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SKULO interval names: Difference between revisions

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SKULO names are the successor to [[SHEFKHED interval names]]. SKULO is an [[Extended-diatonic interval names|extended-diatonic]] system, wherein prefixes are added for sub-chroma alterations from standard Pythagorean diatonic intervals.
SKULO interval names are the successor to [[FKH Extended-diatonic Interval Names]]. SKULO is an [[Extended-diatonic interval names|extended-diatonic]] system, wherein prefixes are added for sub-chroma alterations from standard Pythagorean diatonic intervals. SKULO also exists as a notation system based on a Pythagorean scale on a diatonic staff, with accidentals based on the letters representing the alterations. SKULO interval names observe diatonic interval arithmetic and follow the ''well-ordered'' interval naming scheme, wherein for the interval names of a tuning system no larger interval is named with reference to a smaller interval class.


S signals the septimal intervals, altering a Pythagorean interval by the septimal or Archytas comma, [[64/63]].
S signals the septimal intervals, altering a Pythagorean interval by the septimal or Archytas comma, [[64/63]].
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'''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.
'''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.
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 ==
== K/k, komma-wide/komma-narrow, klassisch intervals ==
Line 52: Line 52:
== SKU ==
== SKU ==
{| class="wikitable"
{| class="wikitable"
|+Basic SKU extended diatonic intervals
|+Basic SKU extended-diatonic intervals
!Short-form
!Short-form
!Mid-form
!Long-form
!Long-form
!JI interval represented
!JI interval represented
|-
|-
|P1
|P1
|prf 1st
|perfect unison
|perfect unison
|1/1
|1/1
|-
|-
|K1
|K1
|k-wde 1st
|komma-wide/comma-wide unison
|komma-wide/comma-wide unison
|81/80
|81/80
|-
|-
|S1
|S1
|sup 1st
|super unison
|super unison
|64/63
|64/63
|-
|-
|U1
|U1
|ubr 1st
|uber unison
|uber unison
|33/32
|33/32
|-
|(kkA1)
|(klassisch/classic augmented unison)
|(25/24)
|-
|kA1
|komma-narrow/comma-narrow augmented unison
|135/128
|-
|A1
|augmented unison
|2187/2048
|-
|SA1
|super augmented unison
|243/224
|-
|-
|sm2
|sm2
|sub-min 2nd
|subminor second
|subminor second
|28/27
|28/27
|-
|-
|m2
|m2
|min 2nd
|minor second
|minor second
|256/243
|256/243
|-
|-
|Km2
|Km2
|kla-min 2nd
|klassisch/classic minor second
|klassisch/classic minor second
|16/15
|16/15
|-
|-
|Um2/n2
|Um2/n2
|ubr-min 2nd / lsr-ntl 2nd
|uber minor second / lesser neutral second
|uber minor second / lesser neutral second
|88/81
|88/81
|-
|-
|uM2/n2
|uM2/N2
|unt-maj 2nd / gtr-ntl 2nd
|unter major second  / greater neutral second
|unter major second  / greater neutral second
|12/11
|12/11
|-
|-
|kM2
|kM2
|k-nrw 2nd
|komma-narrow/comma narrow major second, klassisch/classic major second
|komma-narrow/comma narrow major second
|10/9
|10/9
|-
|-
|M2
|M2
|maj 2nd
|major second
|major second
|9/8
|9/8
|-
|-
|SM2
|SM2
|sup-maj 2nd
|supermajor second
|supermajor second
|8/7
|8/7
|-
|-
|sm3
|sm3
|sub-min 3rd
|subminor third
|subminor third
|7/6
|7/6
|-
|-
|m3
|m3
|min 3rd
|minor third
|minor third
|32/27
|32/27
|-
|-
|Km3
|Km3
|kla-min 3rd
|klassisch/classic minor third
|klassisch/classic minor third
|6/5
|6/5
|-
|-
|Um3/n3
|Um3/n3
|ubr-min 3rd / lsr-ntl 3rd
|uber minor third / lesser neutral third
|uber minor third / lesser neutral third
|11/9
|11/9
|-
|-
|uM3/N3
|uM3/N3
|unt-maj 3rd / gtr-ntl 3rd
|unter major third / greater neutral third
|unter major third / greater neutral third
|27/22
|27/22
|-
|-
|kM3
|kM3
|kla-maj 3rd
|klassisch/classic major third
|klassisch/classic major third
|5/4
|5/4
|-
|-
|M3
|M3
|maj 3rd
|major third
|major third
|81/64
|81/64
|-
|-
|SM3
|SM3
|sup-maj 3rd
|supermajor third
|supermajor third
|9/7
|9/7
|-
|-
|s4
|s4
|sub 4th
|sub fourth
|sub fourth
|21/16
|21/16
|-
|-
|P4
|P4
|prf 4th
|perfect fourth
|perfect fourth
|4/3
|4/3
|-
|-
|K4
|K4
|k-wde 4th
|komma-wide/comma-wide fourth
|komma-wide/comma-wide fourth
|27/20
|27/20
|-
|-
|U4
|U4
|ubr 4th (/ lsr-ntl 4th)
|uber fourth (/ lesser neutral fourth)
|uber fourth (/ lesser neutral fourth)
|11/8
|11/8
|-
|-
|uA4
|uA4
|unt-aug 4th (/ gtr-ntl 4th)
|unter augmented fourth (/ greater neutral fourth)
|unter augmented fourth (/ greater neutral fourth)
|243/176
|243/176
|-
|(kkA4)
|(klassisch/classic augmented fourth)
|(25/18)
|-
|-
|kA4
|kA4
|kla-aug 4th
|komma-narrow/comma-narrow augmented fourth
|klassisch/classic augmented fourth
|45/32
|45/32
|-
|-
|A4
|A4
|aug 4th
|augmented fourth
|augmented fourth
|729/512
|729/512
|-
|-
|SA4
|SA4
|sup-aug 4th
|super augmented fourth
|super augmented fourth
|81/56
|81/56
|-
|-
|sd5
|sd5
|sub-dim 5th
|sub diminished fifth
|sub diminished fifth
|112/81
|112/81
|-
|-
|d5
|d5
|dim 5th
|diminished fifth
|diminished fifth
|1024/729
|1024/729
|-
|-
|Kd5
|Kd5
|kla-dim 5th
|komma-wide/comma-wide diminished fifth
|klassisch/classic diminished fifth
|64/45
|64/45
|-
|(KKd5)
|(klassich/classic diminished fifth)
|(36/25)
|-
|-
|Ud5
|Ud5
|ubr-dim 5th (/lsr-ntl 5th)
|uber diminished fifth (/lesser neutral fifth)
|uber diminished fifth (/lesser neutral fifth)
|352/243
|352/243
|-
|-
|u5
|u5
|unt 5th (/gtr-ntl 5th)
|unter fifth (/greater neutral fifth)
|unter fifth (/greater neutral fifth)
|16/11
|16/11
|-
|-
|k5
|k5
|k-nrw 5th
|komma-narrow/comma-narrow fifth
|komma-narrow/comma-narrow fifth
|40/27
|40/27
|-
|-
|P5
|P5
|prf 5th
|perfect fifth
|perfect fifth
|3/2
|3/2
|-
|-
|S5
|S5
|sup 5th
|super fifth
|super fifth
|32/21
|32/21
|-
|-
|sm6
|sm6
|sub-min 6th
|subminor sixth
|subminor sixth
|14/9
|14/9
|-
|-
|m6
|m6
|min 6th
|minor sixth
|minor sixth
|128/81
|128/81
|-
|-
|Km6
|Km6
|kla-min 6th
|klassisch/classic minor sixth
|klassisch/classic minor sixth
|8/5
|8/5
|-
|-
|Um6/n6
|Um6/n6
|ubr-min 6th / lsr-ntl 6th
|uber minor sixth / lesser neutral sixth
|uber minor sixth / lesser neutral sixth
|44/27
|44/27
|-
|-
|uM6/N6
|uM6/N6
|unt-maj 6th / gtr-ntl 6th
|unter major sixth / greater neutral sixth
|unter major sixth / greater neutral sixth
|18/11
|18/11
|-
|-
|kM6
|kM6
|kla-maj 6th
|klassisch/classic major sixth
|klassisch/classic major sixth
|5/3
|5/3
|-
|-
|M6
|M6
|maj 6th
|major sixth
|major sixth
|27/16
|27/16
|-
|-
|SM6
|SM6
|sup-maj 6th
|supermajor sixth
|supermajor sixth
|12/7
|12/7
|-
|-
|sm7
|sm7
|sub-min 7th
|subminor seventh
|subminor seventh
|7/4
|7/4
|-
|-
|m7
|m7
|min 7th
|minor seventh
|minor seventh
|16/9
|16/9
|-
|-
|Km7
|Km7
|kla-min 7th
|komma-wide/comma-wide minor seventh / klassisch/classic minor seventh
|komma-wide/comma-wide minor seventh / klassisch/classic minor seventh
|9/5
|9/5
|-
|-
|Um7/n7
|Um7/n7
|ubr-min 7th / lsr-ntl 7th
|uber minor seventh / lesser neutral seventh
|uber minor seventh / lesser neutral seventh
|11/6
|11/6
|-
|-
|uM7/N7
|uM7/N7
|unt-maj 7th / gtr-ntl 7th
|unter minor seventh / greater neutral seventh
|unter minor seventh / greater neutral seventh
|81/44
|81/44
|-
|-
|kM7
|kM7
|kla-maj 7th
|klassisch/classic major seventh
|klassisch/classic major seventh
|15/8
|15/8
|-
|-
|M7
|M7
|maj 7th
|major seventh
|major seventh
|243/128
|243/128
|-
|-
|SM7
|SM7
|sup-maj 7th
|supermajor seventh
|supermajor seventh
|27/14
|27/14
|-
|sd8
|sub diminished octave
|448/243
|-
|d8
|diminished octave
|4096/2187
|-
|Kd8
|komma-wide/comma-wide diminished octave
|256/135
|-
|(KKd8)
|(klassisch/classic diminished octave)
|48/25
|-
|-
|u8
|u8
|unt 8ve
|unter octave
|unter octave
|64/33
|64/33
|-
|-
|s8
|s8
|sub 8ve
|sub octave
|sub octave
|63/32
|63/32
|-
|-
|k8
|k8
|k-nrw 8ve
|komma-narrow/comma-narrow octave
|komma-narrow/comma-narrow octave
|160/81
|160/81
|-
|-
|P8
|P8
|pft 8ve
|perfect octave
|perfect octave
|2/1
|2/1
Line 358: Line 341:
'''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.
'''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.
'''41edo''': P1 K1/S1 U1/sm2 m2 Km2 N2 kM2 M2 SM2 sm3 m3 Km3 N3 kM3 M3 SM3 s4 P4 K4 U4 kA4/d5 Kd5/A4 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.
'''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).
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) because it has lower 11-limit error overall:


'''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.
'''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.
Line 373: Line 356:
O alters by 45/44, the undecimal 1/5th tone, where oM2 is 11/10, and Om3 is 40/33.
O alters by 45/44, the undecimal 1/5th tone, where oM2 is 11/10, and Om3 is 40/33.


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


== O/o, On/off, oceanic ==
=== 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.
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.
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 intervals ===
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 and 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.
'''15edo:''' P1 Km2 kM2 SM2/sm3 Km3 kM3 SM3/s4 K4 k5 S5/sm6 Km6 kM6 SM6/sm7 Km7 kM7 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 2nd, comma-wide unison, (classic augmented unison)
|m2, K1, (kkA1)
|256/243, 81/80, (25/24)
|subminor 2nd, uber unison
|sm2, U1
|28/27, 33/32
|-
|2
|109.09
|classic minor 2nd, comma-narrow augmented unison
|Km2, kA1,
|16/15, 135/128
|lesser neutral 2nd, unter augmented unison
|n2, uA1
|88/81, 729/704
|-
|3
|163.64
|classic/comma-narrow major 2nd, augmented unison
|kM2, A1
|10/9, 2187/2048
|greater neutral 2nd, super augmented unison,
|N2, SA1
|12/11, 243/224
|-
|4
|218.18
|major 2nd
|M2
|9/8
|supermajor 2nd
|SM2
|8/7
|-
|5
|272.73
|minor 3rd
|m3
|32/27
|subminor 3rd
|sm3
|7/6
|-
|6
|327.27
|classic minor 3rd
|Km3
|6/5
|lesser neutral 3rd
|n3
|11/9
|-
|7
|381.82
|classic major 3rd
|kM3
|5/4
|greater neutral 3rd
|N3
|27/22
|-
|8
|436.36
|major 3rd
|M3
|81/64
|supermajor 3rd
|SM3
|9/7
|-
|9
|490.91
|perfect 4th
|P4
|4/3
|sub 4th
|s4
|21/16
|-
|10
|545.45
|comma-wide 4th, (classic augmented 4th), diminished 5th
| K4, (kkA4), d5
|27/20, (25/18), 729/512
|uber 4th / lesser neutral 4th, sub diminished 5th
| U4/n4, sd5
|11/8, 112/81
|-
|11
|600
|comma-narrow augmented 4th, comma-wide diminished 5th
|kA4, Kd5
|45/32, 64/45
|unter augmented 4th / greater neutral 4th, uber diminished 5th / lesser neutral 5th
|uA4/N4, Ud5/n5
|243/176, 352/243
|-
|12
|654.55
|comma-narrow 5th, (classic diminished 5th), augmented 4th
|k5, (KKd5), A4
|40/27, (32/25), 1024/729
|unter 5th / greater neutral 5th, super augmented 4th
|u5/N5, SA4
|16/11, 81/56
|-
|13
|709.09
|perfect 5th
|P5
|3/2
|super 5th
|S5
|32/21
|-
|14
|763.64
|minor 6th
|m6
|128/81
|subminor 6th
|sm6
|14/9
|-
|15
|818.18
|classic minor 6th
|Km6
|8/5
|lesser neutral 6th
|n6
|44/27
|-
|16
|872.73
|classic major 6th
|kM6
|5/3
|greater neutral 6th
|N6
|18/11
|-
|17
|927.27
|major 6th
|M6
|27/16
|supermajor 6th
|SM6
|12/7
|-
|18
|981.82
|minor 7th
|m7
|16/9
|subminor 7th
|sm7
|7/4
|-
|19
|1036.36
|classic/comma-wide minor 7th, diminished 8ve
|Km7, d8
|9/5, 4096/2187
| lesser neutral 7th, super diminished 8ve
|n7, sd8
|11/6, 448/243
|-
|20
|1090.91
|classic major 7th, comma-wide diminished 8ve
|kM7, Kd8
|15/8, 256/135
|greater neutral 7th, unter diminished 8ve
|N7, ud8
| 81/44, 1408/729
|-
|21
|1145.45
|major 7th, comma-narrow 8ve, (classic diminished 8ve)
|M7, k8, (KKd8)
|243/128 , 160/81, (48/25)
|supermajor 7th, unter 8ve
|SM7, u8
|27/14, 64/33
|-
|22
|1200
|perfect 8ve
|P8
|2/1
|sub 8ve
|s8
|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.
With the addition of R/r, we can also label the intervals of 87edo where R=1, K=S=2, O=T=H=3, and U=4, with A=8 and m2=6, though R/r needs to be used in both directions.
[[Category:Interval naming]]
[[Category:Interval naming]]
Retrieved from "https://en.xen.wiki/w/SKULO_interval_names"