375edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|375}}
{{ED intro}}


== Theory ==
== Theory ==
375et is only consistent to the [[3-odd-limit]] and the harmonic 3 is about halfway between its steps. It can be used in the 2.9.5.7.13.17.19 [[subgroup]]. Using the patent val, it tempers out 40500000/40353607, 52734375/52706752 and [[6144/6125]] in the 7-limit; 100663296/100656875, 10333575/10307264, 166698/166375, 759375/758912, 151263/151250, [[540/539]], 4302592/4296875, 825000/823543, [[5632/5625]], 16808715/16777216, 1362944/1361367, 4108797/4096000, 67110351/67108864, 805255/802816 and [[1771561/1769472]] in the 11-limit. It [[support]]s [[aufic]] and [[persephone]].
375edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. It can be used as a temperament in the 2.9.5.7.13.17.19 [[subgroup]], in which it [[tempering out|tempers out]] [[250047/250000]], 589824/588245, and {{monzo| 8 7 -13 }} ([[parakleisma]]).
 
For the full 7-limit, the 375cd [[val]] is the best, where it tempers out [[1029/1024]] and [[15625/15552]], [[support]]ing [[tritikleismic]]. In the 11-limit it tempers out [[385/384]] and [[441/440]], supporting undecimal tritikleismic.
 
Using the [[patent val]], it tempers out [[6144/6125]] and 177147/175616, supporting [[aufo]]. In the 11-limit it tempers out [[540/539]] and [[5632/5625]], supporting [[aufic]] and [[persephone]].  


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 13:


=== Subsets and supersets ===
=== Subsets and supersets ===
375 factors into 3 × 5<sup>3<sup> with subset edos {{EDOs|3, 5, 15, 25, 75, and 125}}. [[1175edo]], which triples it, gives a good correction to the harmonic 3 and is consistent to the [[15-odd-limit]].
Since 375 factors into 3 × 5<sup>3</sup>, 375edo has subset edos {{EDOs| 3, 5, 15, 25, 75, and 125 }}. [[1175edo]], which triples it, gives a good correction to the harmonic 3 and is consistent to the [[15-odd-limit]].


== 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" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|1189 -375}}
| {{monzo| 1189 -375 }}
|{{mapping|375 1189}}
| {{mapping| 375 1189 }}
| -0.1404
| −0.1404
| 0.1404
| 0.1404
| 4.39
| 4.39
|-
|-
|2.9.5
| 2.9.5
|{{monzo|8 7 -13}}, {{monzo|97 -24 -9}}
| {{monzo| 8 7 -13 }}, {{monzo| 97 -24 -9 }}
|{{mapping|375 1189 871}}
| {{mapping| 375 1189 871 }}
| -0.2208
| −0.2208
| 0.1615
| 0.1615
| 5.05
| 5.05
|-
|-
|2.9.5.7
| 2.9.5.7
|250047/250000, 26873856/26796875, 26985857024/26904200625
| 250047/250000, 589824/588245, {{monzo| 8 7 -13 }}
|{{mapping|375 1189 871 1053}}
| {{mapping| 375 1189 871 1053 }}
| -0.2345
| −0.2345
| 0.1418
| 0.1418
| 4.43
| 4.43
|}
|}

Latest revision as of 12:19, 21 February 2025

← 374edo 375edo 376edo →
Prime factorization 3 × 53
Step size 3.2 ¢ 
Fifth 219\375 (700.8 ¢) (→ 73\125)
Semitones (A1:m2) 33:30 (105.6 ¢ : 96 ¢)
Dual sharp fifth 220\375 (704 ¢) (→ 44\75)
Dual flat fifth 219\375 (700.8 ¢) (→ 73\125)
Dual major 2nd 64\375 (204.8 ¢)
Consistency limit 3
Distinct consistency limit 3

375 equal divisions of the octave (abbreviated 375edo or 375ed2), also called 375-tone equal temperament (375tet) or 375 equal temperament (375et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 375 equal parts of exactly 3.2 ¢ each. Each step represents a frequency ratio of 21/375, or the 375th root of 2.

Theory

375edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. It can be used as a temperament in the 2.9.5.7.13.17.19 subgroup, in which it tempers out 250047/250000, 589824/588245, and [8 7 -13⟩ (parakleisma).

For the full 7-limit, the 375cd val is the best, where it tempers out 1029/1024 and 15625/15552, supporting tritikleismic. In the 11-limit it tempers out 385/384 and 441/440, supporting undecimal tritikleismic.

Using the patent val, it tempers out 6144/6125 and 177147/175616, supporting aufo. In the 11-limit it tempers out 540/539 and 5632/5625, supporting aufic and persephone.

Odd harmonics

Approximation of odd harmonics in 375edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.16 +0.89 +0.77 +0.89 -0.92 +1.07 -0.27 +0.64 +0.09 -0.38 -1.07
Relative (%) -36.1 +27.7 +24.2 +27.8 -28.7 +33.5 -8.4 +20.1 +2.7 -11.9 -33.6
Steps
(reduced) 		594
(219) 		871
(121) 		1053
(303) 		1189
(64) 		1297
(172) 		1388
(263) 		1465
(340) 		1533
(33) 		1593
(93) 		1647
(147) 		1696
(196)

Subsets and supersets

Since 375 factors into 3 × 53, 375edo has subset edos 3, 5, 15, 25, 75, and 125. 1175edo, which triples it, gives a good correction to the harmonic 3 and is consistent to the 15-odd-limit.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.9 [1189 -375⟩ [⟨375 1189]] −0.1404 0.1404 4.39
2.9.5 [8 7 -13⟩, [97 -24 -9⟩ [⟨375 1189 871]] −0.2208 0.1615 5.05
2.9.5.7 250047/250000, 589824/588245, [8 7 -13⟩ [⟨375 1189 871 1053]] −0.2345 0.1418 4.43
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