FKH Extended-diatonic Interval Names: Difference between revisions

*Any perfect or major interval raised by the apotome, the interval between the major and minor intervals of a single interval-class is labelled 'A' for augmented, and any perfect or minor interval lowered by the same is labelled 'd' for diminished.

*Any augmented interval may be made doubly augmented, with short-hand 'AA' by the further raising of an apotome and any diminished interval made doubly diminished, with short-hand 'dd' by the further lowering of an apotome. This process may be iterated ad nauseum. At this stage we have simply rigorously defined diatonic interval names. Thankfully what remains of the definition leads to more desirable alternatives for most occasions in which one might find these iteratively diminished and augmented intervals.

*Perfect, major and augmented intervals may be given the prefix 'super', with shorthand 'S' which infers an augmentation by the septimal comma, 64/63, whereas perfect, minor and diminished intervals are lowered by the same interval when given the prefix 'sub', with short-form 's'.

*Major and augmented intervals may be given the prefix 'classic' or 'klassisch', with short-form 'k', inferring a diminution by the syntonic comma, 81/80, whereas minor and diminished may also be given the prefix 'classic' or 'klassisch' but with short-hand 'K', inferring an augmentation by 81/80. This results in the labeling of 10/9, 6/5, 5/4, 8/5, 5/3 and 9/5 as classic major second, classic major third, classic minor third, classic minor sixth, classic major sixth and classic minor seventh, as per Keenan's suggestion when a comparison to Pythagorean is needed. 'K' and 'k' are used instead of 'C' and 'c', and the German translation 'klassisch' invoked in order that interval names like Cm7 that are equivalent to common chord names are avoided.

*To extend to the 13-limit, we add that to kP, kM and kA 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 skM3. In tunings where the syntonic comma is tempered out, such that (kP, kM, kA, KP, Km, Kd) = (P, M, A, P, m, d), the 'k' and 'K' prefixes are dropped in the short-form.

Interval names are ranked in 11 tiers:

#Major, minor, A4 and d5.

#'S', 's', 'K', 'K', 'SK' and 'sk' prefixes to major, minor, perfect intervals and to A4 and d5

#Intermediates

#Remaining augmented and diminished intervals

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

31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 hA4 A4 d5 hd5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8

43edo: P1 S1 1-2 sm2 m2 Sm2 sM2 M2 SM2 2-3 sm3 m3 Sm3 sM3 M3 SM3 3-4 s4 P4 S4 sA4 A4 d5 Sd5 s5 P5 S5 5-6 sm6 m6 Sm6 sM6 M6 SM6 6-7 sm7 m7 Sm7 sM7 M7 SM7 7-8 s8 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

72edo: P1 K1 S1 hA1 sm2 nm2 m2 Km2 kN2 N2 KN2 kM2 M2 WM2 SM2 2-3 sm3 nm3 m3 Km3 kN3 N3 KN3 kM3 M3 WM3 SM3 3-4 s4 n4 P4 K4 S4 hA4 KhA4 kA4 A4/d5 Kd5 khd5 hd5 s5 k5 P5 K5 S5 5-6 sm6 nm6 m6 Cm6 kN6 N6 KN6 kM6 M6 KM6 SM6 6-7 sm7 km7 m7 Km7 kN7 N7 KN7 kM7 M7 KM7 SM7 hd8 s8 k8 P8

We can see that

|}The second best mapping of 5 is better by this measure, so we may notate 20edo using this mapping: 20c ('c' here is called a wart, indicating the use of the second best approximations of the third prime, 5).

In 25edo 5/4 is two degrees below the M3, so the interval in-between does not have a separate function using the patent val in the 7-limit. In 25edo the approximation of 5 is excellent, so we check the second best approximations of 7 and 3.

In 25edo 81/80 is represented by 2 degrees rather than by a single degree, so our scheme doesn't completely work for 25edo, but our scheme is based on the diatonic scale, which in 25edo has pretty much completely broken down.

===7''n''-edos===

It is easy to apply our scheme to 14edo:

We can see that 14edo in a Semaphore tuning, and therefore also a barbados tuning. From our secondary interval names:

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:

28edo: P1 kA1 SA1/sm2 Km2 N2 kM2 SM2/sm3 Km3 N3 kM3 SM3/s4 K4 hA4 kA4 SA4/sd5 Kd5 hd5 k5 S5/sm6 Km6 N6 cM6 SM6/sm7 Km7 N7 kM7 SM7/sd8 Kd8 P8