth 12/7
sm7 sub-min 7th subminor seventh 7/4
m7 min 7th minor seventh 16/9
Km7 kla-min 7th komma-wide/comma-wide minor seventh / klassisch/classic minor seventh 9/5
Um7/n7 ubr-min 7th / lsr-ntl 7th uber minor seventh / lesser neutral seventh 11/6
uM7/N7 unt-maj 7th / gtr-ntl 7th unter minor seventh / greater neutral seventh 81/44
kM7 kla-maj 7th klassisch/classic major seventh 15/8
M7 maj 7th major seventh 243/128
SM7 sup-maj 7th supermajor seventh 27/14
u8 unt 8ve unter octave 64/33
s8 sub 8ve sub octave 63/32
k8 k-nrw 8ve komma-narrow/comma-narrow octave 160/81
P8 pft 8ve perfect octave 2/1

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, and Om3 is 40/33.

L/l, Large/little

A capital 'L' signals alteration upward by 896/891, and a lower case 'u' signals alteration downward by 896/891. The alteration is typically applied upward for positive Pythagorean intervals, and downward for negative Pythagorean intervals.

14/11 is a large major third, LM3, and 11/7 is a little minor sixth, lm6.

O/o, On/off, oceanic

A capital 'O' signals alteration upward by 45/44, and a lower case 'o' signals alteration downward by 45/44. The alteration is typically applied downward for positive Pythagorean intervals, and upward for negative Pythagorean intervals.

11/10 is an off major second, or oceanic major second, oM2, and 40/33 is an on minor third, or oceanic minor third, Om3.