13edt: Difference between revisions

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Changing H = 1/1 to J = 1/1 as it seems to be more common
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! Comments
! Comments
! Generator for...
! Generator for...
! [[Arcturus]] nonatonic notation
! [[Arcturus]] nonatonic notation (J = 1/1)
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| H#
| J#
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| 2
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| [[Sirius]]
| [[Sirius]]
| Ib
| Kb
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| 3
| 3
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|  
|  
| [[Bohlen-Pierce|Linear BP]]
| [[Bohlen-Pierce|Linear BP]]
| I
| K
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| [[Canopus]]
| [[Canopus]]
| I#, Jb
| K#, Lb
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| 5
| 5
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| false 3/2
| false 3/2
| false Father
| false Father
| J
| L
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| [[Arcturus]]
| [[Arcturus]]
| K
| M
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| Arcturus
| Arcturus
| L
| N
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| 8
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| false 2/1
| false 2/1
| false Father
| false Father
| L#, Mb
| N#, Ob
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| Canopus
| Canopus
| M
| O
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| Linear BP
| Linear BP
| N
| P
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| Sirius
| Sirius
| O
| Q
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| Tritave
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| H
| J
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Revision as of 03:30, 1 March 2023

← 12edt 13edt 14edt →
Prime factorization 13 (prime)
Step size 146.304 ¢ 
Octave 8\13edt (1170.43 ¢)
Consistency limit 7
Distinct consistency limit 4
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... Arcturus nonatonic notation (J = 1/1)
1 146.3 100 A1/m2 27/25~49/45 J#
2 292.6 200 M2/d3 25/21 Sirius Kb
3 438.9 300 A2/P3/d4 9/7 Linear BP K
4 585.2 400 A3/m4/d5 7/5 Canopus K#, Lb
5 731.5 500 M4/m5 75/49 false 3/2 false Father L
6 877.8 600 A4/M5 5/3 Arcturus M
7 1024.1 700 A5/m6/d7 9/5 Arcturus N
8 1170.4 800 M6/m7 49/25 false 2/1 false Father N#, Ob
9 1316.7 900 A6/M7/d8 15/7 Canopus O
10 1463.0 1000 P8/d9 7/3 Linear BP P
11 1609.3 1100 A8/m9 63/25 Sirius Q
12 1755.7 1200 M9/d10 25/9~135/49 R
13 1902.0 1300 A9/P10 3/1 Tritave J

JI approximation

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Regular temperament properties

Main article: Bohlen-Pierce § Regular temperament properties

See also

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