179edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
Eliora (talk | contribs)
No edit summary
Francium (talk | contribs)
+regular temperament properties
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|179}}
{{EDO intro|179}}
 
== Theory ==
179edo doesn't approximate well any odd harmonic up to 23, best being [[21/16]] with 22% error. Nonetheless, it is consistent in the 7-limit and there are anumber of temperaments to be considered.
179edo doesn't approximate well any odd harmonic up to 23, best being [[21/16]] with 22% error. Nonetheless, it is consistent in the 7-limit and there are anumber of temperaments to be considered.


Line 8: Line 8:
=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|179}}
{{Harmonics in equal|179}}
=== Subsets and supersets ===
=== Subsets and supersets ===
179edo is the 41st [[prime edo]].


179edo is the 41st [[prime edo]].
==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|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}}
|{{val|179 284}}
| -0.6169
| 0.6166
| 9.20
|-
|2.3.5
|{{monzo|20 -17 3}}, {{monzo|28 -3 -10}}
|{{val|179 284 416}}
| -0.7718
| 0.5490
| 8.19
|-
|2.3.5.7
|3136/3125, 4375/4374, 10976/10935
|{{val|179 284 416 503}}
| -0.8673
| 0.5034
| 7.51
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
|1
|35\179
|234.64
|8/7
|[[Rodan]]
|-
|1
|47\179
|315.08
|6/5
|[[Parakleismic]]
|-
|1
|79\167
|529.61
|512/375
|[[Mabila]]
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Prime EDO]]

Revision as of 19:50, 3 November 2023

← 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

Template:EDO intro

Theory

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

179edo tempers out the parakleisma, 1224440064/1220703125 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.

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, 10976/10935 179 284 416 503] -0.8673 0.5034 7.51

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator
(reduced)
Cents
(reduced)
Associated
ratio
Temperaments
1 35\179 234.64 8/7 Rodan
1 47\179 315.08 6/5 Parakleismic
1 79\167 529.61 512/375 Mabila