13edt: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,264 edits
m Removed redundant category
ArrowHead294 (talk | contribs)
15,213 edits
Replace with template
Line 9: Line 9:


== Theory ==
== Theory ==
{| class="wikitable center-all"
{{Harmonics in equal|13|3|1|prec=2|columns=15}}
|+Approximation of prime intervals in 13-EDT
! 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]]

Revision as of 19:01, 14 May 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

alt : Your browser has no SVG support.

Regular temperament properties

Main article: Bohlen-Pierce § Regular temperament properties

See also

Retrieved from "https://en.xen.wiki/index.php?title=13edt&oldid=143148"