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..." |
→Regular temperament properties: + ragitritonic/garitritonic |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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 === | ||
| Line 9: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
475 factors into | 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 == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{ | ! rowspan="2" | [[Comma list]] | ||
|{{ | ! rowspan="2" | [[Mapping]] | ||
| | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{Monzo| 753 -475 }} | |||
| {{Mapping| 475 753 }} | |||
| −0.1138 | |||
| 0.1138 | | 0.1138 | ||
| 4.50 | | 4.50 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{ | | {{Monzo| -14 -19 19 }}, {{monzo| 47 -15 -10 }} | ||
|{{ | | {{Mapping| 475 753 1103 }} | ||
| | | −0.1064 | ||
| 0.0935 | | 0.0935 | ||
| 3.70 | | 3.70 | ||
| Line 39: | Line 42: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+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> | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|157\475 | | 157\475 | ||
|396.63 | | 396.63 | ||
|98304/78125 | | 98304/78125 | ||
|[[Squarschmidt]] | | [[Squarschmidt]] | ||
|- | |- | ||
| | | 1 | ||
| | | 233\475 | ||
| | | 588.63 | ||
| | | 351/250 | ||
| [[Ragitritonic]] (475d) / [[garitritonic]] (475e) | |||
|- | |- | ||
|19 | | 5 | ||
|197\475<br>(3\475) | | 329\475<br>(44\475) | ||
|497.68<br>(7.58) | | 831.16<br>(111.16) | ||
|4/3<br>(225/224) | | 160/99<br>(16/15) | ||
|[[Enneadecal]] | | [[Quintosec]] | ||
|- | |||
| 19 | |||
| 197\475<br>(3\475) | |||
| 497.68<br>(7.58) | |||
| 4/3<br>(225/224) | |||
| [[Enneadecal]] (475d) | |||
|} | |} | ||
<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 | |||
<nowiki>* | |||
Latest revision as of 12:47, 20 May 2026
| ← 474edo | 475edo | 476edo → |
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
| 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
| 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