SHEFKHED interval names: Difference between revisions

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As in Keenan/Fokker and Ups and Downs, intervals may be given multiple names. The following details the order to which certain names are privileged above others.

Interval names are ranked in ~~nine~~ tiers.

Interval names are ranked in 11 tiers:

#Perfect and neutral

#Major, minor, A4 and d5.

#hA4 and hd5

#'S', 's', 'C', 'c', 'SC', 'sc~~', 'cS' and 'Cs~~' prefixes to major, minor, perfect intervals and to A4 and d5

#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to major, minor, perfect intervals and to A4 and d5

#hA1 and hd8 (plus any other hAs and hds if needed)

#Intermediates

#Remaining augmented and diminished intervals ~~(for when the chroma is subtended by more than a single (positive) step of the edo)~~

#Remaining augmented and diminished intervals

#'S', 's', 'C', 'c', 'SC', 'sc', 'cS' and 'Cs' prefixes to ~~augmented~~ and ~~diminished intervals~~

#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to augmented and diminished intervals

#Intervals augmented and diminished more than singularly

#'n' and 'W' prefixes to tier 1-3 interval names

#'n' and 'W' prefixes to tier 4-8 interval names

#Intervals augmented and diminished more than singularly, and 'n' and 'W' prefixes to these intervals.

When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.

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== Application in Regular diatonic edos ==

All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, ~~up to 72edo~~.

All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, with the addition of 53edo which is about as far as this scheme's functional interval names can go, and should, by my opinion. Using he 'function-less' prefixes, 'n' and 'W', 50edo may be named.

12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8

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46edo: P1 C1/S1 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 s4 P4 C4 SC4 scA4/d5 cA4/Cd5 A4/SCd5 SA4/sc5 c5 P5 S5 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 c8/s8 P8

50edo: P1 ~~cS1~~ S1 sm2 ~~Csm2~~ m2 Sm2 sM2 M2 ~~cSM2~~ SM2 sm3 ~~Csm3~~ m3 Sm3 sM3 M3 ~~cSM3~~ SM3 s4 ~~Cs4~~ P4 ~~cS4~~ S4 A4 ~~cSA4~~/~~Csd5~~ d5 s5 ~~Cs5~~ P5 ~~cS5~~ S5 sm6 ~~Csm6~~ m6 Sm6 sM6 M6 ~~cSM6~~ SM6 sm7 ~~Csm7~~ m7 Sm7 sM7 M7 ~~cSM7~~ SM7 s8 ~~Cs8~~ P8

50edo: P1 W1 S1 sm2 nm2 m2 Sm2 sM2 M2 WM2 SM2 sm3 nm3 m3 Sm3 sM3 M3 WM3 SM3 s4 n4 P4 W4 S4 A4 WA4/nd5 d5 s5 n5 P5 WS5 S5 sm6 nm6 m6 Sm6 sM6 M6 WM6 SM6 sm7 nm7 m7 Sm7 sM7 M7 WM7 SM7 s8 n8 P8

53edo: P1 C1/S1 1-2 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 2-3 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 3-4 s4 P4 C4 SC4 scA4 cA4 Cd5 SCd5 SA4/sc5 c5 P5 S5 5-6 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 6-7 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 7-8 c8/s8 P8

~~63edo:~~

72edo may also be given 'functional names' by allowing application of 'comma-wide' and 'comma-narrow' to the neutrals and in the other direction for M, m, A and d. Regular 'wide' and 'narrow' function-less prefixes may be used alternatively. For application to the neutrals, 'c' and 'C' give associations with ratios that are not too complex.

72edo:

72edo: P1 C1 S1 hA1 sm2 nm2 m2 Cm2 cN2 N2 CN2 cM2 M2 WM2 SM2 2-3 sm3 nm3 m3 Cm3 cN3 N3 CN3 cM3 M3 WM3 SM3 3-4 s4 n4 P4 C4 S4 hA4 ChA4 cA4 A4/d5 Cd5 chd5 hd5 s5 c5 P5 C5 S5 5-6 sm6 nm6 m6 Cm6 cN6 N6 CN6 cM6 M6 CM6 SM6 6-7 sm7 cm7 m7 Cm7 cN7 N7 CN7 cM7 M7 CM7 SM7 hd8 s8 c8 P8

We can see that

*17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), and ~~38edo~~ are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[10] 5|4: P1 N2 M2 N3 P4 hd5 P5 N6 m7 N7 P8 in all of these edos.

*17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), 38edo, 41edo and 72edo (through 24edo) are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[10] 5|4: P1 N2 M2 N3 P4 hd5 P5 N6 m7 N7 P8 in all of these edos.

*19edo, 24edo (through 12edo), 26edo, 31edo, 36edo (through 12edo), 38edo (through 19edo) and 43edo are meantone tunings through the use of 'S' and 's'.

*22edo, 27edo and 34edo (through 17edo) are superpythagorean tunings from the use of 'C' and 'c'.

*29edo, 41edo, 46edo and ~~53edo~~ are Pythagorean tunings through the use of both 'S' and 's'; and 'C' and 'c'.

*29edo, 41edo, 46edo, 53edo and 72edo are Pythagorean tunings through the use of both 'S' and 's'; and 'C' and 'c'.

*34edo, 43edo and ~~53edo~~ are barbados tunings through the use of intermediates. We can find the scale Barbados[9] 4|4, P1 1-2 M2 2-3 3-4 P4 P5 5-6 6-7 m7 7-8 P8 in all of those edos, but not necessarily in the primary interval names.

*34edo, 43edo, 53edo and 72edo (through 24edo) are barbados tunings through the use of intermediates. We can find the scale Barbados[9] 4|4, P1 1-2 M2 2-3 3-4 P4 P5 5-6 6-7 m7 7-8 P8 in all of those edos, but not necessarily in the primary interval names.

Every edo in which we see SM2/sm3 also supports barbados, where this interval is the generator, at half a fourth, however rather than 15/13 the generator is more simple represented as 8/7~7/6. The temperament generated by the semi-fourth wherein it represent both SM2 and sm3 (tempering out [[49/48]]) is called Semaphore.

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 As in Keenan/Fokker and Ups and Downs, intervals may be given multiple names. The following details the order to which certain names are privileged above others.
-Interval names are ranked in nine tiers.
+Interval names are ranked in 11 tiers:
 #Perfect and neutral
 #Major, minor, A4 and d5.
 #hA4 and hd5
-#'S', 's', 'C', 'c', 'SC', 'sc', 'cS' and 'Cs' prefixes to major, minor, perfect intervals and to A4 and d5
+#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to major, minor, perfect intervals and to A4 and d5
 #hA1 and hd8 (plus any other hAs and hds if needed)
 #Intermediates
-#Remaining augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
+#Remaining augmented and diminished intervals
-#'S', 's', 'C', 'c', 'SC', 'sc', 'cS' and 'Cs' prefixes to augmented and diminished intervals
+#'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to augmented and diminished intervals
-#Intervals augmented and diminished more than singularly
+#'n' and 'W' prefixes to tier 1-3 interval names
+#'n' and 'W' prefixes to tier 4-8 interval names
+#Intervals augmented and diminished more than singularly, and 'n' and 'W' prefixes to these intervals.
 When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.
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 == Application in Regular diatonic edos ==
-All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, up to 72edo.
+All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, with the addition of 53edo which is about as far as this scheme's functional interval names can go, and should, by my opinion. Using he 'function-less' prefixes, 'n' and 'W', 50edo may be named.
 edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8
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 edo: P1 C1/S1 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 s4 P4 C4 SC4 scA4/d5 cA4/Cd5 A4/SCd5 SA4/sc5 c5 P5 S5 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 c8/s8 P8
-edo: P1 cS1 S1 sm2 Csm2 m2 Sm2 sM2 M2 cSM2 SM2 sm3 Csm3 m3 Sm3 sM3 M3 cSM3 SM3 s4 Cs4 P4 cS4 S4 A4 cSA4/Csd5 d5 s5 Cs5 P5 cS5 S5 sm6 Csm6 m6 Sm6 sM6 M6 cSM6 SM6 sm7 Csm7 m7 Sm7 sM7 M7 cSM7 SM7 s8 Cs8 P8
+edo: P1 W1 S1 sm2 nm2 m2 Sm2 sM2 M2 WM2 SM2 sm3 nm3 m3 Sm3 sM3 M3 WM3 SM3 s4 n4 P4 W4 S4 A4 WA4/nd5 d5 s5 n5 P5 WS5 S5 sm6 nm6 m6 Sm6 sM6 M6 WM6 SM6 sm7 nm7 m7 Sm7 sM7 M7 WM7 SM7 s8 n8 P8
 edo: P1 C1/S1 1-2 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 2-3 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 3-4 s4 P4 C4 SC4 scA4 cA4 Cd5 SCd5 SA4/sc5 c5 P5 S5 5-6 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 6-7 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 7-8 c8/s8 P8
-edo:
+edo may also be given 'functional names' by allowing application of 'comma-wide' and 'comma-narrow' to the neutrals and in the other direction for M, m, A and d. Regular 'wide' and 'narrow' function-less prefixes may be used alternatively. For application to the neutrals, 'c' and 'C' give associations with ratios that are not too complex.
-edo:
+edo: P1 C1 S1 hA1 sm2 nm2 m2 Cm2 cN2 N2 CN2 cM2 M2 WM2 SM2 2-3 sm3 nm3 m3 Cm3 cN3 N3 CN3 cM3 M3 WM3 SM3 3-4 s4 n4 P4 C4 S4 hA4 ChA4 cA4 A4/d5 Cd5 chd5 hd5 s5 c5 P5 C5 S5 5-6 sm6 nm6 m6 Cm6 cN6 N6 CN6 cM6 M6 CM6 SM6 6-7 sm7 cm7 m7 Cm7 cN7 N7 CN7 cM7 M7 CM7 SM7 hd8 s8 c8 P8
 We can see that
-*17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), and 38edo are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[10] 5|4: P1 N2 M2 N3 P4 hd5 P5 N6 m7 N7 P8 in all of these edos.
+*17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), 38edo, 41edo and 72edo (through 24edo) are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[10] 5|4: P1 N2 M2 N3 P4 hd5 P5 N6 m7 N7 P8 in all of these edos.
 *19edo, 24edo (through 12edo), 26edo, 31edo, 36edo (through 12edo), 38edo (through 19edo) and 43edo are meantone tunings through the use of 'S' and 's'.
 *22edo, 27edo and 34edo (through 17edo) are superpythagorean tunings from the use of 'C' and 'c'.
-*29edo, 41edo, 46edo and 53edo are Pythagorean tunings through the use of both 'S' and 's'; and 'C' and 'c'.
+*29edo, 41edo, 46edo, 53edo and 72edo are Pythagorean tunings through the use of both 'S' and 's'; and 'C' and 'c'.
-*34edo, 43edo and 53edo are barbados tunings through the use of intermediates. We can find the scale Barbados[9] 4|4, P1 1-2 M2 2-3 3-4 P4 P5 5-6 6-7 m7 7-8 P8 in all of those edos, but not necessarily in the primary interval names.
+*34edo, 43edo, 53edo and 72edo (through 24edo) are barbados tunings through the use of intermediates. We can find the scale Barbados[9] 4|4, P1 1-2 M2 2-3 3-4 P4 P5 5-6 6-7 m7 7-8 P8 in all of those edos, but not necessarily in the primary interval names.
 Every edo in which we see SM2/sm3 also supports barbados, where this interval is the generator, at half a fourth, however rather than 15/13 the generator is more simple represented as 8/7~7/6. The temperament generated by the semi-fourth wherein it represent both SM2 and sm3 (tempering out [[49/48]]) is called Semaphore.