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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.
'''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.


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 [[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
|}


* [[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...
|-
! [[Arcturus]] nonatonic notation
| 0
| 0
| 0
| P1
| 1/1
| E
|-
|-
| 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
|
| H#
|-
|-
| 2
| 2
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| 200
| 200
| M2/d3
| M2/d3
| 25/21
| [[25/21]] (−9.2{{c}})
|
| F#, Gb
| [[Sirius]]
| Ib
|-
|-
| 3
| 3
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| 300
| 300
| A2/P3/d4
| A2/P3/d4
| 9/7
| [[9/7]] (+3.8{{c}})
|
| G
| [[Bohlen-Pierce|Linear BP]]
| I
|-
|-
| 4
| 4
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| 400
| 400
| A3/m4/d5
| A3/m4/d5
| 7/5
| [[7/5]] (+2.7{{c}})
|
| H
| [[Canopus]]
| I#, Jb
|-
|-
| 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
| J
|-
|-
| 6
| 6
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| 600
| 600
| A4/M5
| A4/M5
| 5/3
| [[5/3]] (−6.5{{c}})
|
| J
| [[Arcturus]]
| K
|-
|-
| 7
| 7
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| 700
| 700
| A5/m6/d7
| A5/m6/d7
| 9/5
| [[9/5]] (+6.5{{c}})
|
| A
| Arcturus
| L
|-
|-
| 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
| L#, Mb
|-
|-
| 9
| 9
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| 900
| 900
| A6/M7/d8
| A6/M7/d8
| 15/7
| [[15/7]] (−2.7{{c}})
|
| B
| Canopus
| M
|-
|-
| 10
| 10
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| 1000
| 1000
| P8/d9
| P8/d9
| 7/3
| [[7/3]] (−3.8{{c}})
|
| C
| Linear BP
| N
|-
|-
| 11
| 11
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| 1100
| 1100
| A8/m9
| A8/m9
| 63/25
| [[63/25]] (+9.2{{c}})
|
| C#, Db
| Sirius
| O
|-
|-
| 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
|
| P
|-
|-
| 13
| 13
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| 1300
| 1300
| A9/P10
| A9/P10
| 3/1
| [[3/1]]
| Tritave
| E
|
| H
|}
|}


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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]]
|}


== 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]]
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