Tetracot: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
+a stripped down version of tuning spectrum for the 2.3.5.11.13 subgroup
→Tunings: +edo tunings; no inversion to the lower semioctave for ratios
Line 95: Line 95:
== Tunings ==
== Tunings ==
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>Generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! Generator<br>(¢)
! Comments
! Comments
|-
|-
|
| 11/10
| 11/10
| 165.004
| 165.004
|  
|  
|-
|-
| 1\7
|
| 171.429
|
|-
|
| 11/9
| 11/9
| 173.704
| 173.704
|  
|  
|-
|-
| 12/11
|  
| 11/6
| 174.894
| 174.894
|  
|  
|-
|-
| 7\48
|
| 175.000
|
|-
|
| 11/8
| 11/8
| 175.132
| 175.132
|  
|  
|-
|-
| 4/3
|  
| 3/2
| 175.489
| 175.489
|  
|  
|-
|-
| 6\41
|
| 175.610
|
|-
|
| 13/11
| 13/11
| 175.899
| 175.899
|  
|  
|-
|-
| 16/15
|  
| 15/8
| 176.021
| 176.021
|  
|  
|-
|-
|
| 5/4
| 5/4
| 176.257
| 176.257
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 18/13
|  
| 13/9
| 176.338
| 176.338
|  
|  
|-
|-
| 5\34
|
| 176.471
|
|-
|
| 15/13
| 15/13
| 176.516
| 176.516
|  
|  
|-
|-
| 6/5
|  
| 5/3
| 176.872
| 176.872
|  
|  
|-
|-
|
| 13/10
| 13/10
| 176.890
| 176.890
|  
|  
|-
|-
|
| 13/12
| 13/12
| 176.905
| 176.905
|  
|  
|-
|-
| 4\27
|
| 177.778
|
|-
|
| 15/11
| 15/11
| 178.984
| 178.984
|  
|  
|-
|-
| 16/13
|  
| 13/8
| 179.736
| 179.736
|  
|  
|-
|-
| 10/9
| 3\20
|
| 180.000
|
|-
|
| 9/5
| 182.404
| 182.404
|  
|  

Revision as of 12:18, 1 January 2024

Tetracot, in this article, is the rank-2 regular temperament for the 2.3.5.11.13 subgroup defined by tempering out 100/99, 144/143, and 243/242.

It can be seen as implying a rank-2 tuning which is generated by a sub-major second of about 176 cents which represents both 10/9 and 11/10. It is so named because the generator is a quarter of fifth: four generators make a fifth which approximates 3/2, which cannot occur in 12edo. Equal temperaments that support tetracot include 27, 34, and 41.

Tetracot has many extensions for the 7-, 11- and 13-limit. See Tetracot extensions.

See Tetracot family or No-sevens subgroup temperaments #Tetracot for more technical data.

Interval chain

Tetracot is considered as a cluster temperament with seven clusters of notes in an octave. The chroma interval between adjacent notes in each cluster represents 40/39 ~ 45/44 ~ 55/54 ~ 65/64 ~ 66/65 ~ 81/80 ~ 121/120 all tempered together. In the following table, odd harmonics and subharmonics 1–15 are in bold.

# Cents* Approximate Ratios
0 0.00 1/1
1 176.3 11/10, 10/9
2 352.7 11/9, 16/13
3 529.0 15/11
4 705.3 3/2
5 881.7 5/3
6 1058.0 11/6, 24/13
7 34.4 55/54, 45/44, 40/39
8 210.7 9/8
9 387.0 5/4
10 563.4 11/8, 18/13
11 739.7 20/13
12 916.0 22/13
13 1092.4 15/8
14 68.7 33/32, 27/26, 25/24
15 245.0 15/13
* in 2.3.5.11.13 subgroup CTE tuning

Scales

Tunings

Tuning spectrum

Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
11/10 165.004
1\7 171.429
11/9 173.704
11/6 174.894
7\48 175.000
11/8 175.132
3/2 175.489
6\41 175.610
13/11 175.899
15/8 176.021
5/4 176.257 5-odd-limit minimax
13/9 176.338
5\34 176.471
15/13 176.516
5/3 176.872
13/10 176.890
13/12 176.905
4\27 177.778
15/11 178.984
13/8 179.736
3\20 180.000
9/5 182.404

Music

Flora Canou
Zhea Erose
Xotla
  • "Electrostat" from Lesser Groove (2020) – Spotify | Bandcamp | YouTube – ambient electro, tetracot[13] in 34edo tuning
Retrieved from "https://en.xen.wiki/index.php?title=Tetracot&oldid=130254"