345edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|345}} == Theory == 345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|345}}
{{ED intro}}


== Theory ==
== Theory ==
345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]], 95703125/95551488 and 359661568/358722675 in the 7-limit; 56723625/56689952, 20155392/20131375, 14348907/14348180, 1835008/1830125, 46656/46585, 2734375/2725888, 2657205/2656192, [[16384/16335]], 514714375/514434888, [[540/539]], 9765625/9732096, 369140625/369098752, [[200704/200475]], 1375/1372, 537109375/536870912, [[19712/19683]], 968000/964467, [[3025/3024]], [[532400/531441]], 202397184/201768035 and 20614528/20588575 in the 11-limit. It provides the optimal patent val for [[quartonic]] and [[kwai]].
345et is only [[consistent]] to the [[5-odd-limit]], though it has a reasonable [[13-limit]] interpretation using the [[patent val]]. It [[tempering out|tempers out]] {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| 47 -15 -10 }} (quintosec comma) in the 5-limit; [[5120/5103]], [[16875/16807]], 2460375/2458624, and 68359375/68024448 in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[16384/16335]], [[19712/19683]], 46656/46585, [[200704/200475]], and [[532400/531441]] in the 11-limit; and [[625/624]] and [[4225/4224]] in the 13-limit. It provides the [[optimal patent val]] for 7-limit [[kwai]].


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


=== Subsets and supersets ===
=== Subsets and supersets ===
345 factors into 3 × 5 × 23, with subset edos {{EDOs|3, 5, 15, 23, 69, and 115}}.
Since 345 factors into {{factorisation|345}}, 345edo has subset edos {{EDOs| 3, 5, 15, 23, 69, and 115 }}.


== 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|547 -345}}
! rowspan="2" | [[Comma list]]
|{{mapping|345 547}}
! rowspan="2" | [[Mapping]]
| -0.2062
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 547 -345 }}
| {{mapping| 345 547 }}
| −0.2062
| 0.2062
| 0.2062
| 5.93
| 5.93
|-
|-
|2.3.5
| 2.3.5
|{{monzo|3 -18 11}}, {{monzo|47 -15 -10}}
| {{monzo| 3 -18 11 }}, {{monzo| 47 -15 -10 }}
|{{mapping|345 547 801}}
| {{mapping| 345 547 801 }}
| -0.1050
| −0.1050
| 0.2210
| 0.2210
| 6.35
| 6.35
|-
|-
|2.3.5.7
| 2.3.5.7
|5120/5103, 16875/16807, 95703125/95551488
| 5120/5103, 16875/16807, 68359375/68024448
|{{mapping|345 547 801 969}}
| {{mapping| 345 547 801 969 }}
| -0.2220
| −0.2220
| 0.2788
| 0.2788
| 8.02
| 8.02
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 3025/3024, 5120/5103, 26796875/26763264
| 540/539, 1375/1372, 5120/5103, 1953125/1940598
|{{mapping|345 547 801 969 1194}}
| {{mapping| 345 547 801 969 1194 }}
| -0.2773
| −0.2773
| 0.2728
| 0.2728
| 7.84
| 7.84
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|540/539, 625/624, 3025/3024, 5120/5103, 4225/4224
| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103
|{{mapping|345 547 801 969 1194 1277}}
| {{mapping| 345 547 801 969 1194 1277 }}
| -0.2857
| −0.2857
| 0.2497
| 0.2497
| 7.18
| 7.18
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=== 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
|38\345
| 13\345
|132.17
| 45.22
|{{monzo|-38 5 13}}
| 250/243
|[[Astro]]
| [[Quartonic]] (5-limit)
|-
| 1
| 38\345
| 132.17
| {{monzo| -38 5 13 }}
| [[Astro]]
|-
|-
|1
| 1
|143\345
| 143\345
|497.39
| 497.39
|4/3
| 4/3
|[[Kwai]]
| [[Kwai]]
|-
|-
|5
| 5
|106\345<br>(32\345)
| 106\345<br />(32\345)
|368.70<br>(111.30)
| 368.70<br />(111.30)
|1024/891<br>(16/15)
| 1024/891<br />(16/15)
|[[Qintosec]]
| [[Quintosec]] (5-limit)
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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
[[Category:Kwai]]

Latest revision as of 13:31, 13 March 2026

← 344edo 345edo 346edo →
Prime factorization 3 × 5 × 23
Step size 3.47826 ¢ 
Fifth 202\345 (702.609 ¢)
Semitones (A1:m2) 34:25 (118.3 ¢ : 86.96 ¢)
Consistency limit 5
Distinct consistency limit 5

345 equal divisions of the octave (abbreviated 345edo or 345ed2), also called 345-tone equal temperament (345tet) or 345 equal temperament (345et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 345 equal parts of about 3.48 ¢ each. Each step represents a frequency ratio of 21/345, or the 345th root of 2.

Theory

345et is only consistent to the 5-odd-limit, though it has a reasonable 13-limit interpretation using the patent val. It tempers out [3 -18 11⟩ (quartonic comma) and [47 -15 -10⟩ (quintosec comma) in the 5-limit; 5120/5103, 16875/16807, 2460375/2458624, and 68359375/68024448 in the 7-limit; 540/539, 1375/1372, 3025/3024, 16384/16335, 19712/19683, 46656/46585, 200704/200475, and 532400/531441 in the 11-limit; and 625/624 and 4225/4224 in the 13-limit. It provides the optimal patent val for 7-limit kwai.

Odd harmonics

Approximation of odd harmonics in 345edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.65 -0.23 +1.61 +1.31 +1.73 +1.21 +0.43 -0.61 +1.62 -1.22 +1.29
Relative (%) +18.8 -6.5 +46.3 +37.6 +49.6 +34.8 +12.3 -17.5 +46.5 -35.0 +37.1
Steps
(reduced) 		547
(202) 		801
(111) 		969
(279) 		1094
(59) 		1194
(159) 		1277
(242) 		1348
(313) 		1410
(30) 		1466
(86) 		1515
(135) 		1561
(181)

Subsets and supersets

Since 345 factors into 3 × 5 × 23, 345edo has subset edos 3, 5, 15, 23, 69, and 115.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [547 -345⟩ [⟨345 547]] −0.2062 0.2062 5.93
2.3.5 [3 -18 11⟩, [47 -15 -10⟩ [⟨345 547 801]] −0.1050 0.2210 6.35
2.3.5.7 5120/5103, 16875/16807, 68359375/68024448 [⟨345 547 801 969]] −0.2220 0.2788 8.02
2.3.5.7.11 540/539, 1375/1372, 5120/5103, 1953125/1940598 [⟨345 547 801 969 1194]] −0.2773 0.2728 7.84
2.3.5.7.11.13 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103 [⟨345 547 801 969 1194 1277]] −0.2857 0.2497 7.18

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 13\345 45.22 250/243 Quartonic (5-limit)
1 38\345 132.17 [-38 5 13⟩ Astro
1 143\345 497.39 4/3 Kwai
5 106\345
(32\345) 		368.70
(111.30) 		1024/891
(16/15) 		Quintosec (5-limit)

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

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