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13edt divides the tritave (3/1) into 13 equal parts of 146.304 cents each, corresponding to 8.202[[edo]]. 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 [[Sensamagic_clan#Bohpier|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.
{{Infobox ET}}
{{ED intro}} It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.
[[File:13edt.png|thumb|alt=13edt.png|A plot of the [[Riemann zeta function#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
[[File:pts-3-5-7-e3-twtop.jpg|thumb|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]


Below is a plot of the [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak edt.
== Theory ==
{{See also| Relationship between Bohlen-Pierce and octave-ful temperaments }}


[[File:13edt.png|alt=13edt.png|13edt.png]]
13edt provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as the rank-2 [[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.


==Intervals==
13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].


{| class="wikitable center-1 right-2 right-3"
In the 3.5.7 subgroup, 13edt maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].
 
=== Odd harmonics ===
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=11}}
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 13edt (continued)}}
 
== Intervals ==
{{Main|Intervals of BP}}
 
{| class="wikitable center-all right-2 right-3"
|-
|-
! Steps
! Steps
! [[Cent]]s
! [[Cent]]s
! [[Hekt]]s
! [[Hekt]]s
! BP nonatonic degree
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! Corresponding JI intervals
! Corresponding<br />3.5.7 subgroup<br />intervals
! Comments
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs, {{nowrap|E {{=}} 1/1}})
! Generator for...
|-
| 0
| 0
| 0
| P1
| 1/1
| E
|-
|-
| 1
| 1
Line 21: Line 40:
| 100
| 100
| A1/m2
| A1/m2
| 27/25~49/45
| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
|
| F
|
|-
|-
| 2
| 2
Line 29: Line 47:
| 200
| 200
| M2/d3
| M2/d3
| 25/21
| [[25/21]] (−9.2{{c}})
|
| F#, Gb
| [[Sirius]]
|-
|-
| 3
| 3
Line 37: Line 54:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| 9/7
| [[9/7]] (+3.8{{c}})
|
| G
| [[Bohlen-Pierce|Linear BP]]
|-
|-
| 4
| 4
Line 45: Line 61:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| 7/5
| [[7/5]] (+2.7{{c}})
|
| H
| [[Canopus]]
|-
|-
| 5
| 5
Line 53: Line 68:
| 500
| 500
| M4/m5
| M4/m5
| 75/49
| [[75/49]] (−5.4{{c}})
| False 3/2
| H#, Jb
| false Father
|-
|-
| 6
| 6
Line 61: Line 75:
| 600
| 600
| A4/M5
| A4/M5
| 5/3
| [[5/3]] (−6.5{{c}})
|
| J
| [[Arcturus]]
|-
|-
| 7
| 7
Line 69: Line 82:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| 9/5
| [[9/5]] (+6.5{{c}})
|
| A
| Arcturus
|-
|-
| 8
| 8
Line 77: Line 89:
| 800
| 800
| M6/m7
| M6/m7
| 49/25
| [[49/25]] (+5.4{{c}})
| False 2/1
| A#, Bb
| false Father
|-
|-
| 9
| 9
Line 85: Line 96:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| 15/7
| [[15/7]] (−2.7{{c}})
|
| B
| Canopus
|-
|-
| 10
| 10
Line 93: Line 103:
| 1000
| 1000
| P8/d9
| P8/d9
| 7/3
| [[7/3]] (−3.8{{c}})
|
| C
| Linear BP
|-
|-
| 11
| 11
Line 101: Line 110:
| 1100
| 1100
| A8/m9
| A8/m9
| 63/25
| [[63/25]] (+9.2{{c}})
|
| C#, Db
| Sirius
|-
|-
| 12
| 12
Line 109: Line 117:
| 1200
| 1200
| M9/d10
| M9/d10
| 25/9~135/49
| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
|
| D
|
|-
|-
| 13
| 13
Line 117: Line 124:
| 1300
| 1300
| A9/P10
| A9/P10
| 3/1
| [[3/1]]
| Tritave
| E
|}
 
== JI approximation ==
[[File:13ed3-17-001.svg|alt=alt : Your browser has no SVG support.]]
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>Equave stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 3.5.7
| 245/243, 3125/3087
| [{{val| 13 19 23 }}] (b13)
| +1.393
| 1.150
| 0.79
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per tritave
! Generator<br />(reduced)
! Cents<br />(reduced)
! Associated<br />ratio
! Temperament
|-
| 1
| 1\13
| 146.30
| 49/45
| [[Procyon]]
|-
| 1
| 2\13
| 292.61
| 25/21
| [[Sirius]]
|-
| 1
| 3\13
| 438.91
| 9/7
| [[BPS]]
|-
| 1
| 4\13
| 585.22
| 7/5
| [[Canopus]]
|-
| 1
| 5\13
| 731.63
| 75/49
|
|
|-
| 1
| 6\13
| 877.83
| 5/3
| [[Arcturus]]
|}
|}


==See also==
== Music ==
*[[Catalog of 3.5.7 subgroup rank two temperaments]]
See also: [[Bohlen–Pierce scale/Music]]
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/dKFwJsxE3VU ''13ed3''] (2026)
 
== See also ==
* [[Bohlen-p_et]]
* [[Catalog of 3.5.7 subgroup rank two temperaments]]
* [[No-twos subgroup temperaments#3.5.7 subgroup temperaments]]
* [[19ed5|19ED5]]: relative ED5
* [[23ed7|23ED7]]: relative ED7


[[Category:3/1]]
[[Category:Tritave]]
[[Category:edt]]
[[Category:Macrotonal]]
[[Category:tritave]]
[[Category:Nonoctave]]
[[category:macrotonal]]
[[Category:Bohlen–Pierce]]
[[category:nonoctave]]

Latest revision as of 06:07, 12 October 2026

← 12edt 13edt 14edt →
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, perfect twelfth, or 3rd harmonic (abbreviated 13edt or 13ed3), is a nonoctave tuning system that divides the interval of 3/1 into 13 equal parts of about 146 ¢ each. Each step represents a frequency ratio of 31/13, or the 13th root of 3. It is best known as the equal-tempered version of the Bohlen–Pierce scale, and therefore has received by far the most attention among equal divisions of the tritave.

13edt.png
A plot of the no-twos Z-function, in terms of which 13edt is the fourth no-twos zeta peak EDT.
Projective tuning space of the 3.5.7 subgroup, showing 13edt near the center.

Theory

See also: Relationship between Bohlen-Pierce and octave-ful temperaments

13edt provides an excellent approximation to the 3.5.7 subgroup, especially for its size, being comparable to 34edo's accuracy in the 5-limit. In this subgroup, it tempers out 245/243 and 3125/3087, the same commas as the rank-2 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.

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 3.5.7 subgroup, 13edt maintains the smallest relative error of any EDT until 258edt and 271edt, and the smallest absolute error until 56edt.

Odd harmonics

Approximation of odd harmonics in 13edt
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.00 -6.53 -3.83 +0.00 -54.80 -51.40 -6.53 +69.39 +23.14 -3.83 -15.02
Relative (%) +0.0 -4.5 -2.6 +0.0 -37.5 -35.1 -4.5 +47.4 +15.8 -2.6 -10.3
Steps
(reduced) 		13
(0) 		19
(6) 		23
(10) 		26
(0) 		28
(2) 		30
(4) 		32
(6) 		34
(8) 		35
(9) 		36
(10) 		37
(11)
Approximation of odd harmonics in 13edt (continued)
Harmonic 25 27 29 31 33 35 37 39 41 43 45 47
Error Absolute (¢) -13.07 +0.00 +22.59 +53.44 -54.80 -10.36 +39.74 -51.40 +8.32 +72.17 -6.53 +64.49
Relative (%) -8.9 +0.0 +15.4 +36.5 -37.5 -7.1 +27.2 -35.1 +5.7 +49.3 -4.5 +44.1
Steps
(reduced) 		38
(12) 		39
(0) 		40
(1) 		41
(2) 		41
(2) 		42
(3) 		43
(4) 		43
(4) 		44
(5) 		45
(6) 		45
(6) 		46
(7)

Intervals

Main article: Intervals of BP
Steps Cents Hekts Enneatonic
degree 		Corresponding
3.5.7 subgroup
intervals 		Lambda
(sLsLsLsLs, E = 1/1)
0 0 0 P1 1/1 E
1 146.3 100 A1/m2 49/45 (−1.1 ¢); 27/25 (+13.1 ¢) F
2 292.6 200 M2/d3 25/21 (−9.2 ¢) F#, Gb
3 438.9 300 A2/P3/d4 9/7 (+3.8 ¢) G
4 585.2 400 A3/m4/d5 7/5 (+2.7 ¢) H
5 731.5 500 M4/m5 75/49 (−5.4 ¢) H#, Jb
6 877.8 600 A4/M5 5/3 (−6.5 ¢) J
7 1024.1 700 A5/m6/d7 9/5 (+6.5 ¢) A
8 1170.4 800 M6/m7 49/25 (+5.4 ¢) A#, Bb
9 1316.7 900 A6/M7/d8 15/7 (−2.7 ¢) B
10 1463.0 1000 P8/d9 7/3 (−3.8 ¢) C
11 1609.3 1100 A8/m9 63/25 (+9.2 ¢) C#, Db
12 1755.7 1200 M9/d10 135/49 (+1.1 ¢); 25/9 (−13.1 ¢) D
13 1902.0 1300 A9/P10 3/1 E

JI approximation

alt : Your browser has no SVG support.

Regular temperament properties

Subgroup Comma list Mapping Optimal
Equave stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
3.5.7 245/243, 3125/3087 [⟨13 19 23]] (b13) +1.393 1.150 0.79

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per tritave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperament
1 1\13 146.30 49/45 Procyon
1 2\13 292.61 25/21 Sirius
1 3\13 438.91 9/7 BPS
1 4\13 585.22 7/5 Canopus
1 5\13 731.63 75/49
1 6\13 877.83 5/3 Arcturus

Music

See also: Bohlen–Pierce scale/Music

Bryan Deister

See also

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