13edt

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13edt.png
A plot of the no-twos Z-function, in terms of which 13edt is the fourth no-twos zeta peak EDT.

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

Approximation of prime intervals in 13-EDT
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

Main article: Intervals of BP
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

JI approximation

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Regular temperament properties

Main article: Bohlen-Pierce #Regular temperament properties

See also