Kite's thoughts on fifthspans: Difference between revisions

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If N-edo's best approximation of a prime P is X edosteps, or X\N, then P's fifthspan is the fifthspan of X\N. Just as an edomapping or [[patent val]] assigns an edostepspan to each prime, a fifthspan mapping assigns a fifthspan to each prime. Prime 2's fifthspan is always 0, and prime 3's fifthspan is always 1. For example, in 12-edo, 5/4 is best approximated by 4\12, which is a major 3rd, which has fifthspan 4. 7/4 is a minor 7th, fifthspan -2. Thus 12edo's fifthspan mapping of 2.3.5.7 is (0 1 4 -2).

The fifthspan mapping can be expressed very concisely as a ~~series~~ of [[uniform solfege]] syllables.

The fifthspan mapping can be expressed very concisely as a string of [[uniform solfege]] syllables. Primes 2 and 3 are always Da and Sa, so those two primes can be omitted. The remaining primes are listed in order. All meantone edos start with Ma, all archy edos have Tha as the 2nd syllable, etc. Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.

{| class="wikitable center-all"

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! prime 11

!prime 13

!solfege

!solfege string

|-

![[19-edo]]

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| -6

| -9

|~~RiThaShaTho~~

|RuThaShaTho

|-

![[31-edo]]

@@ Line 267: / Line 267: @@
 If N-edo's best approximation of a prime P is X edosteps, or X\N, then P's fifthspan is the fifthspan of X\N. Just as an edomapping or [[patent val]] assigns an edostepspan to each prime, a fifthspan mapping assigns a fifthspan to each prime. Prime 2's fifthspan is always 0, and prime 3's fifthspan is always 1. For example, in 12-edo, 5/4 is best approximated by 4\12, which is a major 3rd, which has fifthspan 4. 7/4 is a minor 7th, fifthspan -2. Thus 12edo's fifthspan mapping of 2.3.5.7 is (0 1 4 -2).
-The fifthspan mapping can be expressed very concisely as a series of [[uniform solfege]] syllables.
+The fifthspan mapping can be expressed very concisely as a string of [[uniform solfege]] syllables. Primes 2 and 3 are always Da and Sa, so those two primes can be omitted. The remaining primes are listed in order. All meantone edos start with Ma, all archy edos have Tha as the 2nd syllable, etc. Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.
 {| class="wikitable center-all"
@@ Line 280: / Line 280: @@
 ! prime 11
 !prime 13
-!solfege
+!solfege string
 |-
 ![[19-edo]]
@@ Line 298: / Line 298: @@
 |  -6
 | -9
-|RiThaShaTho
+|RuThaShaTho
 |-
 ![[31-edo]]