327edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|327}} == Theory == 327et tempers out 283115520/282475249, 156250000/155649627, 16875/16807, 19683/19600, 95703125/95551488, 250047/250000..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|327}}
{{ED intro}}


== Theory ==
== Theory ==
327et tempers out 283115520/282475249, 156250000/155649627, [[16875/16807]], [[19683/19600]], 95703125/95551488, [[250047/250000]] and [[2100875/2097152]] in the 7-limit; 95703125/95664294, 56723625/56689952, 55296000/55240493, 20155392/20131375, 46656/46585, 2734375/2725888, 104162436/103984375, 226492416/226474325, [[540/539]], 9765625/9732096, 137781/137500, 5767168/5764801, 85937500/85766121, 1375/1372, [[8019/8000]], 134775333/134217728, 47265625/47258883, [[3025/3024]], 160083/160000, 35937/35840, 202397184/201768035, 102487/102400, 20614528/20588575, 805255/802816, [[1771561/1769472]] and 526153617/524288000 in the 11-limit. It supports [[petrtri]].
327edo is [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation to the full 13-limit in its [[patent val]]. The equal temperament [[tempering out|tempers out]] the [[semicomma]] in the 5-limit; [[16875/16807]], [[19683/19600]], [[250047/250000]], and [[2100875/2097152]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[8019/8000]], 35937/35840, 46656/46585, 102487/102400, 137781/137500, and 160083/160000 in the 11-limit; and [[625/624]], [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4225/4224]], and [[10648/10647]] in the 13-limit. It supports [[mirkat]], [[pnict]], and the subgroup temperament [[petrtri]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
327 factors into 3 × 109, with [[3edo]] and [[109edo]] as its subset EDOs.
Since 327 factors into {{factorisation|327}}, 327edo has [[3edo]] and [[109edo]] as its subsets.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! 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
! rowspan="2" | [[Subgroup]]
|{{monzo|-518 327}}
! rowspan="2" | [[Comma list]]
|{{mapping|327 518}}
! rowspan="2" | [[Mapping]]
| 0.3273
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -518 327 }}
| {{mapping| 327 518 }}
| +0.3273
| 0.3274
| 0.3274
| 8.92
| 8.92
|-
|-
|2.3.5
| 2.3.5
|2109375/2097152, {{monzo|-20 39 -18}}
| 2109375/2097152, {{monzo| -20 39 -18 }}
|{{mapping|327 518 759}}
| {{mapping| 327 518 759 }}
| 0.3608
| +0.3608
| 0.2715
| 0.2715
| 7.40
| 7.40
|-
|-
|2.3.5.7
| 2.3.5.7
|19683/19600, 16875/16807, 2109375/2097152
| 16875/16807, 19683/19600, 2100875/2097152
|{{mapping|327 518 759 918}}
| {{mapping| 327 518 759 918 }}
| 0.2722
| +0.2722
| 0.2807
| 0.2807
| 7.65
| 7.65
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 3025/3024, 8019/8000, 5788125/5767168
| 540/539, 1375/1372, 8019/8000, 2100875/2097152
|{{mapping|327 518 759 918 1131}}
| {{mapping| 327 518 759 918 1131 }}
| 0.2674
| +0.2674
| 0.2512
| 0.2512
| 6.85
| 6.85
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|540/539, 625/624, 3025/3024, 8019/8000, 4225/4224
| 540/539, 625/624, 1575/1573, 2200/2197, 8019/8000
|{{mapping|327 518 759 918 1131 1210}}
| {{mapping| 327 518 759 918 1131 1210 }}
| 0.2301
| +0.2301
| 0.2441
| 0.2441
| 6.65
| 6.65
Line 60: Line 61:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|74\327
| 74\327
|271.56
| 271.56
|75/64
| 75/64
|[[Orson]]
| [[Orson]]
|-
|-
|3
| 3
|44\327
| 44\327
|161.47
| 161.47
|192/175
| 192/175
|[[Pnict]]
| [[Pnict]]
|-
|-
|3
| 3
|50\327
| 50\327
|183.49
| 183.49
|10/9
| 10/9
|[[Mirkat]]
| [[Mirkat]]
|}
|}
 
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Latest revision as of 06:29, 21 February 2025

← 326edo 327edo 328edo →
Prime factorization 3 × 109
Step size 3.66972 ¢ 
Fifth 191\327 (700.917 ¢)
Semitones (A1:m2) 29:26 (106.4 ¢ : 95.41 ¢)
Consistency limit 7
Distinct consistency limit 7

327 equal divisions of the octave (abbreviated 327edo or 327ed2), also called 327-tone equal temperament (327tet) or 327 equal temperament (327et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 327 equal parts of about 3.67 ¢ each. Each step represents a frequency ratio of 21/327, or the 327th root of 2.

Theory

327edo is consistent to the 7-odd-limit, though it has a reasonable approximation to the full 13-limit in its patent val. The equal temperament tempers out the semicomma in the 5-limit; 16875/16807, 19683/19600, 250047/250000, and 2100875/2097152 in the 7-limit; 540/539, 1375/1372, 3025/3024, 8019/8000, 35937/35840, 46656/46585, 102487/102400, 137781/137500, and 160083/160000 in the 11-limit; and 625/624, 1575/1573, 1716/1715, 2200/2197, 4225/4224, and 10648/10647 in the 13-limit. It supports mirkat, pnict, and the subgroup temperament petrtri.

Odd harmonics

Approximation of odd harmonics in 327edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.04 -0.99 -0.02 +1.59 -0.86 -0.16 +1.64 +1.47 -0.27 -1.06 -0.75
Relative (%) -28.3 -27.0 -0.5 +43.5 -23.4 -4.4 +44.7 +40.0 -7.2 -28.8 -20.5
Steps
(reduced) 		518
(191) 		759
(105) 		918
(264) 		1037
(56) 		1131
(150) 		1210
(229) 		1278
(297) 		1337
(29) 		1389
(81) 		1436
(128) 		1479
(171)

Subsets and supersets

Since 327 factors into 3 × 109, 327edo has 3edo and 109edo as its subsets.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-518 327⟩ [⟨327 518]] +0.3273 0.3274 8.92
2.3.5 2109375/2097152, [-20 39 -18⟩ [⟨327 518 759]] +0.3608 0.2715 7.40
2.3.5.7 16875/16807, 19683/19600, 2100875/2097152 [⟨327 518 759 918]] +0.2722 0.2807 7.65
2.3.5.7.11 540/539, 1375/1372, 8019/8000, 2100875/2097152 [⟨327 518 759 918 1131]] +0.2674 0.2512 6.85
2.3.5.7.11.13 540/539, 625/624, 1575/1573, 2200/2197, 8019/8000 [⟨327 518 759 918 1131 1210]] +0.2301 0.2441 6.65

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 74\327 271.56 75/64 Orson
3 44\327 161.47 192/175 Pnict
3 50\327 183.49 10/9 Mirkat

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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