475edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|475}}
{{ED intro}}


== Theory ==
== Theory ==
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=== Subsets and supersets ===
=== Subsets and supersets ===
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.
Since 475 factors into {{factorisation|475}}, 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 ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 753 -475 }}
| {{Monzo| 753 -475 }}
| {{mapping| 475 753 }}
| {{Mapping| 475 753 }}
| &minus;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 }}
| &minus;0.1064
| −0.1064
| 0.0935
| 0.0935
| 3.70
| 3.70
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 98304/78125
| 98304/78125
| [[Squarschmidt]]
| [[Squarschmidt]]
|-
| 1
| 233\475
| 588.63
| 351/250
| [[Ragitritonic]] (475d) / [[garitritonic]] (475e)
|-
|-
| 5
| 5
| 329\475<br />(44\475)
| 329\475<br>(44\475)
| 831.16<br />(111.16)
| 831.16<br>(111.16)
| 160/99<br />(16/15)
| 160/99<br>(16/15)
| [[Quintosec]]
| [[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]] (475d)
| [[Enneadecal]] (475d)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Latest revision as of 12:47, 20 May 2026

← 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

475 equal divisions of the octave (abbreviated 475edo or 475ed2), also called 475-tone equal temperament (475tet) or 475 equal temperament (475et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 475 equal parts of about 2.53 ¢ each. Each step represents a frequency ratio of 21/475, or the 475th root of 2.

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
1 233\475 588.63 351/250 Ragitritonic (475d) / garitritonic (475e)
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 distinct

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