373edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''373 equal divisions of the octave''' ('''373edo'''), or the '''373(-tone) equal temperament''' ('''373tet''', '''373et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 373 parts of about 3.22 [[cent]]s each.
{{EDO intro|373}}


== Theory ==
== Theory ==
373edo is [[consistent]] to the [[15-odd-limit]], with harmonics 3 through 13 all tuned flat. It tempers out {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 2401/2400 ([[breedsma]]), 65625/65536 (horwell), and 43046721/42875000 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, and 496125/495616 in the 11-limit; [[729/728]], [[1001/1000]], [[1716/1715]], [[4225/4224]], and [[10648/10647]] in the 13-limit, enabling [[squbemic chords]] and [[sinbadmic chords]]. It [[support]]s the [[Breedsmic temperaments #Hemitert|hemitert temperament]].
373edo is [[consistency|distinctly consistent]] to the [[15-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 through 13 all tuned flat. The equal temperament [[tempering out|tempers out]] {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 43046721/42875000 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, and 496125/495616 in the 11-limit; [[729/728]], [[1001/1000]], [[1716/1715]], [[4225/4224]], and [[10648/10647]] in the 13-limit, enabling [[squbemic chords]] and [[sinbadmic chords]]. It [[support]]s the [[Breedsmic temperaments #Hemitert|hemitert temperament]].


=== Prime harmonics ===
{{Harmonics in equal|373}}
=== Subsets and supersets ===
373edo is the 74th [[prime edo]].
373edo is the 74th [[prime edo]].
=== Prime harmonics ===
{{Primes in edo|373}}


== 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" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| -591 373 }}
| {{monzo| -591 373 }}
| [{{val| 373 591 }}]
| {{mapping| 373 591 }}
| +0.1939
| +0.1939
| 0.1939
| 0.1939
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| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| -51 19 9 }}
| {{monzo| 8 14 -13 }}, {{monzo| -51 19 9 }}
| [{{val| 373 591 866 }}]
| {{mapping| 373 591 866 }}
| +0.1658
| +0.1658
| 0.1632
| 0.1632
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 65625/65536, 43046721/42875000
| 2401/2400, 65625/65536, 43046721/42875000
| [{{val| 373 591 866 1047 }}]
| {{mapping| 373 591 866 1047 }}
| +0.1654
| +0.1654
| 0.1413
| 0.1413
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 8019/8000, 65625/65536
| 2401/2400, 3025/3024, 8019/8000, 65625/65536
| [{{val| 373 591 866 1047 1290 }}]
| {{mapping| 373 591 866 1047 1290 }}
| +0.2008
| +0.2008
| 0.1449
| 0.1449
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 729/728, 1001/1000, 1716/1715, 3025/3024, 4225/4224
| 729/728, 1001/1000, 1716/1715, 3025/3024, 4225/4224
| [{{val| 373 591 866 1047 1290 1380 }}]
| {{mapping| 373 591 866 1047 1290 1380 }}
| +0.2056
| +0.2056
| 0.1327
| 0.1327
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Untriton]] (5-limit)
| [[Untriton]] (5-limit)
|}
|}
 
<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:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]

Revision as of 14:50, 9 November 2023

← 372edo 373edo 374edo →
Prime factorization 373 (prime)
Step size 3.21716 ¢ 
Fifth 218\373 (701.34 ¢)
Semitones (A1:m2) 34:29 (109.4 ¢ : 93.3 ¢)
Consistency limit 15
Distinct consistency limit 15

Template:EDO intro

Theory

373edo is distinctly consistent to the 15-odd-limit. It has a flat tendency, with harmonics 3 through 13 all tuned flat. The equal temperament tempers out [8 14 -13⟩ (parakleisma) and [-51 19 9⟩ (untriton comma) in the 5-limit; 2401/2400 (breedsma), 65625/65536 (horwell comma), and 43046721/42875000 in the 7-limit; 3025/3024, 8019/8000, 24057/24010, and 496125/495616 in the 11-limit; 729/728, 1001/1000, 1716/1715, 4225/4224, and 10648/10647 in the 13-limit, enabling squbemic chords and sinbadmic chords. It supports the hemitert temperament.

Prime harmonics

Approximation of prime harmonics in 373edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.61 -0.25 -0.46 -1.18 -0.85 +1.21 -1.53 -0.93 -0.09 +0.27
Relative (%) +0.0 -19.1 -7.9 -14.3 -36.8 -26.4 +37.6 -47.7 -28.9 -2.7 +8.5
Steps
(reduced) 		373
(0) 		591
(218) 		866
(120) 		1047
(301) 		1290
(171) 		1380
(261) 		1525
(33) 		1584
(92) 		1687
(195) 		1812
(320) 		1848
(356)

Subsets and supersets

373edo is the 74th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-591 373⟩ [⟨373 591]] +0.1939 0.1939 6.03
2.3.5 [8 14 -13⟩, [-51 19 9⟩ [⟨373 591 866]] +0.1658 0.1632 5.07
2.3.5.7 2401/2400, 65625/65536, 43046721/42875000 [⟨373 591 866 1047]] +0.1654 0.1413 4.39
2.3.5.7.11 2401/2400, 3025/3024, 8019/8000, 65625/65536 [⟨373 591 866 1047 1290]] +0.2008 0.1449 4.50
2.3.5.7.11.13 729/728, 1001/1000, 1716/1715, 3025/3024, 4225/4224 [⟨373 591 866 1047 1290 1380]] +0.2056 0.1327 4.12

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 12\373 38.61 45/44 Hemitert
1 24\373 77.21 256/245 Tertiaseptal
1 98\373 315.28 6/5 Parakleismic (5-limit)
1 111\373 357.10 768/625 Dodifo (5-limit)
1 162\373 521.18 875/648 Maviloid
1 183\373 588.74 45/32 Untriton (5-limit)

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

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