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). | 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. | |||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
|+ Fifthspans of various primes in various edos | |+Fifthspans of various primes in various edos | ||
|- | |- | ||
! | ! | ||
| Line 277: | Line 279: | ||
! prime 7 | ! prime 7 | ||
! prime 11 | ! prime 11 | ||
!prime 13 | |||
!solfege | |||
|- | |- | ||
! [[19-edo]] | ![[19-edo]] | ||
| 0 | | 0 | ||
| 1 | | 1 | ||
| 4 | | 4 | ||
| -9 | | -9 | ||
| 6 | | 6 | ||
| -4 | |||
|MaThoPaFla | |||
|- | |- | ||
! [[22-edo]] | ![[22-edo]] | ||
| 0 | | 0 | ||
| 1 | | 1 | ||
| 9 | | 9 | ||
| -2 | | -2 | ||
| -6 | | -6 | ||
| -9 | |||
|RiThaShaTho | |||
|- | |- | ||
! [[31-edo]] | ![[31-edo]] | ||
| 0 | | 0 | ||
| 1 | | 1 | ||
| 4 | | 4 | ||
| 10 | | 10 | ||
| -13 | | -13 | ||
|15 | |||
|MaLuShoSi | |||
|- | |- | ||
! [[41-edo]] | ![[41-edo]] | ||
| 0 | | 0 | ||
| 1 | | 1 | ||
| -8 | |||
| -14 | |||
| -18 | |||
|20 | |||
|FoDeFlePi | |||
|- | |||
![[53edo|53-edo]] | |||
|0 | |||
|1 | |||
| -8 | | -8 | ||
| -14 | | -14 | ||
| | |23 | ||
|20 | |||
|FoDeRiyuPi | |||
|} | |} | ||
For unsplit rank-2 temperaments, the fifthspan mapping is identical to the 2nd row of the temperament's mapping matrix. Mathematically, the edo's fifthspan mapping is derived by treating the edo as a special case of a specific rank-2 temperament. The 2nd row of this temperament's mapping matrix is the fifthspan mapping. The first row is easily found, it | For unsplit rank-2 temperaments, the fifthspan mapping is identical to the 2nd row of the temperament's mapping matrix. Mathematically, the edo's fifthspan mapping is derived by treating the edo as a special case of a specific rank-2 temperament. The 2nd row of this temperament's mapping matrix is the fifthspan mapping. The first row is easily found, it simply octave-reduces the stacked 5ths. For 12-edo, the temperament is [[Meantone family|Gu & Ru aka Dominant Meantone]]. Here is the full mapping matrix for 12-edo: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
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See also the [[wikipedia:Wicki–Hayden_note_layout|Wicki-Hayden]] layout and the [[wikipedia:Generalized_keyboard|Bosanquet]] keyboard. When playing these instruments, one might want to locate a specific ratio on the keyboard. The dot product of the ratio's monzo with the edo's fifthspan mapping, reduced modulo N, gives the ratio's fifthspan, and hence its location on the instrument. For example, the fifhspan of 7/5 in 31-edo is (0 0 -1 1) ⋅ (0 1 4 10) = -4 + 10 = 6. Whereas in 41-edo, it's (0 0 -1 1) ⋅ (0 1 -8 -14) = -6. Note that this location is based on the indirect (consistent) mapping, not the direct (possibly inconsistent) mapping. The consistent mapping is arguably of greater value on an isomorphic keyboard. | See also the [[wikipedia:Wicki–Hayden_note_layout|Wicki-Hayden]] layout and the [[wikipedia:Generalized_keyboard|Bosanquet]] keyboard. When playing these instruments, one might want to locate a specific ratio on the keyboard. The dot product of the ratio's monzo with the edo's fifthspan mapping, reduced modulo N, gives the ratio's fifthspan, and hence its location on the instrument. For example, the fifhspan of 7/5 in 31-edo is (0 0 -1 1) ⋅ (0 1 4 10) = -4 + 10 = 6. Whereas in 41-edo, it's (0 0 -1 1) ⋅ (0 1 -8 -14) = -6. Note that this location is based on the indirect (consistent) mapping, not the direct (possibly inconsistent) mapping. The consistent mapping is arguably of greater value on an isomorphic keyboard. | ||
See also: [[Antipodes]] | See also: [[Antipodes]], [[Uniform solfege]] | ||
[[Category:Fifth]] | [[Category:Fifth]] | ||
[[Category:Mapping]] | [[Category:Mapping]] | ||