13edt: Difference between revisions
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! Comments | ! Comments | ||
! Generator for... | ! Generator for... | ||
! [[Arcturus]] enneatonic notation | |||
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| H# | |||
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| [[Sirius]] | | [[Sirius]] | ||
| Ib | |||
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| A2/P3/d4 | | A2/P3/d4 | ||
| 9/7 | | 9/7 | ||
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| [[Bohlen-Pierce|Linear BP]] | | [[Bohlen-Pierce|Linear BP]] | ||
| I | |||
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| 4 | | 4 | ||
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| [[Canopus]] | | [[Canopus]] | ||
| I#, Jb | |||
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| 5 | | 5 | ||
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| false 3/2 | | false 3/2 | ||
| false Father | | false Father | ||
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| [[Arcturus]] | | [[Arcturus]] | ||
| K | |||
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| Arcturus | | Arcturus | ||
| L | |||
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| false Father | | false Father | ||
| L#, Mb | |||
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| Canopus | | Canopus | ||
| M | |||
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| Tritave | | Tritave | ||
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Revision as of 04:37, 18 February 2023
| ← 12edt | 13edt | 14edt → |
13 equal divisions of the tritave (13edt) is the nonoctave tuning system derived by dividing the tritave (3/1) into 13 equal steps of 146.3 cents each, or the thirteenth root of 3. It is best known as the equal-tempered version of the Bohlen-Pierce scale.
13edt can be described as approximately 8.202edo. This implies that each step of 13edt can be approximated by 5 steps of 41edo.
In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as bohpier temperament. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 (26edt, 39edt and 52edt) come to the fore.
Theory
| Prime interval | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -29.6 | 0.0 | -6.5 | -3.8 | -54.8 | -51.4 | +69.4 | +23.1 |
| relative (%) | -20 | 0 | -4 | -3 | -37 | -35 | +47 | +16 | |
| Patent val | 8 | 13 | 19 | 23 | 28 | 30 | 34 | 35 | |
| Fifthspan | -1 | 0 | -4 | +2 | +3 | +6 | -1 | -6 | |
Intervals
| Steps | Cents | Hekts | BP nonatonic degree | Corresponding JI intervals | Comments | Generator for... | Arcturus enneatonic notation |
|---|---|---|---|---|---|---|---|
| 1 | 146.3 | 100 | A1/m2 | 27/25~49/45 | H# | ||
| 2 | 292.6 | 200 | M2/d3 | 25/21 | Sirius | Ib | |
| 3 | 438.9 | 300 | A2/P3/d4 | 9/7 | Linear BP | I | |
| 4 | 585.2 | 400 | A3/m4/d5 | 7/5 | Canopus | I#, Jb | |
| 5 | 731.5 | 500 | M4/m5 | 75/49 | false 3/2 | false Father | J |
| 6 | 877.8 | 600 | A4/M5 | 5/3 | Arcturus | K | |
| 7 | 1024.1 | 700 | A5/m6/d7 | 9/5 | Arcturus | L | |
| 8 | 1170.4 | 800 | M6/m7 | 49/25 | false 2/1 | false Father | L#, Mb |
| 9 | 1316.7 | 900 | A6/M7/d8 | 15/7 | Canopus | M | |
| 10 | 1463.0 | 1000 | P8/d9 | 7/3 | Linear BP | N | |
| 11 | 1609.3 | 1100 | A8/m9 | 63/25 | Sirius | O | |
| 12 | 1755.7 | 1200 | M9/d10 | 25/9~135/49 | P | ||
| 13 | 1902.0 | 1300 | A9/P10 | 3/1 | Tritave | H |
JI approximation
Regular temperament properties
See also
- Bohlen-p_et
- Catalog of 3.5.7 subgroup rank two temperaments
- 19ED5: relative ED5
- 23ED7: relative ED7