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'. 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.