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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 [[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, 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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! [[Cent]]s
! [[Cent]]s
! [[Hekt]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! Corresponding
! Corresponding<br />3.5.7 subgroup<br />intervals
3.5.7 subgroup <br>
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs, {{nowrap|E {{=}} 1/1}})
intervals
! [[Lambda ups and downs notation|Lambda]]  
(sLsLsLsLs, <br>
E = 1/1)
|-
|-
| 0
| 0
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| 100
| 100
| A1/m2
| A1/m2
| [[49/45]] (-1.1c); [[27/25]] (+13.1c)
| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
| F
| F
|-
|-
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| 200
| 200
| M2/d3
| M2/d3
| [[25/21]] (-9.2c)
| [[25/21]] (−9.2{{c}})
| F#, Gb
| F#, Gb
|-
|-
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| 300
| 300
| A2/P3/d4
| A2/P3/d4
| [[9/7]] (+3.8c)
| [[9/7]] (+3.8{{c}})
| G
| G
|-
|-
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| 400
| 400
| A3/m4/d5
| A3/m4/d5
| [[7/5]] (+2.7c)
| [[7/5]] (+2.7{{c}})
| H
| H
|-
|-
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| 500
| 500
| M4/m5
| M4/m5
| [[75/49]] (-5.4c)
| [[75/49]] (−5.4{{c}})
| H#, Jb
| H#, Jb
|-
|-
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| 600
| 600
| A4/M5
| A4/M5
| [[5/3]] (-6.5c)
| [[5/3]] (−6.5{{c}})
| J
| J
|-
|-
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| 700
| 700
| A5/m6/d7
| A5/m6/d7
| [[9/5]] (+6.5c)
| [[9/5]] (+6.5{{c}})
| A
| A
|-
|-
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| 800
| 800
| M6/m7
| M6/m7
| [[49/25]] (+5.4c)
| [[49/25]] (+5.4{{c}})
| A#, Bb
| A#, Bb
|-
|-
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| 900
| 900
| A6/M7/d8
| A6/M7/d8
| [[15/7]] (-2.7c)
| [[15/7]] (−2.7{{c}})
| B
| B
|-
|-
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| 1000
| 1000
| P8/d9
| P8/d9
| [[7/3]] (-3.8c)
| [[7/3]] (−3.8{{c}})
| C
| C
|-
|-
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| 1100
| 1100
| A8/m9
| A8/m9
| [[63/25]] (+9.2c)
| [[63/25]] (+9.2{{c}})
| C#, Db
| C#, Db
|-
|-
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| 1200
| 1200
| M9/d10
| M9/d10
| [[135/49]] (+1.1c); [[25/9]] (-13.1c)
| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
| D
| D
|-
|-
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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]]
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