476edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|476}}
{{EDO intro|476}}
== Theory ==
== Theory ==
476edo is consistent to the [[7-odd-limit]] and the [[harmonic]] 3 is about halfway its steps. Using the patent val, it tempers out [[2401/2400]] in the 7-limit; [[4000/3993]], 12005/11979, 117649/117612, 1296000/1294139, [[540/539]], [[441/440]], 352947/352000, 24057/24010, [[8019/8000]], [[9801/9800]] and 160083/160000 in the 11-limit.
476edo is [[consistent]] to the [[7-odd-limit]], but the [[harmonic]] [[3/1|3]] is about halfway its steps, while its [[5/1|5]] and [[7/1|7]] are both tuned flat. To start with, consider the 2.3.5.7 [[patent val]], as well as 2.9.15.21 and 2.9.5.7 [[subgroup]]s.
 
Using the patent val, the equal temperament [[tempering out|tempers out]] [[2401/2400]] and [[19683/19600]] in the 7-limit, [[support]]ing [[harry]]. The 11-limit 476e val tempers out [[3025/3024]] and [[41503/41472]], whereas the patent val tempers out [[243/242]], [[441/440]], [[540/539]], [[4000/3993]], [[8019/8000]], and [[9801/9800]], supporting 11-limit harry.  


=== Odd harmonics ===
=== Odd harmonics ===
Line 8: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
476 factors into 2<sup>2</sup> × 7 × 17, with subset edos {{EDOs|2, 4, 7, 14, 17, 28, 34, 68, 119, and 238}}. [[952edo]], which doubles it, gives a good correction to the harmonic 3, but unfortunately it is unconsistent in the [[5-odd-limit]].
476 factors into 2<sup>2</sup> × 7 × 17, with subset edos {{EDOs| 2, 4, 7, 14, 17, 28, 34, 68, 119, and 238 }}. [[952edo]], which doubles it, gives a good correction to the harmonic 3, but unfortunately it is inconsistent in the [[5-odd-limit]].


== 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|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]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|1509 -476}}
| {{monzo| 1509 -476 }}
|{{mapping|476 1509}}
| {{mapping| 476 1509 }}
| -0.0460
| -0.0460
| 0.0460
| 0.0460
| 1.82
| 1.82
|-
|-
|2.9.5
| 2.9.5
|{{monzo|33 -17 9}}, {{monzo|-65 0 28}}
| {{monzo| 33 -17 9 }}, {{monzo| -65 0 28 }}
|{{mapping|476 1509 1105}}
| {{mapping| 476 1509 1105 }}
| +0.0554
| +0.0554
| 0.1482
| 0.1482
| 5.88
| 5.88
|-
|-
|2.9.5.7
| 2.9.5.7
|703125/702464, 4802000/4782969, 4202539929/4194304000
| 703125/702464, 4802000/4782969, {{monzo| 25 3 -3 8 }}
|{{mapping|476 1509 1105 1336}}
| {{mapping| 476 1509 1105 1336 }}
| +0.1091
| +0.1091
| 0.1586
| 0.1586
Line 47: Line 50:
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)*
! Generator*
! Cents<br>(reduced)*
! Cents*
! Associated<br>Ratio*
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|205\476
| 205\476
|516.81
| 516.81
|27/20
| 27/20
|[[Gravity]]
| [[Larry]] (476)
|-
|-
|2
| 2
|205\476<br>(33\476)
| 205\476<br>(33\476)
|516.81<br>(83.19)
| 516.81<br>(83.19)
|27/20<br>(21/20)
| 27/20<br>(21/20)
|[[Harry]]
| [[Harry]] (11-limit, 476)
|-
|-
|28
| 28
|197\476<br>(6\476)
| 197\476<br>(6\476)
|496.64<br>(15.13)
| 496.64<br>(15.13)
|4/3<br>(105/104)
| 4/3<br>(105/104)
|[[Oquatonic]]
| [[Oquatonic]] (5-limit)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 09:50, 1 January 2024

← 475edo 476edo 477edo →
Prime factorization 22 × 7 × 17
Step size 2.52101 ¢ 
Fifth 278\476 (700.84 ¢) (→ 139\238)
Semitones (A1:m2) 42:38 (105.9 ¢ : 95.8 ¢)
Dual sharp fifth 279\476 (703.361 ¢)
Dual flat fifth 278\476 (700.84 ¢) (→ 139\238)
Dual major 2nd 81\476 (204.202 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

476edo is consistent to the 7-odd-limit, but the harmonic 3 is about halfway its steps, while its 5 and 7 are both tuned flat. To start with, consider the 2.3.5.7 patent val, as well as 2.9.15.21 and 2.9.5.7 subgroups.

Using the patent val, the equal temperament tempers out 2401/2400 and 19683/19600 in the 7-limit, supporting harry. The 11-limit 476e val tempers out 3025/3024 and 41503/41472, whereas the patent val tempers out 243/242, 441/440, 540/539, 4000/3993, 8019/8000, and 9801/9800, supporting 11-limit harry.

Odd harmonics

Approximation of odd harmonics in 476edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.11 -0.60 -0.76 +0.29 +0.78 -1.03 +0.81 +0.93 -0.03 +0.65 -0.54
Relative (%) -44.2 -23.8 -30.1 +11.6 +31.1 -40.9 +32.0 +36.8 -1.3 +25.7 -21.5
Steps
(reduced) 		754
(278) 		1105
(153) 		1336
(384) 		1509
(81) 		1647
(219) 		1761
(333) 		1860
(432) 		1946
(42) 		2022
(118) 		2091
(187) 		2153
(249)

Subsets and supersets

476 factors into 22 × 7 × 17, with subset edos 2, 4, 7, 14, 17, 28, 34, 68, 119, and 238. 952edo, which doubles it, gives a good correction to the harmonic 3, but unfortunately it is inconsistent in the 5-odd-limit.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.9 [1509 -476⟩ [⟨476 1509]] -0.0460 0.0460 1.82
2.9.5 [33 -17 9⟩, [-65 0 28⟩ [⟨476 1509 1105]] +0.0554 0.1482 5.88
2.9.5.7 703125/702464, 4802000/4782969, [25 3 -3 8⟩ [⟨476 1509 1105 1336]] +0.1091 0.1586 6.29

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 205\476 516.81 27/20 Larry (476)
2 205\476
(33\476) 		516.81
(83.19) 		27/20
(21/20) 		Harry (11-limit, 476)
28 197\476
(6\476) 		496.64
(15.13) 		4/3
(105/104) 		Oquatonic (5-limit)

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

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