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[[File:13edt.png|thumb|alt=13edt.png|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]].]]
{{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.]]


'''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 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen-Pierce]] scale.
== 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 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].
13edt can be described as approximately 8.202[[edo]]. 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 [[Sensamagic_clan#Bohpier|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.
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]].


== Theory ==
=== Odd harmonics ===
{| class="wikitable center-all"
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=11}}
|+Approximation of prime intervals in 13-EDT
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 13edt (continued)}}
! 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
|}
 
* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]


== Intervals ==
== Intervals ==
{{Main|Intervals of BP}}
{{Main|Intervals of BP}}


{| class="wikitable center-1 right-2 right-3"
{| 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 81: 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 89: Line 47:
| 200
| 200
| M2/d3
| M2/d3
| 25/21
| [[25/21]] (−9.2{{c}})
|
| F#, Gb
| [[Sirius]]
|-
|-
| 3
| 3
Line 97: 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 105: Line 61:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| 7/5
| [[7/5]] (+2.7{{c}})
|
| H
| [[Canopus]]
|-
|-
| 5
| 5
Line 113: 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 121: Line 75:
| 600
| 600
| A4/M5
| A4/M5
| 5/3
| [[5/3]] (−6.5{{c}})
|
| J
| [[Arcturus]]
|-
|-
| 7
| 7
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| 700
| 700
| A5/m6/d7
| A5/m6/d7
| 9/5
| [[9/5]] (+6.5{{c}})
|
| A
| Arcturus
|-
|-
| 8
| 8
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| 800
| 800
| M6/m7
| M6/m7
| 49/25
| [[49/25]] (+5.4{{c}})
| false 2/1
| A#, Bb
| false Father
|-
|-
| 9
| 9
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| 900
| 900
| A6/M7/d8
| A6/M7/d8
| 15/7
| [[15/7]] (−2.7{{c}})
|
| B
| Canopus
|-
|-
| 10
| 10
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| 1000
| 1000
| P8/d9
| P8/d9
| 7/3
| [[7/3]] (−3.8{{c}})
|
| C
| Linear BP
|-
|-
| 11
| 11
Line 161: Line 110:
| 1100
| 1100
| A8/m9
| A8/m9
| 63/25
| [[63/25]] (+9.2{{c}})
|
| C#, Db
| Sirius
|-
|-
| 12
| 12
Line 169: 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 177: Line 124:
| 1300
| 1300
| A9/P10
| A9/P10
| 3/1
| [[3/1]]
| Tritave
| E
|
|}
|}


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== Regular temperament properties ==
== Regular temperament properties ==
{{Main| Bohlen-Pierce #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]]
|}
 
== Music ==
See also: [[Bohlen–Pierce scale/Music]]
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/dKFwJsxE3VU ''13ed3''] (2026)


== See also ==
== See also ==
* [[Bohlen-p_et]]
* [[Bohlen-p_et]]
* [[Catalog of 3.5.7 subgroup rank two temperaments]]
* [[Catalog of 3.5.7 subgroup rank two temperaments]]
* [[No-twos subgroup temperaments#3.5.7 subgroup temperaments]]
* [[19ed5|19ED5]]: relative ED5
* [[19ed5|19ED5]]: relative ED5
* [[23ed7|23ED7]]: relative ED7
* [[23ed7|23ED7]]: relative ED7


[[Category:3/1]]
[[Category:Edt]]
[[Category:Tritave]]
[[Category:Tritave]]
[[Category:Macrotonal]]
[[Category:Macrotonal]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Bohlen-Pierce]]
[[Category:Bohlen–Pierce]]

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

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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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