Tetracot: Difference between revisions

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== Intervals ==
== Intervals ==
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.
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, prime harmonics are in '''bold'''.  


{| class="wikitable right-1"
{| class="wikitable right-1 right-2"
|-
|-
! #
! #
Line 24: Line 24:
| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
|-
|-
| 1
| 1
Line 32: Line 32:
| 2
| 2
| 352.39
| 352.39
| 11/9, 16/13
| 11/9, '''16/13'''
|-
|-
| 3
| 3
Line 40: Line 40:
| 4
| 4
| 704.79
| 704.79
| 3/2
| '''3/2'''
|-
|-
| 5
| 5
Line 60: Line 60:
| 9
| 9
| 385.77
| 385.77
| 5/4
| '''5/4'''
|-
|-
| 10
| 10
| 561.96
| 561.96
| 11/8, 18/13
| '''11/8''', 18/13
|-
|-
| 11
| 11
Line 89: Line 89:


=== Monkey ===
=== Monkey ===
{| class="wikitable right-1"
{| class="wikitable right-1 right-2"
|-
|-
! #
! #
! Cents<sup>*</sup>
! Cents<nowiki>*</nowiki>
! Approximate Ratios
! Approximate Ratios
|-
|-
Line 105: Line 105:
| 2
| 2
| 351.24
| 351.24
| 11/9, 16/13
| 11/9, '''16/13'''
|-
|-
| 3
| 3
Line 113: Line 113:
| 4
| 4
| 702.49
| 702.49
| 3/2
| '''3/2'''
|-
|-
| 5
| 5
Line 125: Line 125:
| 7
| 7
| 29.36
| 29.36
|  
| 55/54, 45/44, 40/39
|-
|-
| 8
| 8
Line 133: Line 133:
| 9
| 9
| 380.60
| 380.60
| 5/4
| '''5/4'''
|-
|-
| 10
| 10
| 556.22
| 556.22
| 11/8, 18/13
| '''11/8''', 18/13
|-
|-
| 11
| 11
Line 153: Line 153:
| 14
| 14
| 58.71
| 58.71
|  
| 33/32, 27/26, 25/24
|-
|-
| 15
| 15
| 234.34
| 234.34
| 8/7, 15/13
| '''8/7''', 15/13
|-
|-
| 16
| 16
Line 165: Line 165:
| 17
| 17
| 585.58
| 585.58
|  
| 45/32
|-
|-
| 18
| 18
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| 27
| 27
| 1141.80
| 1141.80
|  
| 27/14
|-
|-
| 28
| 28
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=== Bunya ===
=== Bunya ===
{| class="wikitable right-1"
{| class="wikitable right-1 right-2"
|-
|-
! #
! #
! Cents<sup>*</sup>
! Cents<nowiki>*</nowiki>
! Approximate Ratios
! Approximate Ratios
|-
|-
Line 230: Line 230:
| 2
| 2
| 351.77
| 351.77
| 11/9, 16/13
| 11/9, '''16/13'''
|-
|-
| 3
| 3
Line 238: Line 238:
| 4
| 4
| 703.54
| 703.54
| 3/2
| '''3/2'''
|-
|-
| 5
| 5
Line 262: Line 262:
| 10
| 10
| 558.86
| 558.86
| 11/8, 18/13
| '''11/8''', 18/13
|-
|-
| 11
| 11
Line 326: Line 326:
| 26
| 26
| 973.03
| 973.03
| 7/4
| '''7/4'''
|-
|-
| 27
| 27
Line 339: Line 339:


=== Modus ===
=== Modus ===
{| class="wikitable right-1"
{| class="wikitable right-1 right-2"
|-
|-
! #
! #
! Cents<sup>*</sup>
! Cents<nowiki>*</nowiki>
! Approximate Ratios
! Approximate Ratios
|-
|-
Line 355: Line 355:
| 2
| 2
| 353.91
| 353.91
| 11/9, 16/13
| 11/9, '''16/13'''
|-
|-
| 3
| 3
Line 363: Line 363:
| 4
| 4
| 707.81
| 707.81
| 3/2
| '''3/2'''
|-
|-
| 5
| 5
Line 379: Line 379:
| 8
| 8
| 215.63
| 215.63
| 9/8, 8/7
| 9/8, '''8/7'''
|-
|-
| 9
| 9
| 392.58
| 392.58
| 5/4
| '''5/4'''
|-
|-
| 10
| 10
| 569.53
| 569.53
| 11/8, 18/13
| '''11/8''', 18/13
|-
|-
| 11
| 11
Line 436: Line 436:


=== Wollemia ===
=== Wollemia ===
{| class="wikitable right-1"
{| class="wikitable right-1 right-2"
|-
|-
! #
! #
! Cents<sup>*</sup>
! Cents<nowiki>*</nowiki>
! Approximate Ratios
! Approximate Ratios
|-
|-
Line 452: Line 452:
| 2
| 2
| 354.46
| 354.46
| 11/9, 16/13
| 11/9, '''16/13'''
|-
|-
| 3
| 3
Line 460: Line 460:
| 4
| 4
| 708.92
| 708.92
| 3/2
| '''3/2'''
|-
|-
| 5
| 5
Line 472: Line 472:
| 7
| 7
| 40.62
| 40.62
|  
| 55/54, 45/44, 40/39
|-
|-
| 8
| 8
Line 480: Line 480:
| 9
| 9
| 395.08
| 395.08
| 5/4, 14/11
| '''5/4''', 14/11
|-
|-
| 10
| 10
| 572.31
| 572.31
| 11/8, 18/13, 7/5
| '''11/8''', 18/13, 7/5
|-
|-
| 11
| 11
Line 500: Line 500:
| 14
| 14
| 81.24
| 81.24
|  
| 27/26, 25/24
|-
|-
| 15
| 15
Line 520: Line 520:
| 19
| 19
| 967.39
| 967.39
| 7/4
| '''7/4'''
|-
|-
| 20
| 20
Line 748: Line 748:
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! [[Eigenmonzo|eigenmonzo<br>(Unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! Generator<br>(¢)
! Comments
! Comments
Line 950: Line 950:


== Scales ==
== Scales ==
* [[Tetracot7]] - [[6L 1s]] scale
* [[Tetracot7]] – [[6L 1s]] scale
* [[Tetracot13]] - improper [[7L 6s]]
* [[Tetracot13]] – improper [[7L 6s]]
* [[Tetracot20]] - improper [[7L 13s]]
* [[Tetracot20]] – improper [[7L 13s]]


[[Category:Temperaments]]
[[Category:Temperaments]]

Revision as of 07:21, 5 September 2023

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 7, 11 and 13-limit include monkey (34 & 41), bunya (34d & 41), modus (27e & 34d) and wollemia (27e & 34).

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

Intervals

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, prime harmonics are in bold.

# Cents* Approximate Ratios
0 0.00 1/1
1 176.20 11/10, 10/9
2 352.39 11/9, 16/13
3 528.59 15/11
4 704.79 3/2
5 880.98 5/3
6 1057.18 11/6, 24/13
7 33.38 55/54, 45/44, 40/39
8 209.57 9/8
9 385.77 5/4
10 561.96 11/8, 18/13
11 738.16 20/13
12 914.36 22/13
13 1090.55 15/8
14 66.75 33/32, 27/26, 25/24
15 242.95 15/13
* in 2.3.5.11.13 POTE tuning

Monkey

# Cents* Approximate Ratios
0 0.00 1/1
1 175.62 11/10, 10/9
2 351.24 11/9, 16/13
3 526.87 15/11
4 702.49 3/2
5 878.11 5/3
6 1053.73 11/6, 24/13
7 29.36 55/54, 45/44, 40/39
8 204.98 9/8
9 380.60 5/4
10 556.22 11/8, 18/13
11 731.85 20/13
12 907.47 22/13
13 1083.09 13/7, 15/8
14 58.71 33/32, 27/26, 25/24
15 234.34 8/7, 15/13
16 409.96
17 585.58 45/32
18 761.20
19 936.83 12/7
20 1112.45
21 88.07
22 263.69
23 439.31 9/7
24 614.94 10/7
25 790.56 11/7
26 966.18
27 1141.80 27/14
28 117.43 15/14
* in 13-limit POTE tuning

Bunya

# Cents* Approximate Ratios
0 0.00 1/1
1 175.89 11/10, 10/9
2 351.77 11/9, 16/13
3 527.66 15/11
4 703.54 3/2
5 879.43 5/3
6 1055.31 11/6, 24/13
7 31.20 56/55, 55/54, 45/44, 40/39
8 207.09 9/8
9 382.97 5/4
10 558.86 11/8, 18/13
11 734.74 20/13
12 910.63 22/13
13 1086.52 28/15, 15/8
14 62.40 33/32, 28/27, 27/26, 25/24
15 238.29 15/13
16 414.17 14/11
17 590.06 7/5
18 765.94 14/9
19 941.83
20 1117.72 21/11
21 93.60 21/20
22 269.49 7/6
23 445.37
24 621.26
25 797.15
26 973.03 7/4
27 1148.92 35/18
28 124.80 14/13
* in 13-limit POTE tuning

Modus

# Cents* Approximate Ratios
0 0.00 1/1
1 176.95 11/10, 10/9
2 353.91 11/9, 16/13
3 530.86 15/11
4 707.81 3/2
5 884.77 5/3
6 1061.72 11/6, 24/13, 13/7
7 38.67 55/54, 45/44, 40/39, 36/35
8 215.63 9/8, 8/7
9 392.58 5/4
10 569.53 11/8, 18/13
11 746.49 20/13
12 923.44 22/13, 12/7
13 1100.39 15/8, 40/21
14 77.35 27/26, 25/24, 22/21
15 254.30 15/13
16 431.25 9/7
17 608.20 10/7
18 785.16 11/7
19 962.11
20 1139.06 27/14
21 116.02 15/14
* in 13-limit POTE tuning

Wollemia

# Cents* Approximate Ratios
0 0.00 1/1
1 177.23 11/10, 10/9
2 354.46 11/9, 16/13
3 531.69 15/11
4 708.92 3/2
5 886.16 5/3
6 1063.39 11/6, 24/13, 28/15
7 40.62 55/54, 45/44, 40/39
8 217.85 9/8
9 395.08 5/4, 14/11
10 572.31 11/8, 18/13, 7/5
11 749.54 20/13, 14/9
12 926.77 22/13
13 1104.01 15/8
14 81.24 27/26, 25/24
15 258.47 15/13, 7/6
16 435.70
17 612.93
18 790.16
19 967.39 7/4
20 1144.62
21 121.86 14/13
* in 13-limit POTE tuning

Tuning spectra

Monkey

Gencom: [2 10/9; 100/99 105/104 144/143 243/242]

Gencom mapping: [⟨1 1 1 5 2 4], ⟨0 4 9 -15 10 -2]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
11/10 165.004
11/9 173.704
14/13 174.746
12/11 174.894
11/8 175.132
14/11 175.300 11-odd-limit minimax
8/7 175.412
7/6 175.428
9/7 175.438
4/3 175.489
15/14 175.694
7/5 175.729 7, 9, 13 and 15-odd-limit minimax
13/11 175.899
16/15 176.021
5/4 176.257 5-odd-limit minimax
18/13 176.338
15/13 176.516
6/5 176.872
13/10 176.890
13/12 176.905
15/11 178.984
16/13 179.736
10/9 182.404

Bunya

Gencom: [2 10/9; 100/99 144/143 225/224 243/242]

Gencom mapping: [⟨1 1 1 -1 2 4], ⟨0 4 9 26 10 -2]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
11/10 165.004
11/9 173.704
12/11 174.894
11/8 175.132
15/14 175.427
7/5 175.442 11-odd-limit minimax
4/3 175.489
8/7 175.724
7/6 175.767 7-odd-limit minimax
9/7 175.829 9-odd-limit minimax
13/11 175.899 13 and 15-odd-limit minimax
14/13 176.011
16/15 176.021
14/11 176.094
5/4 176.257 5-odd-limit minimax
18/13 176.338
15/13 176.516
6/5 176.872
13/10 176.890
13/12 176.905
15/11 178.984
16/13 179.736
10/9 182.404

Modus

Gencom: [2 10/9; 64/63 78/77 100/99 144/143]

Gencom mapping: [⟨1 1 1 4 2 4], ⟨0 4 9 -8 10 -2]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
11/10 165.004
11/9 173.704
12/11 174.894
11/8 175.132
4/3 175.489
13/11 175.899
16/15 176.021
5/4 176.257 5-odd-limit minimax
18/13 176.338
15/13 176.516
14/11 176.805 11, 13 and 15-odd-limit minimax
6/5 176.872
13/10 176.890
13/12 176.905
15/14 177.116
9/7 177.193 9-odd-limit minimax
7/5 177.499 7-odd-limit minimax
7/6 177.761
14/13 178.617
8/7 178.897
15/11 178.984
16/13 179.736
10/9 182.404

Wollemia

Gencom: [2 10/9; 56/55 91/90 100/99 243/242]

Gencom mapping: [⟨1 1 1 0 2 4], ⟨0 4 9 19 10 -2]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
11/10 165.004
11/9 173.704
12/11 174.894
11/8 175.132
4/3 175.489
13/11 175.899
16/15 176.021
5/4 176.257 5-odd-limit minimax
18/13 176.338
15/13 176.516
6/5 176.872
13/10 176.890
13/12 176.905
8/7 177.307 7, 9, 11, 13 and 15-odd-limit minimax
14/13 177.538
7/6 177.791
7/5 178.251
9/7 178.629
15/11 178.984
14/11 179.723
16/13 179.736
15/14 180.093
10/9 182.404

Scales

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