152edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Jerdle (talk | contribs)
autopatrolled
212 edits
mNo edit summary
Adopt template: EDO intro; cleanup; note escapade comma
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
The '''152 equal divisions of the octave''' ('''152edo''') or '''152(-tone) equal temperament''' ('''152tet''', '''152et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 152 [[equal]]ly sized parts of about 7.89 [[cent]]s each.
{{EDO intro|152}}


== Theory ==
== Theory ==
152et is a strong 11-limit system, with the 3, 5, 7, and 11 slightly sharp. It tempers out 1600000/1594323, the [[amity comma]], in the 5-limit; [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[Mirkwai comma|16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[4000/3993]], 5632/5625 and [[9801/9800]] in the 11-limit.  
152et is a strong 11-limit system, with the 3, 5, 7, and 11 slightly sharp. It tempers out 1600000/1594323 ([[amity comma]]) and {{monzo| 32 -7 -9 }} ([[escapade comma]]) in the 5-limit; [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[Mirkwai comma|16875/16807]] in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[5632/5625]] and [[9801/9800]] in the 11-limit.  


It has two reasonable mappings for 13, with the 152f val scoring much better. The patent val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1001/1000]], [[1573/1568]], and [[4096/4095]]. The 152f val tempers out [[352/351]], [[625/624]], [[640/637]], [[729/728]], [[847/845]], [[1188/1183]], [[1575/1573]], [[1716/1715]] and [[2080/2079]].  
It has two reasonable mappings for 13, with the 152f val scoring much better. The [[patent val]] tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1001/1000]], [[1573/1568]], and [[4096/4095]]. The 152f val tempers out [[352/351]], [[625/624]], [[640/637]], [[729/728]], [[847/845]], [[1188/1183]], [[1575/1573]], [[1716/1715]] and [[2080/2079]].  


It provides the [[optimal patent val]] for the 11-limit [[Mirkwai clan #Grendel|grendel]] and [[Mirkwai clan #Kwai|kwai]] linear temperaments, the 13-limit rank two temperament [[Ragismic microtemperaments #Octoid-Octopus|octopus]], the 11-limit planar temperament [[Hemifamity family #Laka|laka]], and the rank five temperament tempering out 169/168.  
It provides the [[optimal patent val]] for the 11-limit [[grendel]] and [[kwai]] linear temperaments, the 13-limit rank-2 temperament [[Ragismic microtemperaments #Octoid-Octopus|octopus]], the 11-limit planar temperament [[laka]], and the rank-5 temperament tempering out 169/168.  


[[Paul Erlich]] has suggested that 152edo could be considered a sort of [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3038.html#3041 universal tuning].
[[Paul Erlich]] has suggested that 152edo could be considered a sort of [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3038.html#3041 universal tuning].


152 = 8 × 19, with divisors 2, 4, 8, 19, 38, 76.
=== Prime harmonics ===
{{Harmonics in equal|152}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Harmonics in equal|152|columns=11}}
Since 152 factors into 2<sup>3</sup> × 19, 152edo has subset edos {{EDOs| 2, 4, 8, 19, 38, 76 }}.


== 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]] (¢)

Revision as of 15:29, 1 September 2023

← 151edo 152edo 153edo →
Prime factorization 23 × 19
Step size 7.89474 ¢ 
Fifth 89\152 (702.632 ¢)
Semitones (A1:m2) 15:11 (118.4 ¢ : 86.84 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

152et is a strong 11-limit system, with the 3, 5, 7, and 11 slightly sharp. It tempers out 1600000/1594323 (amity comma) and [32 -7 -9⟩ (escapade comma) in the 5-limit; 4375/4374, 5120/5103, 6144/6125 and 16875/16807 in the 7-limit; 540/539, 1375/1372, 3025/3024, 4000/3993, 5632/5625 and 9801/9800 in the 11-limit.

It has two reasonable mappings for 13, with the 152f val scoring much better. The patent val tempers out 169/168, 325/324, 351/350, 364/363, 1001/1000, 1573/1568, and 4096/4095. The 152f val tempers out 352/351, 625/624, 640/637, 729/728, 847/845, 1188/1183, 1575/1573, 1716/1715 and 2080/2079.

It provides the optimal patent val for the 11-limit grendel and kwai linear temperaments, the 13-limit rank-2 temperament octopus, the 11-limit planar temperament laka, and the rank-5 temperament tempering out 169/168.

Paul Erlich has suggested that 152edo could be considered a sort of universal tuning.

Prime harmonics

Approximation of prime harmonics in 152edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.68 +0.53 +2.23 +1.31 -3.69 -2.32 +2.49 +3.30 -3.26 -0.30
Relative (%) +0.0 +8.6 +6.7 +28.2 +16.6 -46.7 -29.4 +31.5 +41.9 -41.3 -3.8
Steps
(reduced) 		152
(0) 		241
(89) 		353
(49) 		427
(123) 		526
(70) 		562
(106) 		621
(13) 		646
(38) 		688
(80) 		738
(130) 		753
(145)

Subsets and supersets

Since 152 factors into 23 × 19, 152edo has subset edos 2, 4, 8, 19, 38, 76.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [241 -152⟩ [⟨152 241]] -0.213 0.213 2.70
2.3.5 1600000/1594323, [32 -7 -9⟩ [⟨152 241 353]] -0.218 0.174 2.21
2.3.5.7 4375/4374, 5120/5103, 16875/16807 [⟨152 241 353 427]] -0.362 0.291 3.69
2.3.5.7.11 540/539, 1375/1372, 4000/3993, 5120/5103 [⟨152 241 353 427 526]] -0.365 0.260 3.30
2.3.5.7.11.13 352/351, 540/539, 625/624, 729/728, 1575/1573 [⟨152 241 353 427 526 563]] (152f) -0.494 0.373 4.73

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 7\152 55.26 33/32 Escapade / alphaquarter
1 31\152 244.74 15/13 Subsemifourth
1 39\152 307.89 3200/2673 Familia
1 43\152 339.47 243/200 Amity
1 49\152 386.84 5/4 Grendel
1 63\152 497.37 4/3 Kwai
1 71\152 560.53 242/175 Whoosh / whoops
2 7\152 55.26 33/32 Biscapade
2 9\152 71.05 25/24 Vishnu / acyuta (152f) / ananta (152)
2 43\152
(33\152) 		339.47
(260.53) 		243/200
(64/55) 		Hemiamity
2 55\152
(21\152) 		434.21
(165.79) 		9/7
(11/10) 		Supers
4 63\152
(13\152) 		497.37
(102.63) 		4/3
(35/33) 		Undim / unlit
8 63\152
(6\152) 		497.37
(47.37) 		4/3
(36/35) 		Twilight
8 74\152
(2\152) 		584.21
(15.79) 		7/5
(126/125) 		Octoid (152f) / octopus (152)
19 63\152
(1\152) 		497.37
(7.89) 		4/3
(225/224) 		Enneadecal
38 63\152
(1\152) 		497.37
(7.89) 		4/3
(225/224) 		Hemienneadecal
Retrieved from "https://en.xen.wiki/index.php?title=152edo&oldid=123146"