475edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! 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 | | 2.3 | ||
| {{monzo| 753 -475 }} | | {{monzo| 753 -475 }} | ||
| {{mapping| 475 753 }} | | {{mapping| 475 753 }} | ||
| | | −0.1138 | ||
| 0.1138 | | 0.1138 | ||
| 4.50 | | 4.50 | ||
| Line 26: | Line 35: | ||
| {{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 | ||
| 0.0935 | | 0.0935 | ||
| 3.70 | | 3.70 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 51: | Line 67: | ||
| 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 | |||
Revision as of 13:13, 16 November 2024
| ← 474edo | 475edo | 476edo → |
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 |
| 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