179edo: Difference between revisions

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**Imported revision 262961194 - Original comment: **
 
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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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-09 09:56:57 UTC</tt>.<br>
: The original revision id was <tt>262961194</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 179 equal division divides the octave into 179 steps of 6.704 cents each. It tempers out the parakleisma, |8 14 -13&gt;, in the 5-limit, and supports parakleismic and its extensions, providing the optimal patent val for 11- and 13-limit [[Ragismic microtemperaments#Parakleismic-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.


</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
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.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;179edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 179 equal division divides the octave into 179 steps of 6.704 cents each. It tempers out the parakleisma, |8 14 -13&amp;gt;, in the 5-limit, and supports parakleismic and its extensions, providing the optimal patent val for 11- and 13-limit &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Parakleismic-Parkleismic"&gt;parkleismic temperament&lt;/a&gt;. 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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 ===
{{Harmonics in equal|179}}
 
=== Subsets and supersets ===
179edo is the 41st [[prime edo]].
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 284 -179 }}
| {{Mapping| 179 284 }}
| −0.6169
| 0.6166
| 9.20
|-
| 2.3.5
| {{Monzo| 20 -17 3 }}, {{monzo| 28 -3 -10 }}
| {{Mapping| 179 284 416 }}
| −0.7718
| 0.5490
| 8.19
|-
| 2.3.5.7
| 3136/3125, 4375/4374, 65536/64827
| {{Mapping| 179 284 416 503 }}
| −0.8673
| 0.5034
| 7.51
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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)
|}
<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 ==
; [[Francium]]
* "in a hopeless situation" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/787NGqkI5FHjQcfEOu4y4w Spotify] | [https://francium223.bandcamp.com/track/in-a-hopeless-situation Bandcamp] | [https://www.youtube.com/watch?v=kGdWmP81jwc YouTube]
 
[[Category:Listen]]

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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