31edo: Difference between revisions
m →Music: list formatting, a bit of space to make the wikitext readable as well |
added M2, m2 and A1 to the template, moved the primes-error table up to the top |
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| Prime factorization = 31 | | Prime factorization = 31 | ||
| Subgroup = 2.3.5.7.11 | | Subgroup = 2.3.5.7.11 | ||
| Step size = 38. | | Step size = 38.710¢ | ||
| Fifth type = meantone 13\31 696.77¢ | | Fifth type = meantone 13\31 = 696.77¢ | ||
| Major 2nd = 5\31 = 194¢ | |||
| Minor 2nd = 3\31 = 116¢ | |||
| Augmented 1sn = 2\31 = 77¢ | |||
| Common uses = ~1/4-comma meantone<br/>general-purpose 11-limit system<br/>extended harmony utilizing dieses | | Common uses = ~1/4-comma meantone<br/>general-purpose 11-limit system<br/>extended harmony utilizing dieses | ||
| Important MOS = diatonic (meantone) 5L2s 5553553 (18\31, 1\1)<br/>whole tone ([[hemithirds]]) 1L5s 555556 (5\31, 1\1)<br/>[[mohajira]] heptatonic 3L4s 5454544 (9\31, 1\1)<br/>[[mohajira]] decatonic 7L3s 4441441441 (9\31, 1\1)<br/>oneirotonic ([[A-Team]]) 5L3s 55255252 (12\31, 1\1) | | Important MOS = diatonic (meantone) 5L2s 5553553 (18\31, 1\1)<br/>whole tone ([[hemithirds]]) 1L5s 555556 (5\31, 1\1)<br/>[[mohajira]] heptatonic 3L4s 5454544 (9\31, 1\1)<br/>[[mohajira]] decatonic 7L3s 4441441441 (9\31, 1\1)<br/>oneirotonic ([[A-Team]]) 5L3s 55255252 (12\31, 1\1) | ||
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== Theory == | == Theory == | ||
Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit. | {| class="wikitable center-all" | ||
! colspan="2" | | |||
!prime 2 | |||
!prime 3 | |||
!prime 5 | |||
!prime 7 | |||
!prime 11 | |||
!prime 13 | |||
!prime 17 | |||
!prime 19 | |||
!prime 23 | |||
|- | |||
! rowspan="2" |Error | |||
!absolute (¢) | |||
| 0 | |||
| -5.18 | |||
| +0.8 | |||
| -1.1 | |||
| -9.4 | |||
| +11.1 | |||
| +11.2 | |||
| +12.2 | |||
| -8.9 | |||
|- | |||
![[Relative error|relative]] (%) | |||
| 0 | |||
| -13 | |||
| +2 | |||
| -3 | |||
| -24 | |||
| +29 | |||
| +29 | |||
| +31 | |||
| -23 | |||
|- | |||
! colspan="2" |[[nearest edomapping]] | |||
|31 | |||
|18 | |||
|10 | |||
|25 | |||
|14 | |||
|22 | |||
|3 | |||
|8 | |||
|16 | |||
|- | |||
! colspan="2" |[[fifthspan]] | |||
| 0 | |||
| +1 | |||
| +4 | |||
| +10 | |||
| -13 | |||
| +15 | |||
| -5 | |||
| -3 | |||
| -6 | |||
|}Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit. | |||
31edo's 12\31 generator (an approximate 21/16) supports [[A-Team]] and yields [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale|8-note "oneirotonic" scales similar to those in 13edo]] but with the 9/8 and 5/4 better in tune; this temperament is also represented by [[13edo]], [[18edo]] and [[44edo]]. | 31edo's 12\31 generator (an approximate 21/16) supports [[A-Team]] and yields [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale|8-note "oneirotonic" scales similar to those in 13edo]] but with the 9/8 and 5/4 better in tune; this temperament is also represented by [[13edo]], [[18edo]] and [[44edo]]. | ||
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=== Selected just intervals === | === Selected just intervals === | ||
==== 15-odd-limit interval mappings ==== | ==== 15-odd-limit interval mappings ==== | ||