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Created page with "{{Infobox ET}} {{EDO intro|613}} == Theory == 613edo is only consistent to the 3-odd-limit and the harmonic 3 is about halfway its steps. It can be used i..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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== 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" |[[Comma list | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
|- | ! colspan="2" | Tuning error | ||
![[TE error|Absolute]] (¢) | |- | ||
![[TE simple badness|Relative]] (%) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.9 | | 2.9 | ||
| {{monzo|-1943 613}} | | {{monzo|-1943 613}} | ||
| {{mapping|613 1943}} | | {{mapping|613 1943}} | ||
| 0.0506 | | +0.0506 | ||
| 0.0506 | | 0.0506 | ||
| 2.58 | | 2.58 | ||
| Line 32: | Line 33: | ||
| 32805/32768, {{monzo|-97 -69 136}} | | 32805/32768, {{monzo|-97 -69 136}} | ||
| {{mapping|613 1943 1423}} | | {{mapping|613 1943 1423}} | ||
| 0.1299 | | +0.1299 | ||
| 0.1194 | | 0.1194 | ||
| 6.10 | | 6.10 | ||
| Line 39: | Line 40: | ||
| 32805/32768, 40500000/40353607, {{monzo|-23 -7 11 7}} | | 32805/32768, 40500000/40353607, {{monzo|-23 -7 11 7}} | ||
| {{mapping|613 1943 1423 1721}} | | {{mapping|613 1943 1423 1721}} | ||
| 0.0814 | | +0.0814 | ||
| 0.1331 | | 0.1331 | ||
| 6.80 | | 6.80 | ||
|} | |} | ||
== Music == | |||
; [[Francium]] | |||
* "Donut" from ''You Are A...'' (2024) – [https://open.spotify.com/track/4vYmtSdt7sU38eYIeUYFf1 Spotify] | [https://francium223.bandcamp.com/track/donut Bandcamp] | [https://www.youtube.com/watch?v=QhAI2F4taxU YouTube] – porcupinefish[15] in 613edo tuning | |||
Latest revision as of 13:19, 21 February 2025
| ← 612edo | 613edo | 614edo → |
613 equal divisions of the octave (abbreviated 613edo or 613ed2), also called 613-tone equal temperament (613tet) or 613 equal temperament (613et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 613 equal parts of about 1.96 ¢ each. Each step represents a frequency ratio of 21/613, or the 613th root of 2.
Theory
613edo is only consistent to the 3-odd-limit and the harmonic 3 is about halfway its steps. It can be used in the 2.9.5.7.13.19.23.29.31.37 subgroup, tempering out 1521/1520, 875/874, 1863/1862, 38475/38416, 2205/2204, 1520/1519, 186875/186624, 1665/1664 and 194560/194481.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.818 | -0.669 | +0.179 | -0.321 | +0.721 | -0.723 | +0.149 | +0.754 | +0.040 | -0.960 | +0.111 |
| Relative (%) | +41.8 | -34.2 | +9.1 | -16.4 | +36.8 | -37.0 | +7.6 | +38.5 | +2.0 | -49.1 | +5.7 | |
| Steps (reduced) |
972 (359) |
1423 (197) |
1721 (495) |
1943 (104) |
2121 (282) |
2268 (429) |
2395 (556) |
2506 (54) |
2604 (152) |
2692 (240) |
2773 (321) | |
Subsets and supersets
613edo is the 112th prime EDO. 1226edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1943 613⟩ | [⟨613 1943]] | +0.0506 | 0.0506 | 2.58 |
| 2.9.5 | 32805/32768, [-97 -69 136⟩ | [⟨613 1943 1423]] | +0.1299 | 0.1194 | 6.10 |
| 2.9.5.7 | 32805/32768, 40500000/40353607, [-23 -7 11 7⟩ | [⟨613 1943 1423 1721]] | +0.0814 | 0.1331 | 6.80 |