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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}}
[[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]].]]


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.
'''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, and therefore has received by far the most attention among equal divisions of the tritave.  


[[File:13edt.png|alt=13edt.png|13edt.png]]
It 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 [[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.
 
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 [[no-2]] [[3/1]]-[[equave]]-[[7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].


== Theory ==
== Theory ==
{| class="wikitable center-all"
{{Harmonics in equal|13|3|1|prec=2|intervals=odd}}
|+Approximation of prime intervals in 13-EDT
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}}
! 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
|}


* [[Intervals of BP]]
* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]
* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]


==Intervals==
== Intervals ==
{{See also|Bohlen-p_et}}
{{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 80: Line 39:
| 100
| 100
| A1/m2
| A1/m2
| 27/25~49/45
| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
|
| F
|
|-
|-
| 2
| 2
Line 88: Line 46:
| 200
| 200
| M2/d3
| M2/d3
| 25/21
| [[25/21]] (−9.2{{c}})
|
| F#, Gb
| [[Sirius]]
|-
|-
| 3
| 3
Line 96: Line 53:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| 9/7
| [[9/7]] (+3.8{{c}})
|
| G
| [[Bohlen-Pierce|Linear BP]]
|-
|-
| 4
| 4
Line 104: Line 60:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| 7/5
| [[7/5]] (+2.7{{c}})
|
| H
| [[Canopus]]
|-
|-
| 5
| 5
Line 112: Line 67:
| 500
| 500
| M4/m5
| M4/m5
| 75/49
| [[75/49]] (−5.4{{c}})
| False 3/2
| H#, Jb
| false Father
|-
|-
| 6
| 6
Line 120: Line 74:
| 600
| 600
| A4/M5
| A4/M5
| 5/3
| [[5/3]] (−6.5{{c}})
|
| J
| [[Arcturus]]
|-
|-
| 7
| 7
Line 128: Line 81:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| 9/5
| [[9/5]] (+6.5{{c}})
|
| A
| Arcturus
|-
|-
| 8
| 8
Line 136: Line 88:
| 800
| 800
| M6/m7
| M6/m7
| 49/25
| [[49/25]] (+5.4{{c}})
| False 2/1
| A#, Bb
| false Father
|-
|-
| 9
| 9
Line 144: Line 95:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| 15/7
| [[15/7]] (−2.7{{c}})
|
| B
| Canopus
|-
|-
| 10
| 10
Line 152: Line 102:
| 1000
| 1000
| P8/d9
| P8/d9
| 7/3
| [[7/3]] (−3.8{{c}})
|
| C
| Linear BP
|-
|-
| 11
| 11
Line 160: Line 109:
| 1100
| 1100
| A8/m9
| A8/m9
| 63/25
| [[63/25]] (+9.2{{c}})
|
| C#, Db
| Sirius
|-
|-
| 12
| 12
Line 168: Line 116:
| 1200
| 1200
| M9/d10
| M9/d10
| 25/9~135/49
| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
|
| D
|
|-
|-
| 13
| 13
Line 176: Line 123:
| 1300
| 1300
| A9/P10
| A9/P10
| 3/1
| [[3/1]]
| Tritave
| E
|
|}
|}


==JI approximation==
== JI approximation ==
[[File:13ed3-17-001.svg|alt=alt : Your browser has no SVG support.]]
[[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==
== See also ==
*[[Catalog of 3.5.7 subgroup rank two temperaments]]
* [[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 16:47, 22 May 2026

← 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, and therefore has received by far the most attention among equal divisions of the tritave.

It 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 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 no-2 3/1-equave-7-limit, 13edt maintains the smallest relative error of any EDT until 258edt and 271edt, and the smallest absolute error until 56edt.

Theory

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
Harmonic 25 27 29 31 33 35 37 39 41 43 45
Error Absolute (¢) -13.07 +0.00 +22.59 +53.44 -54.80 -10.36 +39.74 -51.40 +8.32 +72.17 -6.53
Relative (%) -8.9 +0.0 +15.4 +36.5 -37.5 -7.1 +27.2 -35.1 +5.7 +49.3 -4.5
Steps
(reduced) 		38
(12) 		39
(0) 		40
(1) 		41
(2) 		41
(2) 		42
(3) 		43
(4) 		43
(4) 		44
(5) 		45
(6) 		45
(6)

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

See also

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