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FKH Extended-diatonic Interval Names: Difference between revisions

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72edo may also be given 'functional names' by allowing application of 'komma-wide' and 'komma-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, 'k' and 'K' give associations with ratios that are not too complex.
72edo may also be given 'functional names' by allowing application of 'komma-wide' and 'komma-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, 'k' and 'K' give associations with ratios that are not too complex.


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
72edo: P1 K1 S1 hA1 sm2 km2 m2 Km2 kN2 N2 KN2 kM2 M2 KM2 SM2 2-3 sm3 nm3 m3 Km3 kN3 N3 KN3 kM3 M3 KM3 SM3 3-4 s4 k4 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
 
80edo: P1 S1 K1 SK1 sm2 m2 Wm2 Km2 SKm2 _ skM2 kM2 nM2 M2 SM2 sm3


We can see that
We can see that
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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


15edo: P1 K1/Km2 cM2 M2/m3 Km3 cM3 P4 K4 k5 P5 Km6 kM6 M6/m7 Km7 kM7/k8 P8
15edo: P1 K1/Km2 kM2 M2/m3 Km3 kM3 P4 K4 k5 P5 Km6 kM6 M6/m7 Km7 kM7/k8 P8


The remaining 5''n-''edos are difficult, however.
The remaining 5''n-''edos are difficult, however.
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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'. If we use neutrals (hA4 and hd5) then we can still build a well-ordered interval names set using alterations of 81/80:
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'. If we use neutrals (hA4 and hd5) then we can still build a well-ordered interval names set using alterations of 81/80:


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
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 kM6 SM6/sm7 Km7 N7 kM7 SM7/sd8 Kd8 P8
===Super-flat edos===
===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 sc _ ≤ s/c_ ≤ _ ≤ S/C_ ≤ SC _ (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.
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 sk _ ≤ s/k_ ≤ _ ≤ S/K_ ≤ SK _ (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 cla M = 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:
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 cla M = 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:
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Applying our enharmonic equivalences our primary well-ordered interval names for Mavila[9] 4|4 and 9edo are:
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.
P1 M2 Km3 kM3 P4 P5 Km6 kM6 m7 P8.


11edo has diatonic interval names:
11edo has diatonic interval names:
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