13edt: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
[[File:13edt.png|thumb|alt=13edt.png|A plot of the [[ | {{ED intro}} 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|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.]] | |||
== Theory == | |||
{{See also| Relationship between Bohlen-Pierce and octave-ful temperaments }} | |||
13edt 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 the rank-2 [[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 3.5.7 subgroup, 13edt maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]]. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=11}} | |||
{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 13edt (continued)}} | |||
== Intervals == | == Intervals == | ||
{{Main|Intervals of BP}} | {{Main|Intervals of BP}} | ||
{| class="wikitable center- | {| class="wikitable center-all right-2 right-3" | ||
|- | |- | ||
! Steps | ! Steps | ||
! [[Cent]]s | ! [[Cent]]s | ||
! [[Hekt]]s | ! [[Hekt]]s | ||
! | ! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree | ||
! Corresponding | ! 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 | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 32: | Line 40: | ||
| 100 | | 100 | ||
| A1/m2 | | A1/m2 | ||
| 27/25 | | [[49/45]] (−1.1{{c}}); [[27/25]] (+13.1{{c}}) | ||
| F | |||
| | |||
|- | |- | ||
| 2 | | 2 | ||
| Line 41: | Line 47: | ||
| 200 | | 200 | ||
| M2/d3 | | M2/d3 | ||
| 25/21 | | [[25/21]] (−9.2{{c}}) | ||
| F#, Gb | |||
| | |||
|- | |- | ||
| 3 | | 3 | ||
| Line 50: | Line 54: | ||
| 300 | | 300 | ||
| A2/P3/d4 | | A2/P3/d4 | ||
| 9/7 | | [[9/7]] (+3.8{{c}}) | ||
| G | |||
| | |||
|- | |- | ||
| 4 | | 4 | ||
| Line 59: | Line 61: | ||
| 400 | | 400 | ||
| A3/m4/d5 | | A3/m4/d5 | ||
| 7/5 | | [[7/5]] (+2.7{{c}}) | ||
| H | |||
| | |||
|- | |- | ||
| 5 | | 5 | ||
| Line 68: | Line 68: | ||
| 500 | | 500 | ||
| M4/m5 | | M4/m5 | ||
| 75/49 | | [[75/49]] (−5.4{{c}}) | ||
| | | H#, Jb | ||
|- | |- | ||
| 6 | | 6 | ||
| Line 77: | Line 75: | ||
| 600 | | 600 | ||
| A4/M5 | | A4/M5 | ||
| 5/3 | | [[5/3]] (−6.5{{c}}) | ||
| J | |||
| | |||
|- | |- | ||
| 7 | | 7 | ||
| Line 86: | Line 82: | ||
| 700 | | 700 | ||
| A5/m6/d7 | | A5/m6/d7 | ||
| 9/5 | | [[9/5]] (+6.5{{c}}) | ||
| | | A | ||
|- | |- | ||
| 8 | | 8 | ||
| Line 95: | Line 89: | ||
| 800 | | 800 | ||
| M6/m7 | | M6/m7 | ||
| 49/25 | | [[49/25]] (+5.4{{c}}) | ||
| | | A#, Bb | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 104: | Line 96: | ||
| 900 | | 900 | ||
| A6/M7/d8 | | A6/M7/d8 | ||
| 15/7 | | [[15/7]] (−2.7{{c}}) | ||
| | | B | ||
|- | |- | ||
| 10 | | 10 | ||
| Line 113: | Line 103: | ||
| 1000 | | 1000 | ||
| P8/d9 | | P8/d9 | ||
| 7/3 | | [[7/3]] (−3.8{{c}}) | ||
| | | C | ||
|- | |- | ||
| 11 | | 11 | ||
| Line 122: | Line 110: | ||
| 1100 | | 1100 | ||
| A8/m9 | | A8/m9 | ||
| 63/25 | | [[63/25]] (+9.2{{c}}) | ||
| | | C#, Db | ||
|- | |- | ||
| 12 | | 12 | ||
| Line 131: | Line 117: | ||
| 1200 | | 1200 | ||
| M9/d10 | | M9/d10 | ||
| 25/9 | | [[135/49]] (+1.1{{c}}); [[25/9]] (−13.1{{c}}) | ||
| D | |||
| | |||
|- | |- | ||
| 13 | | 13 | ||
| Line 140: | Line 124: | ||
| 1300 | | 1300 | ||
| A9/P10 | | A9/P10 | ||
| 3/1 | | [[3/1]] | ||
| | | E | ||
|} | |} | ||
| Line 170: | Line 152: | ||
=== 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 | ||
|- | |- | ||
| Line 201: | Line 184: | ||
| [[Canopus]] | | [[Canopus]] | ||
|- | |- | ||
|1 | | 1 | ||
|5\13 | | 5\13 | ||
|731.63 | | 731.63 | ||
|75/49 | | 75/49 | ||
| | | | ||
|- | |- | ||
| Line 213: | Line 196: | ||
| [[Arcturus]] | | [[Arcturus]] | ||
|} | |} | ||
== Music == | |||
See also: [[Bohlen–Pierce scale/Music]] | |||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/dKFwJsxE3VU ''13ed3''] (2026) | |||
== See also == | == See also == | ||
| Line 221: | Line 210: | ||
* [[23ed7|23ED7]]: relative ED7 | * [[23ed7|23ED7]]: relative ED7 | ||
[[Category:Tritave]] | [[Category:Tritave]] | ||
[[Category:Macrotonal]] | [[Category:Macrotonal]] | ||
[[Category:Nonoctave]] | [[Category:Nonoctave]] | ||
[[Category: | [[Category:Bohlen–Pierce]] | ||