315edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Skip disambiguation
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
Line 12: Line 12:


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 42: Line 34:
| 0.2659
| 0.2659
| 6.98
| 6.98
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 60: Line 46:
|-
|-
| 5
| 5
| 131\315<br>(5\315)
| 131\315<br />(5\315)
| 499.05<br>(19.05)
| 499.05<br />(19.05)
| 4/3<br>(81/80)
| 4/3<br />(81/80)
| [[Pental (temperament)|Pental]] (5-limit)
| [[Pental (temperament)|Pental]] (5-limit)
|-
|-
| 9
| 9
| 83\315<br>(13\315)
| 83\315<br />(13\315)
| 316.19<br>(49.52)
| 316.19<br />(49.52)
| 6/5<br>(36/35)
| 6/5<br />(36/35)
| [[Ennealimmal]]
| [[Ennealimmal]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 02:25, 16 November 2024

← 314edo 315edo 316edo →
Prime factorization 32 × 5 × 7
Step size 3.80952 ¢ 
Fifth 184\315 (700.952 ¢)
Semitones (A1:m2) 28:25 (106.7 ¢ : 95.24 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

315edo is consistent to the 7-odd-limit with a flat tendency in the harmonics 3, 5, and 7. The equal temperament tempers out 2401/2400, 4375/4374 and 35595703125/35246833664. Using the 315e val in the 11-limit (⟨315 ​499 ​731​ 884​ 1089]), it tempers out 385/384, 1375/1372, 4375/4374 and 644204/643125, supporting beyla and ennealiminal.

Odd harmonics

Approximation of odd harmonics in 315edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.00 -1.55 -1.21 +1.80 +1.06 +1.38 +1.26 +1.71 -0.37 +1.60 +0.30
Relative (%) -26.3 -40.7 -31.7 +47.4 +27.9 +36.1 +32.9 +44.9 -9.7 +42.0 +7.8
Steps
(reduced) 		499
(184) 		731
(101) 		884
(254) 		999
(54) 		1090
(145) 		1166
(221) 		1231
(286) 		1288
(28) 		1338
(78) 		1384
(124) 		1425
(165)

Subsets and supersets

Since 315 factors into 32 × 5 × 7, 315edo has subset edos 3, 5, 7, 9, 15, 21, 35, 45, 63, and 105. 945edo, which triples it, gives a good correction to the harmonic 11.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [-499 315⟩ | [⟨315 499]] | 0.3163 | 0.3164 | 8.31 |- | 2.3.5 | [-27 -2 13⟩, [-28 25 -5⟩ | [⟨315 499 731]] | 0.4337 | 0.3071 | 8.06 |- | 2.3.5.7 | 2401/2400, 4375/4374, [-21 6 11 -5⟩ | [⟨315 499 731 884]] | 0.4328 | 0.2659 | 6.98 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 107\315 | 407.62 | 15625/12288 | Ditonic |- | 5 | 131\315
(5\315) | 499.05
(19.05) | 4/3
(81/80) | Pental (5-limit) |- | 9 | 83\315
(13\315) | 316.19
(49.52) | 6/5
(36/35) | Ennealimmal Template:Rank-2 end Template:Orf

Retrieved from "https://en.xen.wiki/index.php?title=315edo&oldid=165751"