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SKULO interval names: Difference between revisions

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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.
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.
=== Larger edos ===
If we allow for application of SKULO in any direction we can label much larger edos. In order to produce well-ordered interval names using only single alterations all we need is for S, K, U, L, and O to represent degrees 1 through floor(''N/2),'' where ''N'' is the number of degrees to either the diatonic or chromatic semitone, whichever is larger.
For example, for 72edo, S=2, K=1, U=3, L=1, O=2, with A=6 and m2=6, so only U needs to be used in both directions (72edo could alternatively be notated using only S, K, and U, with all three alterations used in both directions).
'''72edo:''' P1 K1/L1 S2/O2 U1/um2 sm2 lm2 m2 Km2 Om2 N2 oM2 kM2 M2 LM2 SM2 UM2/um3 sm3 lm3 m3 Km3 Om3 N3 oM3 kM3 M3 LM3 SM3 UM3/u4 s4 l4 P4 K4 O4 U4/uA4 oA4/sd5 kA4/ld5 A4/d5 LA4/Kd5 SA4/Od5 Ud5/u5 o5 k5 P5 L5 S5 U5/um6 sm6 lm6 m6 Km6 Om6 N6 oM6 kM6 M6 LM6 SM6 UM6/um7 sm7 lm7 m7 Km7 Om7 N7 oM7 kM7 M7 LM7 SM7 UM7/u8 s8/o8 k8/l8 P8
118edo, an important 5-limit and 11-limit edo, can be labelled with SKULO where S=3, K=2, U=5, L=1, and O=4, with A=11 and m2=9.
80edo can be labelled with S=1, K=2, U=4, and O=3 (where L=0, A=9, and m2=5), using all of S, K, and U in both directions, but it may perhaps more usefully be labelled with an additional pair that take SKULO into the 13-limit. Similarly, 94edo could be labelled with S=2, K=2, U=5, L=1, and O=4, using L, O, and U in both directions, but may benefit from additional 13-limit prefixes. Even with the additional prefixes, however, 80edo and 94edo cannot be labelled without using any prefixes twice, 72edo seems to be the largest edos in which that's possible.
== SKULOTH ==
T and H can also be added to extend into the 13-limit, which is important for naming intervals in many edos such as 36, 37, 43, 50, 53, 80, and 94edo.
T alters by the tridecimal comma, 1053/1024, where 13/8 is labelled Tm6.
H alters by 40/39, where 15/13 is labelled HM2, and 13/10 is labelled h4.
=== T/t, Tall/tiny, tridecimal ===
A capital 'T' signals alteration upward by 1053/1024, and a lower case 't' signals alteration downward by 1053/1024. The alteration is typically applied downwards for positive Pythagorean intervals, and upwards for negative Pythagorean intervals.
13/8 is a tall or tridecimal minor sixth, Tm6, and 16/13 is a tiny or tridecimal major third, tM3.
=== H/h, Hyper/hypo ===
A capital 'H' signals alteration upward by 40/39, and a lower case 'h' signals alteration downward by 40/39. The alteration is typically applied upwards for positive Pythagorean intervals, and downwards for negative Pythagorean intervals.
=== 2.3.5.7.13 edos ===
36edo can be labelled with S and T, and 53edo can be notated with S, K, T, and H.
'''36edo:''' P1 S1/T1 sm2 m2 Tm2 tM2 M2 SM2 sm3 m3 Tm3 tM3 M3 SM3 s4 P4 T4 tA4 A4/d5 Td5 t5 P5 S5 sm6 m6 Tm6 tM6 M6 SM6 sm7 m7 Tm7 tM7 M7 SM7 s8/t8 P8.
'''53edo:''' P1 K1/S1 T1/H1/hm2 sm2 m2 Km2 Tm2 tM2 kM2 M2 SM2 HM2/hm3 sm3 m3 Km3 Tm3 tM3 kM3 M3 SM3 HM3/h4 s4 P4 K4 T4 tA4 kA4 Kd5 Td5 t5 k5 P5 S5 H5/hm6 sm6 m6 Km6 Tm6 tM6 kM6 M6 SM6 HM6/hm7 sm7 m7 Km7 Tm7 tM7 kM7 M7 SM7 HM7/h8/t8 k8/s8 P8.
=== 13-limit edos ===
We can now label 38edo and 45edo without using augmented and diminished 2nds, 3rds, 6ths, or 7ths, and introduce labels for 43edo, 50edo, 55edo, and 58edo:
'''38edo''': P1 S1 A1 sm2 m2 N2 M2 SM2 HM2/hm3 sm3 m3 N3 M3 SM3 HM3/h4 s4 P4 U4 A4 SA4/sd5 d5 u5 P5 S5 H5/hm6 sm6 m6 N6 M6 SM6 HM6/hm7 sm7 m7 N7 M7 SM7 d8 s8 P8.
'''45edo''': P1 S1/U1 uA1 A1 sm2 m2 n2 N2 M2 SM2 HM2/hm3 sm3 m3 n3 N3 M3 SM3 HM3/h4 s4 P4 U4 A4 SA4 sd5 d5 u5 P5 S5 H5/hm6 sm6 m6 n6 N6 M6 SM6 HM6/hm7 sm7 m7 n7 N7 M7 SM7 d8 Ud8 s8/u8 P8.
'''43edo:''' P1 S1/T1 H1/hm2 sm2 m2 Tm2 tM2 M2 SM2 HM2/hm3 sm3 m3 Tm3 tM3 M3 SM3 HM3/h4 s4 P4 T4 tA4 A4 d5 Td5 t5 P5 S5 H5/hm6 sm6 m6 Tm6 tM6 M6 SM6 HM6/hm7 sm7 m7 Tm7 tM7 M7 SM7 HM7/h8 s8/t8 P8
'''50edo:''' P1 L1 S1/T1 sm2 lm2 Km2 Tm2 tM2 kM2 LM2 SM2 sm3 lm3 Km3 Tm3 tM3 kM3 LM3 SM3 s4 l4 P4 T4 tA4 kA4 LA4/ld5 Kd5 Td5 t5 P5 L5 S5 sm6 lm6 Km6 Tm6 tM6 kM6 LM6 SM6 sm7 lm7 Km7 Tm7 tM7 kM7 LM7 SM7 s8/t8 l8 P8.
'''55edo:''' P1 L1/O1 S1 sm2 lm2 m2 Om2 n2/N2 oM2 M2 LM2 SM2 sm3 lm3 m3 Om3 n3/N3 oM3 M3 SM3 s4 l4 P4 O4 M4 oA4/sd5 A4/ld5 LA4/d5 SA4/Od5 m5 o5 P5 L5 S5 sm6 lm6 m6 Om6 n6/N6 oM6 M6 LM6 SM6 sm7 lm7 m7 Om7 n7/N7 oM7 M7 LM7 SM7 s8 l8/o8 P8.
'''58edo:''' P1 K1/S1 O1/H1/hm2 U1/sm2 m2 Km2 Om2 n2/N2 oM2 kM2 M2 SM2 HM2/hm3 sm3 m3 Km3 Om3 n3/N3 oM3 kM3 M3 SM3 HM3/h4 s4 P4 K4 O4 M4 oA4/d5 kA4/Kd5 A4/Od5 m5 o5 k5 P5 S5 H5/hm6 sm6 m6 Km6 Om6 n6/N6 oM6 kM6 M6 SM6 HM6/hm7 sm7 m7 Km7 Om7 n7/N7 oM7 kM7 M7 SM7/u8 HM7/h8/o8 k8/s8 P8.
72edo may also be labelled now without having to use any prefixes in both directions, i.e.,
'''72edo:''' P1 K1 S1/O1 U1/H1/hm2 sm2 lm2 m2 Km2 Om2 n2/N2 oM2 kM2 M2 LM2 SM2 HM2/hm3 sm3 lm3 m3 Km3 Om3 n3/N3 oM3 kM3 M3 LM3 SM3 HM3/h4 s4 l4 P4 K4 O4 M4 oA4/sd5 kA4 A4/d5 Kd5 SA4/Od5 m5 o5 k5 P5 L5 S5 H5/hm6 sm6 lm6 m6 Km6 Om6 n6/N6 oM6 kM6 M6 LM6 SM6 HM6/hm7 sm7 lm7 m7 Km7 Om7 N7 oM7 kM7 M7 LM7 SM7 HM7/h8/u8 s8/o8 k8 P8.
Fans of 37edo may be noting its absence at this stage. In 37edo the major intervals are equivalent to supermajor and hypermajor intervals. We call 37edo a Hyper-Pythagorean tuning, tempering out 416/405, where 17, 22, 27 and 37 are Super-Pythagorean tunings, which temper out 64/63, i.e., SM=M, and 7, 12, 19, 26, 31, 43 and 50 are meantone tunings, which temper out 81/80, i.e., kM=M. Unlike in the supermajor tunings 17, 22, and 27, for which kM is one degree below M, this is not the case for 37, where LM is actually between kM and M. Since LM is one degree below M, 896/891 is -1 degrees in 37edo. Labelling LM as smaller than M is confusing, and we do not consider an interval name list to be well-ordered if it does this. However, we already know we should write SM/sm and s4/S5 instead of M/m and P4/P5, so we don’t have a problem.
'''37edo:''' P1 sm2 lm2 Km2 Tm2/tM2 kM2 LM2 SM2 sm3 lm3 Km3 Tm3/tM3 kM3 LM3 SM3 s4 l4 K4 T4 t5 k5 L5 S5 sm6 lm6 Km6 tM6/Tm6 kM6 LM6 SM6 sm7 lm7 Km7 Tm7/tM7 kM7 LM7 SM7 P8.
=== Larger edos ===
SKULOTH is able to produce well-ordered interval names for 130edo, where S=3, K=2, U=6, L=1, O=4, T=5 and H=5, with A=12, m2=10.
== Alternatives ==
Other edos may be labelled in a similar way to 37edo (i.e., labeling the diatonic intervals with sub and super prefixes) if accuracy to JI is preferenced over simplicity, for example,
'''10edo:''' P1 Tm2/tM2 SM2/sm3 Tm3/tM3 SM3/s4 T4/t5 S5/sm6 Tm6/tM6 SM6/sm7 Tm7/tM7 P8.
'''27edo:''' P1 sm2 Km2 Tm2/tM2 kM2 SM2 sm3 Km3 Tm3/tM3 kM3 SM3 P4 K4 T4 t5 k5 P5 sm6 Km6 tM6/Tm6 kM6 SM6 sm7 Km7 Tm7/tM7 kM7 SM7 P8.
The following table details the alternative labels that may be used for 22edo, where the compromise between accuracy and simplicity is arguably not as simple, which we could argue is a feature of 22edo.
{| class="wikitable"
|+22edo (patent 11-limit val)
!Degree
!Cents
!5-limit interval name
!Short-form
!JI ratio
!Alternative SKU interval name
!Short-form
!JI ratio
|-
|0
|0
|perfect unison
|P1
|1/1
|super unison
|S1
|64/63
|-
|1
|54.55
|minor second / comma-wide unison
|m2/K1
|256/243, 81/80
|sub minor second / uber unison
|sm2/U1
|28/27, 33/32
|-
|2
|109.09
|classic minor second
|Km2
|16/15
|lesser neutral second
|n2
|88/81
|-
|3
|163.64
|classic major second
|kM2
|10/9
|greater neutral second
|N2
|12/11
|-
|4
|218.18
|major second
|M2
|9/8
|super major second
|SM2
|8/7
|-
|5
|272.73
|minor third
|m3
|32/27
|super minor third
|sm3
|7/6
|-
|6
|327.27
|classic minor third
|Km3
|6/5
|lesser neutral third
|n3
|11/9
|-
|7
|381.82
|classic major third
|kM3
|5/4
|greater neutral third
|N3
|27/22
|-
|8
|436.36
|major third
|M3
|81/64
|super major third
|SM3
|9/7
|-
|9
|490.91
|perfect fourth
|P4
|4/3
|sub fourth
|s4
|21/16
|-
|10
|545.45
|comma-wide fourth
|K4
|27/20
|uber fourth
|U4
|11/8
|-
|11
|600
|classic augmented fourth
classic diminished fifth
|kA4
Kd5
|45/32
64/45
|unter augmented fourth
uber diminished fifth
|uA4
Ud5
|243/176
352/243
|-
|12
|654.55
|comma-narrow fifth
|k5
|40/27
|unter fifth
|u5
|16/11
|-
|13
|709.09
|perfect fifth
|P5
|3/2
|super fifth
|S5
|32/21
|-
|14
|763.64
|minor sixth
|m6
|128/81
|sub minor sixth
|sm6
|14/9
|-
|15
|818.18
|classic minor sixth
|Km6
|8/5
|less neutral sixth
|n6
|44/27
|-
|16
|872.73
|classic major sixth
|kM6
|5/3
|greater neutral sixth
|N6
|18/11
|-
|17
|927.27
|major sixth
|M6
|27/16
|super major sixth
|SM6
|12/7
|-
|18
|981.82
|minor seventh
|m7
|16/9
|sub minor seventh
|sm7
|7/4
|-
|19
|1036.36
|classic minor seventh
|Km7
|9/5
|lesser neutral seventh
|n7
|11/6
|-
|20
|1090.91
|classic major seventh
|kM7
|15/8
|greater neutral seventh
|N7
|81/44
|-
|21
|1145.45
|major seventh / comma-narrow octave
|M7 / k8
|243/128, 160/81
|super major seventh / unter octave
|SM7/k8
|27/14, 64/33
|-
|22
|1200
|perfect octave
|P8
|2/1
|sub octave
|s8
|2/1, 63/32
|}
== WOFED interval names ==
Well-ordered, functional, extended diatonic interval names.
I have reference the 'well ordered' naming principle above. Namely, it is a principle in which no larger interval is named with reference to a smaller interval class, within a single tuning system. All lists of interval names above follow this principle. In order to label the intervals of 94edo following this principle, a further alteration is added - R/r - for rastmic intervals. R/r signals alteration by the rastma, 243/242, such that the rastmic minor second, Rm2 is 128/121, and the rastmic major third, rM3, is 121/96. is Such an interval naming system was called WOFED interval names in an earlier draft of my interval naming scheme. The interval names for 94edo it results in are shown at [[94edo#Intervals]].
S=64/63, K=81/80, U=33/32, L=896/891, O=45/44, R=243/242, T=1053/1024, H=40/39
S=K=2, O=H=3, U(=T)=4, L=R=1
'''94edo:''' P1 L1/R1 K1/S1 O1/H1 U1/hm2 uA1/sm2 oA1/lm2 kA1/m2 rA1/Rm2 A1/Km2 LA1/Om2 SA1/n2 N2 oM2 kM2 rM2 M2 LM2 SM2 HM2 hm3 sm3 lm3 m3 Rm3 Km3 Om3 n3 N3 oM3 kM3 rM3 M3 LM3 SM3 HM3 h4 s4 l4 P4 R4 K4 O4 U4/hd5 uA4/sd5 oA4/ld5 kA4/d5 rA4/Rd5 A4/Kd5 LA4/Od5 SA4/Ud5 HA4/u5 o5 k5 r5 P5 L5 S5 H5 hm6 sm6 lm6 m6 Rm6 Km6 Om6 n6 N6 oM6 kM6 rM6 M6 LM6 SM6 HM6 hm7 sm7 lm7 m7 Rm7 Km7 Om7 n7 N7/sd8 oM7/ld8 kM7/d8 rM7/Rd8 M7/Kd8 LM7/Od8 SM7/Ud8 HM7/u8 o8/h8 k8/s8 l8/r8 P8.
[[Category:Interval naming]]
[[Category:Interval naming]]
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