144edo: Difference between revisions

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Expand on theory; +RTT table and rank-2 temperaments
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== Theory ==
== Theory ==
144edo is closely related to [[72edo]], but the [[patent val]]s differ on the mapping for 13 and 17. It is [[contorted]] in the 11-limit, tempering out [[225/224]], [[243/242]], [[385/384]], [[441/440]], and [[4000/3993]]. Using the patent val, it tempers out [[847/845]], [[1188/1183]], 1701/1690, 1875/1859, and [[4225/4224]] in the 13-limit; [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]] in the 17-limit; 210/209, 325/323, [[343/342]], [[363/361]], [[400/399]], [[513/512]], and 665/663 in the 19-limit.
144edo is closely related to [[72edo]], but the [[patent val]]s differ on the mapping for [[13/1|13]] and [[17/1|17]]. It is [[enfactoring|enfactored]] in the 11-limit, [[tempering out]] [[225/224]], [[243/242]], [[385/384]], [[441/440]], and [[4000/3993]]. Using the [[patent val]], it tempers out [[847/845]], [[1188/1183]], 1701/1690, 1875/1859, and [[4225/4224]] in the 13-limit.  


The 144gh val is a tuning for semihemisecordite temperament, while 144eff val is a tuning for hemimiracle. 144ee is a tuning for oracle. 144cf val supports necromanteion.  
Although the patent val comes out on top accuracy in the 13-limit, in the 17-limit 144 falls behind to 144g. The 144g val tempers out [[170/169]], [[289/288]], [[375/374]], [[561/560]], [[595/594]]. It supports [[semihemisecordite]], the 62 & 82f temperament. The patent val tempers out [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]]. It [[support]]s [[hemisecordite]], the 41 & 103 temperament, though [[103edo]] is better suited for this purpose. In the 19-limit the patent val tempers out 210/209, 325/323, [[343/342]], [[363/361]], [[400/399]], [[513/512]], and 665/663.
 
Besides all these, the 144eff val supports hemimiracle, the 41 & 103e temperament. 144ee supports oracle, the 31 & 113e temperament. 144cf supports necromanteion, the 31 & 113cf temperament.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Approximation to φ ===
=== Approximation to φ ===
144edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci EDO, it can produce extremely precise approximation of the [[Logarithmic phi|logarithmic golden ratio]] at 89 steps. In addition, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps.
144edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci edo, it can produce extremely precise approximation of the [[Logarithmic phi|logarithmic golden ratio]] at 89 steps. In addition, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps.
 
== 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]] (%)
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 385/384, 847/845, 1875/1859
| {{mapping| 144 228 334 404 498 533 }}
| +0.560
| 0.595
| 7.13
|-
| style="border-top: double;" | 2.3.5.7.11.13.17
| style="border-top: double;" | 170/169, 225/224, 243/242, 289/288, 375/374, 385/384
| style="border-top: double;" | {{mapping| 144 228 334 404 498 533 588 }} (144g)
| style="border-top: double;" | +0.653
| style="border-top: double;" | 0.596
| style="border-top: double;" | 7.15
|-
| style="border-top: double;" | 2.3.5.7.11.13.17
| style="border-top: double;" | 225/224, 243/242, 273/272, 325/324, 847/845, 875/867
| style="border-top: double;" | {{mapping| 144 228 334 404 498 533 589 }} (144)
| style="border-top: double;" | +0.362
| style="border-top: double;" | 0.734
| style="border-top: double;" | 8.80
|}
 
=== 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
! Temperament
|-
| 1
| 7\144
| 58.33
| 27/26
| [[Hemisecordite]] (144)
|-
| 2
| 7\144
| 58.33
| 27/26
| [[Semihemisecordite]] (144g)
|}


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] by [[5 hideya]]
; [[Amano Hideya]]
* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] (2022)

Revision as of 06:55, 10 October 2023

← 143edo 144edo 145edo →
Prime factorization 24 × 32
Step size 8.33333 ¢ 
Fifth 84\144 (700 ¢) (→ 7\12)
Semitones (A1:m2) 12:12 (100 ¢ : 100 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

144edo is closely related to 72edo, but the patent vals differ on the mapping for 13 and 17. It is enfactored in the 11-limit, tempering out 225/224, 243/242, 385/384, 441/440, and 4000/3993. Using the patent val, it tempers out 847/845, 1188/1183, 1701/1690, 1875/1859, and 4225/4224 in the 13-limit.

Although the patent val comes out on top accuracy in the 13-limit, in the 17-limit 144 falls behind to 144g. The 144g val tempers out 170/169, 289/288, 375/374, 561/560, 595/594. It supports semihemisecordite, the 62 & 82f temperament. The patent val tempers out 273/272, 715/714, 833/832, 875/867, 891/884, and 1275/1274. It supports hemisecordite, the 41 & 103 temperament, though 103edo is better suited for this purpose. In the 19-limit the patent val tempers out 210/209, 325/323, 343/342, 363/361, 400/399, 513/512, and 665/663.

Besides all these, the 144eff val supports hemimiracle, the 41 & 103e temperament. 144ee supports oracle, the 31 & 113e temperament. 144cf supports necromanteion, the 31 & 113cf temperament.

Prime harmonics

Approximation of odd harmonics in 144edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.96 -2.98 -2.16 -3.91 -1.32 +1.14 +3.40 +3.38 +2.49 -4.11 -3.27
Relative (%) -23.5 -35.8 -25.9 -46.9 -15.8 +13.7 +40.8 +40.5 +29.8 -49.4 -39.3
Steps
(reduced) 		228
(84) 		334
(46) 		404
(116) 		456
(24) 		498
(66) 		533
(101) 		563
(131) 		589
(13) 		612
(36) 		632
(56) 		651
(75)

Subsets and supersets

144edo is the square of world-dominant 12edo. One step of it is called farab.

Since 144 factors into 24 × 32, 144edo has subset edos 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72.

Approximation to φ

144edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci edo, it can produce extremely precise approximation of the logarithmic golden ratio at 89 steps. In addition, it also excellently represents the acoustic golden ratio by 100 steps.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5.7.11.13 225/224, 243/242, 385/384, 847/845, 1875/1859 [⟨144 228 334 404 498 533]] +0.560 0.595 7.13
2.3.5.7.11.13.17 170/169, 225/224, 243/242, 289/288, 375/374, 385/384 [⟨144 228 334 404 498 533 588]] (144g) +0.653 0.596 7.15
2.3.5.7.11.13.17 225/224, 243/242, 273/272, 325/324, 847/845, 875/867 [⟨144 228 334 404 498 533 589]] (144) +0.362 0.734 8.80

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperament
1 7\144 58.33 27/26 Hemisecordite (144)
2 7\144 58.33 27/26 Semihemisecordite (144g)

Music

Amano Hideya
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