179edo: Difference between revisions

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179edo does not approximate well any odd [[harmonic]] up to 23, best being [[21/16]] with 22% error. Nonetheless, it is [[consistent]] in the [[7-odd-limit]] and there are a number of temperaments to be considered.
179edo does not approximate well any odd [[harmonic]] up to 23, best being [[21/16]] with 22% error. Nonetheless, it is [[consistent]] in the [[7-odd-limit]] and there are a number of temperaments to be considered.


The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }} in the 5-limit, and [[support]]s [[parakleismic]] and its [[extension]]s, providing the [[optimal patent val]] for 11- and 13-limit [[Ragismic microtemperaments #parkleismic|parkleismic]] temperament. In the 7-limit it tempers out [[3136/3125]], [[4375/4374]] and [[10976/10935]], in the 11-limit [[176/175]] and 1375/1372 and in the 13-limit [[169/168]], [[325/324]], [[351/350]] and [[352/351]], providing the optimal patent val for 11- and 13-limit [[ulmo]] temperament.
The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }} in the 5-limit, and [[support]]s [[parakleismic]] and its [[extension]]s, providing the [[optimal patent val]] for 11- and 13-limit [[Ragismic microtemperaments #parkleismic|parkleismic]] temperament. In the 7-limit it tempers out [[3136/3125]], [[4375/4374]] and [[10976/10935]], in the 11-limit [[176/175]] and 1375/1372 and in the 13-limit [[169/168]], [[325/324]], [[351/350]] and [[352/351]], providing the optimal patent val for 11- and 13-limit [[ulmo]] temperament. It is additionally the optimal patent val for 7-limit [[cohemimabila]].


=== Odd harmonics ===
=== Odd harmonics ===
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! 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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 284 -179 }}
| {{Monzo| 284 -179 }}
| {{mapping| 179 284 }}
| {{Mapping| 179 284 }}
| −0.6169
| −0.6169
| 0.6166
| 0.6166
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 20 -17 3 }}, {{monzo| 28 -3 -10 }}
| {{Monzo| 20 -17 3 }}, {{monzo| 28 -3 -10 }}
| {{mapping| 179 284 416 }}
| {{Mapping| 179 284 416 }}
| −0.7718
| −0.7718
| 0.5490
| 0.5490
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| 2.3.5.7
| 2.3.5.7
| 3136/3125, 4375/4374, 65536/64827
| 3136/3125, 4375/4374, 65536/64827
| {{mapping| 179 284 416 503 }}
| {{Mapping| 179 284 416 503 }}
| −0.8673
| −0.8673
| 0.5034
| 0.5034
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|+ 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
|-
|-
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| 6/5
| 6/5
| [[Parakleismic]]
| [[Parakleismic]]
|-
| 1
| 71\179
| 475.98
| 21/16
| [[Subfourth]] (179ef)
|-
|-
| 1
| 1
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| [[Mabila]] (5-limit)
| [[Mabila]] (5-limit)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|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 ==

Latest revision as of 13:30, 13 March 2026

← 178edo 179edo 180edo →
Prime factorization 179 (prime)
Step size 6.70391 ¢ 
Fifth 105\179 (703.911 ¢)
Semitones (A1:m2) 19:12 (127.4 ¢ : 80.45 ¢)
Consistency limit 7
Distinct consistency limit 7

179 equal divisions of the octave (abbreviated 179edo or 179ed2), also called 179-tone equal temperament (179tet) or 179 equal temperament (179et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 179 equal parts of about 6.7 ¢ each. Each step represents a frequency ratio of 21/179, or the 179th root of 2.

Theory

179edo does not approximate well any odd harmonic up to 23, best being 21/16 with 22% error. Nonetheless, it is consistent in the 7-odd-limit and there are a number of temperaments to be considered.

The equal temperament tempers out the parakleisma, [8 14 -13⟩ in the 5-limit, and supports parakleismic and its extensions, providing the optimal patent val for 11- and 13-limit parkleismic temperament. In the 7-limit it tempers out 3136/3125, 4375/4374 and 10976/10935, in the 11-limit 176/175 and 1375/1372 and in the 13-limit 169/168, 325/324, 351/350 and 352/351, providing the optimal patent val for 11- and 13-limit ulmo temperament. It is additionally the optimal patent val for 7-limit cohemimabila.

Odd harmonics

Approximation of odd harmonics in 179edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.96 +2.51 +3.24 -2.79 -1.60 -2.54 -2.24 +2.31 -2.54 -1.51 +1.89
Relative (%) +29.2 +37.5 +48.3 -41.7 -23.8 -37.9 -33.3 +34.4 -37.9 -22.5 +28.2
Steps
(reduced) 		284
(105) 		416
(58) 		503
(145) 		567
(30) 		619
(82) 		662
(125) 		699
(162) 		732
(16) 		760
(44) 		786
(70) 		810
(94)

Subsets and supersets

179edo is the 41st prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [284 -179⟩ [⟨179 284]] −0.6169 0.6166 9.20
2.3.5 [20 -17 3⟩, [28 -3 -10⟩ [⟨179 284 416]] −0.7718 0.5490 8.19
2.3.5.7 3136/3125, 4375/4374, 65536/64827 [⟨179 284 416 503]] −0.8673 0.5034 7.51

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 35\179 234.64 8/7 Rodan (179d)
1 47\179 315.08 6/5 Parakleismic
1 71\179 475.98 21/16 Subfourth (179ef)
1 79\179 529.61 512/375 Mabila (5-limit)

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

Music

Francium
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