Prime factorization
|
13 (prime)
|
Step size
|
146.304¢
|
Octave
|
8\13edt (1170.43¢)
|
Consistency limit
|
7
|
Distinct consistency limit
|
4
|
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
Regular temperament properties
- Main article: Bohlen-Pierce #Regular temperament properties
See also