402653184edo: Difference between revisions
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{{Software tuning}} | |||
{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
'''402,653,184edo''' is an [[equal-step tuning|equal-step]] [[tuning system]] that allows for finer subdivisions within the standard 128-note [[MIDI]] scale based on [[12edo]], using 32-bit data. | |||
It could be applied through the 7.25 pitch attribute introduced in MIDI 2.0. | Since 128 notes equal 2<sup>7</sup>, this leaves 25 additional bits for further subdivision of the 12-tone equal temperament. By calculating 12 × 2<sup>25</sup>, we arrive at a system with 402,653,184 equally spaced divisions of the octave. | ||
It could be applied through the [https://amei.or.jp/midistandardcommittee/MIDI2.0/MIDI2.0-DOCS/M2-104-UM_v1-0_UMP_and_MIDI_2-0_Protocol_Specification.pdf#page=34 7.25 pitch attribute] introduced in MIDI 2.0. | |||
The step size is about 2.98 × 10<sup>-6</sup> cents, meaning that intervals rounded to this system have a precision of 1.49 × 10<sup>-6</sup> cents. At 20000 Hz, the phase precision is approximately 16 hours. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|402653184|columns=11|intervals=prime}} | |||
{{Harmonics in equal|402653184|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 402653184edo (continued)|intervals=prime}} | |||
=== Subsets and supersets === | |||
402653184 factorizes into {{factorization|402653184}}. | |||
[[Category:Equal divisions of the octave]] | |||
Latest revision as of 07:23, 15 September 2026
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This page presents a primarily software-related tuning.
This tuning is more notable for use in software rather than for its musical properties. |
| ← 402653183edo | 402653184edo | 402653185edo → |
402653184 equal divisions of the octave (abbreviated 402653184edo or 402653184ed2), also called 402653184-tone equal temperament (402653184tet) or 402653184 equal temperament (402653184et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 402653184 equal parts of about 2.98e-06 ¢ each. Each step represents a frequency ratio of 21/402653184, or the 402653184th root of 2.
Theory
402,653,184edo is an equal-step tuning system that allows for finer subdivisions within the standard 128-note MIDI scale based on 12edo, using 32-bit data.
Since 128 notes equal 27, this leaves 25 additional bits for further subdivision of the 12-tone equal temperament. By calculating 12 × 225, we arrive at a system with 402,653,184 equally spaced divisions of the octave.
It could be applied through the 7.25 pitch attribute introduced in MIDI 2.0.
The step size is about 2.98 × 10-6 cents, meaning that intervals rounded to this system have a precision of 1.49 × 10-6 cents. At 20000 Hz, the phase precision is approximately 16 hours.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000000 | -0.00000130 | -0.00000127 | +0.00000005 | -0.00000022 | +0.00000087 | +0.00000146 | +0.00000093 | +0.00000111 | -0.00000149 | +0.00000138 |
| Relative (%) | +0.0 | -43.6 | -42.5 | +1.5 | -7.5 | +29.0 | +48.9 | +31.1 | +37.2 | -50.0 | +46.2 | |
| Steps (reduced) |
402653184 (0) |
638190197 (235537013) |
934931740 (129625372) |
1130390398 (325084030) |
1392951156 (184991604) |
1489993835 (282034283) |
1645829928 (35217192) |
1710441539 (99828803) |
1821426625 (210813889) |
1956081515 (345468779) |
1994822919 (384210183) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000128 | -0.00000031 | +0.00000087 | +0.00000018 | +0.00000081 | -0.00000048 | -0.00000130 | +0.00000010 | -0.00000138 | -0.00000010 | -0.00000146 |
| Relative (%) | +43.0 | -10.5 | +29.2 | +6.1 | +27.1 | -16.1 | -43.7 | +3.4 | -46.3 | -3.4 | -49.1 | |
| Steps (reduced) |
2097602985 (84337065) |
2157235373 (143969453) |
2184902781 (171636861) |
2236572887 (223306967) |
2306365409 (293099489) |
2368664954 (355399034) |
2388030272 (374764352) |
2442530127 (26611023) |
2476215258 (60296154) |
2492352567 (76433463) |
2538237389 (122318285) | |
Subsets and supersets
402653184 factorizes into 227 × 3.