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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.


==Intervals==
13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].


{| class="wikitable"
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 ==
{{Harmonics in equal|13|3|1|prec=2|intervals=odd}}
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}}
 
* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]
 
== Intervals ==
{{Main|Intervals of BP}}
 
{| class="wikitable center-all right-2 right-3"
|-
! Steps
! [[Cent]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! Corresponding<br />3.5.7 subgroup<br />intervals
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs, {{nowrap|E {{=}} 1/1}})
|-
| 0
| 0
| 0
| P1
| 1/1
| E
|-
|-
! | Steps
| 1
! | Cents
| 146.3
! | BP nonatonic degree
| 100
! | Corresponding JI intervals
| A1/m2
! | Comments
| [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}})
! | Generator for...
| F
|-
|-
| | 1
| 2
| | 146.3
| 292.6
| | A1/m2
| 200
| | 27/25~49/45
| M2/d3
| |
| [[25/21]] (−9.2{{c}})
| |
| F#, Gb
|-
|-
| | 2
| 3
| | 292.6
| 438.9
| | M2/d3
| 300
| | 25/21
| A2/P3/d4
| |
| [[9/7]] (+3.8{{c}})
| | [[Sirius]]
| G
|-
|-
| | 3
| 4
| | 438.9
| 585.2
| | A2/P3/d4
| 400
| | 9/7
| A3/m4/d5
| |
| [[7/5]] (+2.7{{c}})
| | [[Bohlen-Pierce|Linear BP]]
| H
|-
|-
| | 4
| 5
| | 585.2
| 731.5
| | A3/m4/d5
| 500
| | 7/5
| M4/m5
| |
| [[75/49]] (−5.4{{c}})
| | [[Canopus]]
| H#, Jb
|-
|-
| | 5
| 6
| | 731.5
| 877.8
| | M4/m5
| 600
| | 75/49
| A4/M5
| | False 3/2
| [[5/3]] (−6.5{{c}})
| | false Father
| J
|-
|-
| | 6
| 7
| | 877.8
| 1024.1
| | A4/M5
| 700
| | 5/3
| A5/m6/d7
| |  
| [[9/5]] (+6.5{{c}})
| | [[Arcturus]]
| A
|-
| 8
| 1170.4
| 800
| M6/m7
| [[49/25]] (+5.4{{c}})
| A#, Bb
|-
| 9
| 1316.7
| 900
| A6/M7/d8
| [[15/7]] (−2.7{{c}})
| B
|-
| 10
| 1463.0
| 1000
| P8/d9
| [[7/3]] (−3.8{{c}})
| C
|-
| 11
| 1609.3
| 1100
| A8/m9
| [[63/25]] (+9.2{{c}})
| C#, Db
|-
| 12
| 1755.7
| 1200
| M9/d10
| [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}})
| D
|-
| 13
| 1902.0
| 1300
| A9/P10
| [[3/1]]
| E
|}
 
== JI approximation ==
[[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
|-
|-
| | 7
! Periods<br />per tritave
| | 1024.1
! Generator<br />(reduced)
| | A5/m6/d7
! Cents<br />(reduced)
| | 9/5
! Associated<br />ratio
| |
! Temperament
| | Arcturus
|-
|-
| | 8
| 1
| | 1170.4
| 1\13
| | M6/m7
| 146.30
| | 49/25
| 49/45
| | False 2/1
| [[Procyon]]
| | false Father
|-
|-
| | 9
| 1
| | 1316.7
| 2\13
| | A6/M7/d8
| 292.61
| | 15/7
| 25/21
| |
| [[Sirius]]
| | Canopus
|-
|-
| | 10
| 1
| | 1463.0
| 3\13
| | P8/d9
| 438.91
| | 7/3
| 9/7
| |
| [[BPS]]
| | Linear BP
|-
|-
| | 11
| 1
| | 1609.3
| 4\13
| | A8/m9
| 585.22
| | 63/25
| 7/5
| |
| [[Canopus]]
| | Sirius
|-
|-
| | 12
| 1
| | 1755.7
| 5\13
| | M9/d10
| 731.63
| | 25/9~135/49
| 75/49
| |
|
| |  
|-
|-
| | 13
| 1
| | 1902.0
| 6\13
| | A9/P10
| 877.83
| | 3/1
| 5/3
| | Tritave
| [[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]]
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