373edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
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"
Tags: Mobile edit Mobile web edit
Update
Line 3: Line 3:


== Theory ==
== Theory ==
373edo is [[consistency|distinctly consistent]] to the [[15-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 through 13 all tuned flat. As an equal temperament, it [[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 also [[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. As an equal temperament, it [[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 also [[support]]s the [[hemitert]] temperament.  


=== Prime harmonics ===
=== Prime harmonics ===
Line 20: Line 20:
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
Line 27: Line 27:
|-
|-
| 2.3
| 2.3
| {{monzo| -591 373 }}
| {{Monzo| -591 373 }}
| {{mapping| 373 591 }}
| {{Mapping| 373 591 }}
| +0.1939
| +0.1939
| 0.1939
| 0.1939
Line 34: Line 34:
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| -51 19 9 }}
| {{Monzo| 8 14 -13 }}, {{monzo| -51 19 9 }}
| {{mapping| 373 591 866 }}
| {{Mapping| 373 591 866 }}
| +0.1658
| +0.1658
| 0.1632
| 0.1632
Line 42: Line 42:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 65625/65536, 43046721/42875000
| 2401/2400, 65625/65536, 43046721/42875000
| {{mapping| 373 591 866 1047 }}
| {{Mapping| 373 591 866 1047 }}
| +0.1654
| +0.1654
| 0.1413
| 0.1413
Line 49: Line 49:
| 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
| {{mapping| 373 591 866 1047 1290 }}
| {{Mapping| 373 591 866 1047 1290 }}
| +0.2008
| +0.2008
| 0.1449
| 0.1449
Line 56: Line 56:
| 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
| {{mapping| 373 591 866 1047 1290 1380 }}
| {{Mapping| 373 591 866 1047 1290 1380 }}
| +0.2056
| +0.2056
| 0.1327
| 0.1327
Line 66: Line 66:
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 88: Line 88:
| 315.28
| 315.28
| 6/5
| 6/5
| [[Parakleismic]] (5-limit)
| [[Counterpara]]
|-
|-
| 1
| 1
Line 108: Line 108:
| [[Untriton]] (5-limit)
| [[Untriton]] (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/>* [[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


== Music ==
== Music ==

Revision as of 08:45, 20 August 2026

← 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

373 equal divisions of the octave (abbreviated 373edo or 373ed2), also called 373-tone equal temperament (373tet) or 373 equal temperament (373et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 373 equal parts of about 3.22 ¢ each. Each step represents a frequency ratio of 21/373, or the 373rd root of 2.

Theory

373edo is distinctly consistent to the 15-odd-limit. It has a flat tendency, with harmonics 3 through 13 all tuned flat. As an equal temperament, it 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 also 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.

Intervals

Main article: Table of 373edo intervals

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 Counterpara
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 distinct

Music

Francium
Retrieved from "https://en.xen.wiki/index.php?title=373edo&oldid=236078"