524edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
524edo is only [[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps | 524edo is only [[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps. Otherwise it has a good approximation to harmonics [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], [[15/1|15]], [[17/1|17]], and [[19/1|19]], making it suitable for a 2.9.15.7.11.13.17.19 [[subgroup]] interpretation. The 2.9.7.13.19 subgroup is particularly good, where it [[support]]s [[Eliora]]'s [[ostara]], the 93 & 524 temperament. The generator 293\524 represents the [[28/19]] interval in the 2.7.19 subgroup and it serves as ostara's generator in the no-threes 19-limit. Being around 671 cents, can also be used to as a generator for [[mavila]] or [[pelog]]. | ||
In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655. | In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655. | ||
| Line 9: | Line 9: | ||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|524}} | {{Harmonics in equal|524}} | ||
=== Subsets and supersets === | |||
Since 524 factors into {{factorization|524}}, 524edo has subset edos {{EDOs| 2, 4, 131, and 262 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
Based on treating 524edo as a no-threes system: | Based on treating 524edo as a no-threes system: | ||
{| 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|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
| Line 25: | Line 30: | ||
| {{monzo| 1217 -524 }} | | {{monzo| 1217 -524 }} | ||
| {{mapping| 524 1217 }} | | {{mapping| 524 1217 }} | ||
| | | −0.152 | ||
| 0.153 | | 0.153 | ||
| 6.67 | | 6.67 | ||
| Line 32: | Line 37: | ||
| {{monzo| 33 -13 -1 }}, {{monzo| -4 -43 37 }} | | {{monzo| 33 -13 -1 }}, {{monzo| -4 -43 37 }} | ||
| {{mapping| 524 1217 1471 }} | | {{mapping| 524 1217 1471 }} | ||
| | | −0.087 | ||
| 0.155 | | 0.155 | ||
| 6.79 | | 6.79 | ||
| Line 39: | Line 44: | ||
| 1835008/1830125, {{monzo| 3 7 3 -8 }}, {{monzo| -13 -5 10 -1 }} | | 1835008/1830125, {{monzo| 3 7 3 -8 }}, {{monzo| -13 -5 10 -1 }} | ||
| {{mapping| 524 1217 1471 1813 }} | | {{mapping| 524 1217 1471 1813 }} | ||
| | | −0.108 | ||
| 0.139 | | 0.139 | ||
| 6.09 | | 6.09 | ||
| Line 46: | Line 51: | ||
| 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488 | | 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488 | ||
| {{mapping| 524 1217 1471 1813 1939 }} | | {{mapping| 524 1217 1471 1813 1939 }} | ||
| | | −0.082 | ||
| 0.135 | | 0.135 | ||
| 5.88 | | 5.88 | ||
| Line 53: | Line 58: | ||
| 1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610 | | 1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610 | ||
| {{mapping| 524 1217 1471 1813 1939 2142 }} | | {{mapping| 524 1217 1471 1813 1939 2142 }} | ||
| | | −0.084 | ||
| 0.122 | | 0.122 | ||
| 5.37 | | 5.37 | ||
| Line 59: | Line 64: | ||
== Scales == | == Scales == | ||
* Ostara[7]: 62 62 62 107 62 62 107 | * Ostara[7]: 62 62 62 107 62 62 107 – [[2L 5s]] | ||
Latest revision as of 22:59, 20 February 2025
| ← 523edo | 524edo | 525edo → |
524 equal divisions of the octave (abbreviated 524edo or 524ed2), also called 524-tone equal temperament (524tet) or 524 equal temperament (524et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 524 equal parts of about 2.29 ¢ each. Each step represents a frequency ratio of 21/524, or the 524th root of 2.
Theory
524edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise it has a good approximation to harmonics 7, 9, 11, 13, 15, 17, and 19, making it suitable for a 2.9.15.7.11.13.17.19 subgroup interpretation. The 2.9.7.13.19 subgroup is particularly good, where it supports Eliora's ostara, the 93 & 524 temperament. The generator 293\524 represents the 28/19 interval in the 2.7.19 subgroup and it serves as ostara's generator in the no-threes 19-limit. Being around 671 cents, can also be used to as a generator for mavila or pelog.
In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.10 | +0.71 | -0.12 | -0.09 | +0.59 | -0.07 | -0.48 | +0.39 | +0.20 | +0.97 | -0.79 |
| Relative (%) | +48.0 | +31.0 | -5.4 | -4.1 | +25.8 | -3.0 | -21.1 | +16.9 | +8.6 | +42.6 | -34.6 | |
| Steps (reduced) |
831 (307) |
1217 (169) |
1471 (423) |
1661 (89) |
1813 (241) |
1939 (367) |
2047 (475) |
2142 (46) |
2226 (130) |
2302 (206) |
2370 (274) | |
Subsets and supersets
Since 524 factors into 22 × 131, 524edo has subset edos 2, 4, 131, and 262.
Regular temperament properties
Based on treating 524edo as a no-threes system:
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.5 | [1217 -524⟩ | [⟨524 1217]] | −0.152 | 0.153 | 6.67 |
| 2.5.7 | [33 -13 -1⟩, [-4 -43 37⟩ | [⟨524 1217 1471]] | −0.087 | 0.155 | 6.79 |
| 2.5.7.11 | 1835008/1830125, [3 7 3 -8⟩, [-13 -5 10 -1⟩ | [⟨524 1217 1471 1813]] | −0.108 | 0.139 | 6.09 |
| 2.5.7.11.13 | 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488 | [⟨524 1217 1471 1813 1939]] | −0.082 | 0.135 | 5.88 |
| 2.5.7.11.13.17 | 1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610 | [⟨524 1217 1471 1813 1939 2142]] | −0.084 | 0.122 | 5.37 |
Scales
- Ostara[7]: 62 62 62 107 62 62 107 – 2L 5s