1553edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
1553edo is only [[consistent]] to the 5-odd-limit and harmonic 3 is about halfway between its steps. It has a reasonable approximation of the 2.9.5.7.13 [[subgroup]], where it notably tempers out [[4096/4095]] and 140625/140608. | 1553edo is only [[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps. It has a reasonable approximation of the 2.9.5.7.13 [[subgroup]], where it notably tempers out [[4096/4095]] and 140625/140608. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 10: | Line 11: | ||
1553edo is the 245th [[prime edo]]. 3106edo, which doubles it, provides a good correction to the harmonic 3. | 1553edo is the 245th [[prime edo]]. 3106edo, which doubles it, provides a good correction to the harmonic 3. | ||
==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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 23: | Line 25: | ||
| 2.9 | | 2.9 | ||
| {{monzo| 4923 -1553 }} | | {{monzo| 4923 -1553 }} | ||
| {{ | | {{mapping| 1553 4923 }} | ||
| | | −0.0130 | ||
| 0.0130 | | 0.0130 | ||
| 1.68 | | 1.68 | ||
| Line 30: | Line 32: | ||
| 2.9.5 | | 2.9.5 | ||
| {{monzo| 93 -33 5 }}, {{monzo| -36 -26 51 }} | | {{monzo| 93 -33 5 }}, {{monzo| -36 -26 51 }} | ||
| {{ | | {{mapping| 1553 4923 3606 }} | ||
| | | −0.0137 | ||
| 0.0106 | | 0.0106 | ||
| 1.38 | | 1.38 | ||
| Line 37: | Line 39: | ||
| 2.9.5.7 | | 2.9.5.7 | ||
| {{monzo| -5 5 5 -8 }}, {{monzo| 2 -10 14 -1 }}, {{monzo| 37 1 -4 -11 }} | | {{monzo| -5 5 5 -8 }}, {{monzo| 2 -10 14 -1 }}, {{monzo| 37 1 -4 -11 }} | ||
| {{ | | {{mapping| 1553 4923 3606 4360 }} | ||
| | | −0.0225 | ||
| 0.0178 | | 0.0178 | ||
| 2.31 | | 2.31 | ||
| Line 44: | Line 46: | ||
| 2.9.5.7.13 | | 2.9.5.7.13 | ||
| 4096/4095, 140625/140608, 28829034/28824005, {{monzo| 4 10 -9 0 -4 }} | | 4096/4095, 140625/140608, 28829034/28824005, {{monzo| 4 10 -9 0 -4 }} | ||
| {{ | | {{mapping| 1553 4923 3606 4360 5372 }} | ||
| | | −0.0271 | ||
| 0.0184 | | 0.0184 | ||
| 2.38 | | 2.38 | ||
|} | |} | ||
==Music== | |||
* [https://www.youtube.com/watch?v=gdxwRJSLyvw Stumbling Over Mystery] | == Music == | ||
; [[Francium]] | |||
* [https://www.youtube.com/watch?v=gdxwRJSLyvw ''Stumbling Over Mystery''] (2023) | |||
[[Category:Listen]] | |||
Latest revision as of 13:11, 21 February 2025
| ← 1552edo | 1553edo | 1554edo → |
1553 equal divisions of the octave (abbreviated 1553edo or 1553ed2), also called 1553-tone equal temperament (1553tet) or 1553 equal temperament (1553et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1553 equal parts of about 0.773 ¢ each. Each step represents a frequency ratio of 21/1553, or the 1553rd root of 2.
Theory
1553edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. It has a reasonable approximation of the 2.9.5.7.13 subgroup, where it notably tempers out 4096/4095 and 140625/140608.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.345 | +0.035 | +0.137 | +0.082 | -0.384 | +0.168 | -0.310 | +0.132 | -0.024 | -0.208 | -0.071 |
| Relative (%) | -44.7 | +4.6 | +17.8 | +10.6 | -49.7 | +21.7 | -40.1 | +17.0 | -3.1 | -26.9 | -9.2 | |
| Steps (reduced) |
2461 (908) |
3606 (500) |
4360 (1254) |
4923 (264) |
5372 (713) |
5747 (1088) |
6067 (1408) |
6348 (136) |
6597 (385) |
6821 (609) |
7025 (813) | |
Subsets and supersets
1553edo is the 245th prime edo. 3106edo, which doubles it, provides a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [4923 -1553⟩ | [⟨1553 4923]] | −0.0130 | 0.0130 | 1.68 |
| 2.9.5 | [93 -33 5⟩, [-36 -26 51⟩ | [⟨1553 4923 3606]] | −0.0137 | 0.0106 | 1.38 |
| 2.9.5.7 | [-5 5 5 -8⟩, [2 -10 14 -1⟩, [37 1 -4 -11⟩ | [⟨1553 4923 3606 4360]] | −0.0225 | 0.0178 | 2.31 |
| 2.9.5.7.13 | 4096/4095, 140625/140608, 28829034/28824005, [4 10 -9 0 -4⟩ | [⟨1553 4923 3606 4360 5372]] | −0.0271 | 0.0184 | 2.38 |
Music
- Stumbling Over Mystery (2023)