475edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|475}} == Theory == 475et is only consistent to the 5-odd-limit. It can be considered for the 2.3.5.11.13.19.23 subgroup, tempering out 23..."
 
Review
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== Theory ==
== Theory ==
475et is only consistent to the [[5-odd-limit]]. It can be considered for the 2.3.5.11.13.19.23 [[subgroup]], tempering out [[2376/2375]], 11132/11115, [[3250/3249]], 42757/42750, 11979/11960 and 14300/14283. It [[support]]s [[will]], [[countritonic]] and [[cotoneum]].
475edo is only [[consistent]] to the [[5-odd-limit]]. The equal temperament [[tempering out|tempers out]] {{monzo| -14 -19 19 }} ([[enneadeca]]) and {{monzo| 47 -15 -10 }} (quintosec comma) in the 5-limit. In the 7-limit, the 475d [[val]] [[support]]s [[enneadecal]] and the [[patent val]] supports [[cotoneum]].
 
It can be considered for the 2.3.5.11.13.19.23 [[subgroup]], tempering out [[2376/2375]], [[3250/3249]], 11132/11115, 11979/11960, 14300/14283 and 42757/42750.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
475 factors into 5<sup>2</sup> × 19, with subset edos {{EDOs|5, 19, 25, and 95}}. [[950edo]], which doubles it, gives a good correction to the harmonic 7.
Since 475 factors into 5<sup>2</sup> × 19, 475edo has subset edos {{EDOs| 5, 19, 25, and 95 }}. [[950edo]], which doubles it, gives a good correction to the harmonic 7.


== 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]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|753 -475}}
| {{monzo| 753 -475 }}
|{{mapping|475 753}}
| {{mapping| 475 753 }}
| -0.1138
| -0.1138
| 0.1138
| 0.1138
| 4.50
| 4.50
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-14 -19 19}}, {{monzo|47 -15 -10}}
| {{monzo| -14 -19 19 }}, {{monzo| 47 -15 -10 }}
|{{mapping|475 753 1103}}
| {{mapping| 475 753 1103 }}
| -0.1064
| -0.1064
| 0.0935
| 0.0935
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! Temperaments
! Temperaments
|-
|-
|1
| 1
|157\475
| 157\475
|396.63
| 396.63
|98304/78125
| 98304/78125
|[[Squarschmidt]]
| [[Squarschmidt]]
|-
|-
|5
| 5
|329\475<br>(44\475)
| 329\475<br>(44\475)
|831.16<br>(111.16)
| 831.16<br>(111.16)
|80/49<br>(15/14)
| 160/99<br>(16/15)
|[[Qintosec]]
| [[Quintosec]]
|-
|-
|19
| 19
|197\475<br>(3\475)
| 197\475<br>(3\475)
|497.68<br>(7.58)
| 497.68<br>(7.58)
|4/3<br>(225/224)
| 4/3<br>(225/224)
|[[Enneadecal]]
| [[Enneadecal]] (475d)
|}
|}
<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 13:30, 19 February 2024

← 474edo 475edo 476edo →
Prime factorization 52 × 19
Step size 2.52632 ¢ 
Fifth 278\475 (702.316 ¢)
Semitones (A1:m2) 46:35 (116.2 ¢ : 88.42 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

475edo is only consistent to the 5-odd-limit. The equal temperament tempers out [-14 -19 19⟩ (enneadeca) and [47 -15 -10⟩ (quintosec comma) in the 5-limit. In the 7-limit, the 475d val supports enneadecal and the patent val supports cotoneum.

It can be considered for the 2.3.5.11.13.19.23 subgroup, tempering out 2376/2375, 3250/3249, 11132/11115, 11979/11960, 14300/14283 and 42757/42750.

Prime harmonics

Approximation of prime harmonics in 475edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.36 +0.21 -1.25 -0.58 +0.74 +1.15 +0.59 +0.78 +1.16 -0.61
Relative (%) +0.0 +14.3 +8.4 -49.4 -23.0 +29.1 +45.5 +23.4 +30.8 +45.9 -24.3
Steps
(reduced) 		475
(0) 		753
(278) 		1103
(153) 		1333
(383) 		1643
(218) 		1758
(333) 		1942
(42) 		2018
(118) 		2149
(249) 		2308
(408) 		2353
(453)

Subsets and supersets

Since 475 factors into 52 × 19, 475edo has subset edos 5, 19, 25, and 95. 950edo, which doubles it, gives a good correction to the harmonic 7.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [753 -475⟩ [⟨475 753]] -0.1138 0.1138 4.50
2.3.5 [-14 -19 19⟩, [47 -15 -10⟩ [⟨475 753 1103]] -0.1064 0.0935 3.70

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 157\475 396.63 98304/78125 Squarschmidt
5 329\475
(44\475) 		831.16
(111.16) 		160/99
(16/15) 		Quintosec
19 197\475
(3\475) 		497.68
(7.58) 		4/3
(225/224) 		Enneadecal (475d)

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

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