13edt: Difference between revisions

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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"
|+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 ==

Revision as of 11:06, 28 August 2024

← 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.

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 7-limit, 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.

Theory

Approximation of harmonics in 13edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -29.57 +0.00 -59.13 -6.53 -29.57 -3.83 +57.61 +0.00 -36.10 -54.80 -59.13 -51.40 -33.39 -6.53 +28.04
Relative (%) -20.2 +0.0 -40.4 -4.5 -20.2 -2.6 +39.4 +0.0 -24.7 -37.5 -40.4 -35.1 -22.8 -4.5 +19.2
Steps
(reduced) 		8
(8) 		13
(0) 		16
(3) 		19
(6) 		21
(8) 		23
(10) 		25
(12) 		26
(0) 		27
(1) 		28
(2) 		29
(3) 		30
(4) 		31
(5) 		32
(6) 		33
(7)

Intervals

Main article: Intervals of BP
Steps Cents Hekts BP nonatonic degree Corresponding JI intervals Comments Generator for... Arcturus nonatonic notation (J = 1/1)
1 146.3 100 A1/m2 27/25~49/45 J#
2 292.6 200 M2/d3 25/21 Sirius Kb
3 438.9 300 A2/P3/d4 9/7 Linear BP K
4 585.2 400 A3/m4/d5 7/5 Canopus K#, Lb
5 731.5 500 M4/m5 75/49 false 3/2 false Father L
6 877.8 600 A4/M5 5/3 Arcturus M
7 1024.1 700 A5/m6/d7 9/5 Arcturus N
8 1170.4 800 M6/m7 49/25 false 2/1 false Father N#, Ob
9 1316.7 900 A6/M7/d8 15/7 Canopus O
10 1463.0 1000 P8/d9 7/3 Linear BP P
11 1609.3 1100 A8/m9 63/25 Sirius Q
12 1755.7 1200 M9/d10 25/9~135/49 R
13 1902.0 1300 A9/P10 3/1 Tritave J

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

Retrieved from "https://en.xen.wiki/index.php?title=13edt&oldid=152904"