SHEFKHED interval names: Difference between revisions
No edit summary |
No edit summary |
||
| Line 465: | Line 465: | ||
*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. Accordingly 16/13 is 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' or 'C' prefixes are dropped in the short-form. | *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. Accordingly 16/13 is 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' or 'C' 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' and mid-form 'unicond'. Similarly the interval half-way between M7 and P8 is given the short-form '7-8', long-form 'seventh-octave' and mid-form 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', long-form 'second-third', and mid-form 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', long-form 'sixth-seventh', and mid-form 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', long-form 'third-fourth' and mid-form 'thourth', and it's octave-complement, the interval half-way between P5 and m6 is given the short-form '5-6', long-form 'fifth-sixth' and mid-form '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' and mid-form 'unicond'. Similarly the interval half-way between M7 and P8 is given the short-form '7-8', long-form 'seventh-octave' and mid-form 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', long-form 'second-third', and mid-form 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', long-form 'sixth-seventh', and mid-form 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', long-form 'third-fourth' and mid-form 'thourth', and it's octave-complement, the interval half-way between P5 and m6 is given the short-form '5-6', long-form 'fifth-sixth' and mid-form '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, '4-5', with long-form 'fourth-fifth' and short-form '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, | *For completeness, '4-5', with long-form 'fourth-fifth' and short-form '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 'up' and 'down' prefixes, with short form '^' and 'v' 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, 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 'up' and 'down' prefixes, with short form '^' and 'v' 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. | ||
| Line 596: | Line 596: | ||
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'. | ||
On top of this, well-ordered interval-name sets are desired, leading to interval names in lower tires being used in preference to higher-tier names in some cases. | Where the same interval may be named c4 or s4, s4 is preferred and where the same interval may be named C4 or S4, C4 is preferred. Similarly, where the same interval may be named C5 or S5, S5 is preferred and where the same interval may be named c5 or s5, c5 is preferred. This is to ensure the interval is named after the simpler ratio. On top of this, well-ordered interval-name sets are desired, leading to interval names in lower tires being used in preference to higher-tier names in some cases. | ||
=== Regular diatonic edos === | === Regular diatonic edos === | ||
| Line 1,447: | Line 1,447: | ||
Where the M6/m7 represents both 7/4 and 12/7, we know that 5edo is a superpythagorean tuning, tempering out 64/63, and a semaphore tuning, tempering out 49/48. It is therefore also a barbados tuning, tempering out 676/675. We may write 5edo then as | Where the M6/m7 represents both 7/4 and 12/7, we know that 5edo is a superpythagorean tuning, tempering out 64/63, and a semaphore tuning, tempering out 49/48. It is therefore also a barbados tuning, tempering out 676/675. We may write 5edo then as | ||
P1 SM2/sm3 P4 P5 SM6/m7 P8 to express it as a semaphore tuning, or | P1 SM2/sm3 P4 P5 SM6/m7 P8 to express it as a semaphore tuning (equivalent to Semaphore[5] 2|2), or | ||
P1 2-3 P4 P5 6-7 P8 to express it as a barbados tuning, where secondary names for P1 are sm2 and 1-2 respectively, etc. | P1 2-3 P4 P5 6-7 P8 to express it as a barbados tuning (equivalent to Barbados[5] 2|2), where secondary names for P1 are sm2 and 1-2 respectively, etc. | ||
Up to | Up to 30edo, for all 5''n''-edos the 3\5 fifth (3 degrees of 5edo) is the best fifth. 10edo and 15edo may be easily named: | ||
10edo: P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8 | 10edo: P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8 | ||
| Line 1,532: | Line 1,532: | ||
|0.82 | |0.82 | ||
|} | |} | ||
The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may employ ups and downs to | The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may employ ups and downs to name the intervals that do not carry a separate function under this mapping: | ||
25edo: P1 ^P1/^m2 Cm2 cM2 vM2 M2/m3 ^m3 Cm3 cM3 vM3/v4 P4 ^4 C4 c5 v5 P5 ^5/^m6 Cm6 cM6 vM6 M6/m7 ^m7 Cm7 cM7 vM7/vP8 P8 | 25edo: P1 ^P1/^m2 Cm2 cM2 vM2 M2/m3 ^m3 Cm3 cM3 vM3/v4 P4 ^4 C4 c5 v5 P5 ^5/^m6 Cm6 cM6 vM6 M6/m7 ^m7 Cm7 cM7 vM7/vP8 P8 | ||
| Line 1,539: | Line 1,539: | ||
=== 7''n''-edos === | === 7''n''-edos === | ||
At the other limit, in 7edo the large and small steps of the diatonic scale are the same size, and the apotome is tempered out, hence every degree may be written as a neutral, however the interval-name ordering rules lead us to the following primary interval name set: | |||
P1 N2 N3 P4 P5 N6 N7 P8, equivalent to Neutral[7] 3|3. | |||
It is easy to apply our scheme to 14edo: | |||
P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8 | |||
We can see that 14edo in a Semaphore tuning, and therefore also a barbados tuning. From our secondary interval names: | |||
N1 1-2 m2/M2 2-3 m3/M3 3-4 N4 4-5 N5 5-6 m6/M6 6-7 m7/M7 7-8 N8, along with our first, we can see Samaphore[9] and Barbados[9] as subsets of 14edo. | |||
In 21edo, 81/80 is subtended by a single degree, but in the wrong direction. We use alterations of 64/63 to name the intervals below m and above M just as we do normally, however as these intervals are equivalent, and are also neutral, they are labelled neutral: | |||
21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8 | |||
In 28edo, 81/80 is also subtended by -1 degrees, but since 64/63 is subtended by 2 degrees we cannot label all of our intervals using 'S' and 's'. Considering the diatonic scale is all neutral, we can still build a well-ordered interval names set using alterations of 81/80: | |||
28edo: P1 cA1 SA1/sm2 Cm2 N2 cM2 SM2/sm3 Cm3 N3 cM3 SM3/s4 C4 N4 cA4 SA4/sd5 Cd5 N5 c5 S5/sm6 Cm6 N6 cM6 SM6/sm7 Cm7 N7 cM7 SM7/sd8 Cd8 P8 | |||
=== Super-flat edos === | |||
There are edos whose best fifth is flatter even than 4\7. In such edos major intervals are smaller than minor intervals, augmented smaller than major and diminished larger than minor. We expand our definition of well-ordered intervals to include that within each degree... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s/c_ ≤ _ ≤ S/C_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...). In order to obtain well-ordered interval-name sets, we use enharmonic equivalences, replacing diatonic intervals with altered intervals. | |||
In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cM = m, where meantone can be defined by cM = M, and schismatic by cM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM=M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names: | |||
P1 M2 M3 m3 P4 P5 M6 m6 m7 P8. | |||
9edo has diatonic interval names: | |||
P1 M2 m2/M3 m3 P4 P5 M6 m6/M7 m7 P8. | |||
Applying our enharmonic equivalences our primary well-ordered interval names for Mavila[9] 4|4 and 9edo are: | |||
P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8. | |||
11edo has diatonic interval names: | |||
P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8. | |||
Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are: | |||
P1 Cm2 N2 N3 cM3 P4 P5 Cm6 N6 N7 cM7 P8, in which we can see Neutral[7] 3|3. | |||
16edo has diatonic interval names: | |||
P1 d1 M2 m2 M3 m3 A4 P4 d4/A5 P5 d5 M6 m6 M7 m7 A8 P8. | |||
In 16edo 81/80 is represented by -1 degrees and 64/63 by 1 degree, so m = sM = SM | |||
It's primary well-ordered interval names are: | |||
P1 S1 Cm2 cM2 Cm3 cM3 s4 P4 S4/s5 P5 S5 Cm6 cM6 Cm7 cM7 s8 P8 | |||
Similarly, the primary well-ordered interval names for 23edo are: | |||
P1 S1 1-2 Cm2 cM2 2-3 Cm3 cM3 3-4 s4 P4 S4 s5 P5 S5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8, from which we can see it is a Barbados tuning. | |||