13edt: Difference between revisions
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→Intervals: fix table formatting, set column alignment |
Added table of approximation of prime intervals |
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[[File:13edt.png|alt=13edt.png|13edt.png]] | [[File:13edt.png|alt=13edt.png|13edt.png]] | ||
== Theory == | |||
{| class="wikitable center-all" | |||
|+Approximation of prime intervals in 13-EDT | |||
! colspan="2" |Prime interval | |||
! 2 | |||
! 3 | |||
! 5 | |||
! 7 | |||
! 11 | |||
! 13 | |||
! 17 | |||
! 19 | |||
|- | |||
! rowspan="2" |Error | |||
! absolute ([[cent|¢]]) | |||
| -29.6 | |||
| 0.0 | |||
| -6.5 | |||
| -3.8 | |||
| -54.8 | |||
| -51.4 | |||
| +69.4 | |||
| +23.1 | |||
|- | |||
! [[Relative error|relative]] (%) | |||
| -20 | |||
| 0 | |||
| -4 | |||
| -3 | |||
| -37 | |||
| -35 | |||
| +47 | |||
| +16 | |||
|- | |||
! colspan="2" |[[Patent val]] | |||
| 8 | |||
| 13 | |||
| 19 | |||
| 23 | |||
| 28 | |||
| 30 | |||
| 34 | |||
| 35 | |||
|- | |||
! colspan="2" |[[Fifthspan]] | |||
| -1 | |||
| 0 | |||
| -4 | |||
| +2 | |||
| +3 | |||
| +6 | |||
| -1 | |||
| -6 | |||
|} | |||
* [[Intervals of BP]] | |||
* [[Relationship between Bohlen-Pierce and octave-ful temperaments]] | |||
==Intervals== | ==Intervals== | ||
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| | | | ||
|} | |} | ||
==JI approximation== | |||
==See also== | ==See also== | ||
Revision as of 15:36, 15 July 2021
13edt divides the tritave (3/1) into 13 equal parts of 146.304 cents each, corresponding to 8.202edo. It is the equal-tempered version of the well-known Bohlen-Pierce scale. In the 7-limit, it tempers out 245/243 and 3125/3087, the same commas as bohpier temperament. It is less impressive in higher p-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.
Below is a plot of the no-twos Z-function, in terms of which 13edt is the fourth no-twos zeta peak edt.
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... |
|---|---|---|---|---|---|---|
| 1 | 146.3 | 100 | A1/m2 | 27/25~49/45 | ||
| 2 | 292.6 | 200 | M2/d3 | 25/21 | Sirius | |
| 3 | 438.9 | 300 | A2/P3/d4 | 9/7 | Linear BP | |
| 4 | 585.2 | 400 | A3/m4/d5 | 7/5 | Canopus | |
| 5 | 731.5 | 500 | M4/m5 | 75/49 | False 3/2 | false Father |
| 6 | 877.8 | 600 | A4/M5 | 5/3 | Arcturus | |
| 7 | 1024.1 | 700 | A5/m6/d7 | 9/5 | Arcturus | |
| 8 | 1170.4 | 800 | M6/m7 | 49/25 | False 2/1 | false Father |
| 9 | 1316.7 | 900 | A6/M7/d8 | 15/7 | Canopus | |
| 10 | 1463.0 | 1000 | P8/d9 | 7/3 | Linear BP | |
| 11 | 1609.3 | 1100 | A8/m9 | 63/25 | Sirius | |
| 12 | 1755.7 | 1200 | M9/d10 | 25/9~135/49 | ||
| 13 | 1902.0 | 1300 | A9/P10 | 3/1 | Tritave |