13edt: Difference between revisions

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{{Infobox ET}}
{{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]].]]
[[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]].]]


'''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.  
'''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.  


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.
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]].
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]].
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 ==
{{Harmonics in equal|13|3|1|prec=2}}
{{Harmonics in equal|13|3|1|prec=2|intervals=odd}}
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=16}}
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}}


* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]
* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]
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{{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...
! [[4L 5s (3/1-equivalent)#Notation|BPS enneatonic notation]] (J = 1/1)
|-
|-
| 0
| 0
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| P1
| P1
| 1/1
| 1/1
|
| E
|
| J
|-
|-
| 1
| 1
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| 100
| 100
| A1/m2
| A1/m2
| 27/25, 49/45
| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
|
| F
|
| K
|-
|-
| 2
| 2
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| 200
| 200
| M2/d3
| M2/d3
| 25/21
| [[25/21]] (−9.2{{c}})
|
| F#, Gb
| [[Sirius]]
| K#, Lb
|-
|-
| 3
| 3
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| 300
| 300
| A2/P3/d4
| A2/P3/d4
| 9/7
| [[9/7]] (+3.8{{c}})
|
| G
| [[Bohlen-Pierce-Stearns|Linear BP]]
| L
|-
|-
| 4
| 4
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| 400
| 400
| A3/m4/d5
| A3/m4/d5
| 7/5
| [[7/5]] (+2.7{{c}})
|
| H
| [[Canopus]]
| M
|-
|-
| 5
| 5
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| 500
| 500
| M4/m5
| M4/m5
| 75/49
| [[75/49]] (−5.4{{c}})
| false 3/2
| H#, Jb
| false Father
| M#, Nb
|-
|-
| 6
| 6
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| 600
| 600
| A4/M5
| A4/M5
| 5/3
| [[5/3]] (−6.5{{c}})
|
| J
| [[Arcturus]]
| N
|-
|-
| 7
| 7
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| 700
| 700
| A5/m6/d7
| A5/m6/d7
| 9/5
| [[9/5]] (+6.5{{c}})
|
| A
| Arcturus
| O
|-
|-
| 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
| O#, Pb
|-
|-
| 9
| 9
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| 900
| 900
| A6/M7/d8
| A6/M7/d8
| 15/7
| [[15/7]] (−2.7{{c}})
|
| B
| Canopus
| P
|-
|-
| 10
| 10
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| 1000
| 1000
| P8/d9
| P8/d9
| 7/3
| [[7/3]] (−3.8{{c}})
|
| C
| Linear BP
| Q
|-
|-
| 11
| 11
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| 1100
| 1100
| A8/m9
| A8/m9
| 63/25
| [[63/25]] (+9.2{{c}})
|
| C#, Db
| Sirius
| Q#, Rb
|-
|-
| 12
| 12
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| 1200
| 1200
| M9/d10
| M9/d10
| 25/9, 135/49
| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
|
| D
|
| R
|-
|-
| 13
| 13
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| 1300
| 1300
| A9/P10
| A9/P10
| 3/1
| [[3/1]]
| Tritave
| E
|
| J
|}
|}


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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all right-3 left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per tritave
|-
! Generator<br>(reduced)
! Periods<br />per tritave
! Cents<br>(reduced)
! Generator<br />(reduced)
! Associated<br>ratio
! Cents<br />(reduced)
! Associated<br />ratio
! Temperament
! Temperament
|-
|-
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| [[Canopus]]
| [[Canopus]]
|-
|-
|1
| 1
|5\13
| 5\13
|731.63
| 731.63
|75/49
| 75/49
|
|
|-
|-
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* [[23ed7|23ED7]]: relative ED7
* [[23ed7|23ED7]]: relative ED7


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

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

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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

See also

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