SHEFKHED interval names: Difference between revisions
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*Where neutral intervals split the apotome, to pair with neutral when acting on perfect intervals are 'hemi augmented' and 'hemi diminished', with short form 'hA' and 'hd'. 'Hemi' is used instead of 'semi' of 'half' because 'half diminished' is a type of chord, and 'semi' begins with the letter 's', which has been associated with alterations of 64/63. In all cases it's presence implies neutral temperament and the tempering out of 243/242. Accordingly it implies a diminution from perfect, major, augmented of 33/32, as well as an augmentation from perfect, minor or diminished of 33/32, but may not be used to imply those alterations in any other cases. | *Where neutral intervals split the apotome, to pair with neutral when acting on perfect intervals are 'hemi augmented' and 'hemi diminished', with short form 'hA' and 'hd'. 'Hemi' is used instead of 'semi' of 'half' because 'half diminished' is a type of chord, and 'semi' begins with the letter 's', which has been associated with alterations of 64/63. In all cases it's presence implies neutral temperament and the tempering out of 243/242. Accordingly it implies a diminution from perfect, major, augmented of 33/32, as well as an augmentation from perfect, minor or diminished of 33/32, but may not be used to imply those alterations in any other cases. | ||
*To extend to the 13-limit, we add that to cP, cM and cA intervals may be added the 'sub' or 's' prefix, in this instance indicating a diminution of [[65/64]], and that to CP, Cm and Cd intervals may be added the 'super' or 'S' prefix, indication an augmentation of the same interval. Accordingly the difference between 65/64 and 64/63, 4096/4095, the ''tridecimal schisma'', is tempered out. | *To extend to the 13-limit, we add that to cP, cM and cA intervals may be added the 'sub' or 's' prefix, in this instance indicating a diminution of [[65/64]], and that to CP, Cm and Cd intervals may be added the 'super' or 'S' prefix, indication an augmentation of the same interval. Accordingly the difference between 65/64 and 64/63, 4096/4095, the ''tridecimal schisma'', is tempered out. 16/13 can then be labelled a 'sub classic major third', or scM3. In tunings where the syntonic comma is tempered out, such that (cP, cM, cA, CP, Cm, Cd) = (P, M, A, P, m, d), the 'c' and 'C' prefixes are dropped in the short-form. | ||
*Similarly, to extend the 11-limit, we add that to SP, SM and SA intervals may be added the 'comma-narrow' or 'c' prefix, in this case indicating a diminution of 99/98, the 7-11 comma, and that to sP, sm and sd intervals may be added the 'comma-wide' of 'C' prefix, in this case indicating an augmentation of the same interval. Accordingly the difference between 81/80 and 99/98, 441/440, is tempered out. 14/11 can then be labelled a 'comma-narrow super major third' or cSM3. In tunings where the septimal comma is tempered out, such that (SP, SM, SA, sP, sm, sd) = (P, M, A, P, m, d), the 'S' and 's' prefixes are dropped in the short form. | |||
*Where N indicates a splitting of the apotome and of the perfect fifth, interval names indicating the splitting of the limma and of the perfect fourth are included for remaining unnamed intervals, reflecting limited, but existing practice. The interval half-way between P1 and m2 is given the short-form '1-2', and long-form 'unison-second' that may be said 'unind'. Similarly the interval half-way between M7 and P8 is given the short-form '1-2' and long-form 'seventh-octave' that may be said 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', with long-form 'second-third' that may be said 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', with long-form 'sixth-seventh', that may be said 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', with long-form 'third-fourth', that may be said 'thourth', and it's octave-complement, the interval half-way between P5 and m6 has short-form '5-6', with long-form 'fifth-sixth', that may be said 'fixth'. These interval names can be associated with [[The Archipelago|Barbados]] temperament, indicating the tempering out of 676/675, generated by 2-3, half of the fourth, associated with the ratio 15/13. These ''intermediates'' lie 40/39 above major intervals and the perfect unison and fifth, and below minor intervals and the perfect fourth and octave. 3-4, for example, is associated with the ratio 13/10. | *Where N indicates a splitting of the apotome and of the perfect fifth, interval names indicating the splitting of the limma and of the perfect fourth are included for remaining unnamed intervals, reflecting limited, but existing practice. The interval half-way between P1 and m2 is given the short-form '1-2', and long-form 'unison-second' that may be said 'unind'. Similarly the interval half-way between M7 and P8 is given the short-form '1-2' and long-form 'seventh-octave' that may be said 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', with long-form 'second-third' that may be said 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', with long-form 'sixth-seventh', that may be said 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', with long-form 'third-fourth', that may be said 'thourth', and it's octave-complement, the interval half-way between P5 and m6 has short-form '5-6', with long-form 'fifth-sixth', that may be said 'fixth'. These interval names can be associated with [[The Archipelago|Barbados]] temperament, indicating the tempering out of 676/675, generated by 2-3, half of the fourth, associated with the ratio 15/13. These ''intermediates'' lie 40/39 above major intervals and the perfect unison and fifth, and below minor intervals and the perfect fourth and octave. 3-4, for example, is associated with the ratio 13/10. | ||
*For completeness, the interval '4-5', long form 'fourth-fifth' that may be said 'firth' is added, though it is separate to the other intermediates, splitting not the limma, but the dieses (between A4 and d5), or the octave. It does not map to any particular ratios and is not needed as a primary interval name, apart from in 16edo, and is included mostly to be used as an optional secondary interval name when there are no others. | *For completeness, the interval '4-5', long form 'fourth-fifth' that may be said 'firth' is added, though it is separate to the other intermediates, splitting not the limma, but the dieses (between A4 and d5), or the octave. It does not map to any particular ratios and is not needed as a primary interval name, apart from in 16edo, and is included mostly to be used as an optional secondary interval name when there are no others. | ||
*In any prefix is used before 'P' then 'P' is removed in both the short-form and long-form names. | *In any prefix is used before 'P' then 'P' is removed in both the short-form and long-form names. | ||
*The prefixes so far take us as far as 53edo, which is considered a 'commatic' scale by many, and as far as extended-diatonic function, which I hope to reflect with this scheme, could be considered to apply. Keenan's functional names take us to 31edo, after which 'narrow' and 'wide' prefixes are added to differentiate different intervals in medium to large sized edos of the same function. Ups and Downs takes function as far as regular diatonic and mids (equivalent to neutrals), which will give us most of a well-ordered interval name set for 17edo (if mids were extended as I have extended neutrals, all the notes would be obtainable) without up or down prefixes, and only functional names, or all of 19edo or 26edo, since these are meantone edos with the apotome subtended by a single degree and may be given a well-ordered interval names set using only regular diatonic interval names. The up and down prefixes are not functional, and specify movement instead by a single step of an edo. If the naming of systems with more than one interval per function is desired, then 'wide' and 'narrow' prefixes, with short form 'W' and 'n' respectively are to be employed. This also allows the notation of intervals for which intermediates are the only available functional interval name. Note: For regular diatonic intervals, I consider function only to go as far as singly diminished or augmented intervals, and never use multiply diminished or augmented intervals for my interval names. | *The prefixes so far take us as far as 53edo (72edo), which is considered a 'commatic' scale by many, and as far as extended-diatonic function, which I hope to reflect with this scheme, could be considered to apply. Keenan's functional names take us to 31edo, after which 'narrow' and 'wide' prefixes are added to differentiate different intervals in medium to large sized edos of the same function. Ups and Downs takes function as far as regular diatonic and mids (equivalent to neutrals), which will give us most of a well-ordered interval name set for 17edo (if mids were extended as I have extended neutrals, all the notes would be obtainable) without up or down prefixes, and only functional names, or all of 19edo or 26edo, since these are meantone edos with the apotome subtended by a single degree and may be given a well-ordered interval names set using only regular diatonic interval names. The up and down prefixes are not functional, and specify movement instead by a single step of an edo. If the naming of systems with more than one interval per function is desired, then 'wide' and 'narrow' prefixes, with short form 'W' and 'n' respectively are to be employed. This also allows the notation of intervals for which intermediates are the only available functional interval name. Note: For regular diatonic intervals, I consider function only to go as far as singly diminished or augmented intervals, and never use multiply diminished or augmented intervals for my interval names. | ||
*Where 'c', 'a' and 'd' are also note names, in some contexts short-form interval names may be confused as short-form chord names, such as Cm7, which is a minor seventh chord rooted on C. Normally context differentiates between, or it can simple be added 'the interval' or 'the chord', but if alternative abbreviations for interval-names that may not be confused with chord names are desired, such 'mid-form' abbreviations are provided in the following tables, which summarise the prefixes listed in this section. | *Where 'c', 'a' and 'd' are also note names, in some contexts short-form interval names may be confused as short-form chord names, such as Cm7, which is a minor seventh chord rooted on C. Normally context differentiates between, or it can simple be added 'the interval' or 'the chord', but if alternative abbreviations for interval-names that may not be confused with chord names are desired, such 'mid-form' abbreviations are provided in the following tables, which summarise the prefixes listed in this section. | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 87: | Line 88: | ||
|C | |C | ||
|Co-W | |Co-W | ||
|up 81/80 | |up 81/80 (or 99/98) | ||
|- | |- | ||
|comma-narrow | |comma-narrow | ||
|c | |c | ||
|co-n | |co-n | ||
|down 81/80 | |down 81/80 (or 99/98) | ||
|- | |- | ||
|wide | |wide | ||
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#Major, minor, A4 and d5. | #Major, minor, A4 and d5. | ||
#hA4 and hd5 | #hA4 and hd5 | ||
#'S', 's', 'C', 'c', 'SC' and ' | #'S', 's', 'C', 'c', 'SC', 'sc', 'cS' and 'Cs' prefixes to major, minor, perfect intervals and to A4 and d5 | ||
#hA1 and hd8 (plus any other hAs and hds if needed) | #hA1 and hd8 (plus any other hAs and hds if needed) | ||
#Intermediates | #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 (for when the chroma is subtended by more than a single (positive) step of the edo) | ||
#'S', 's', 'C', 'c', 'SC' and ' | #'S', 's', 'C', 'c', 'SC', 'sc', 'cS' and 'Cs' prefixes to augmented and diminished intervals | ||
#Intervals augmented and diminished more than singularly | #Intervals augmented and diminished more than singularly | ||
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'. | 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 == | == 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, | 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. | ||
12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8 | 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 | 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 | |||
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 | 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: | |||
We can see that | We can see that | ||