Modus: Difference between revisions

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Complete interval table
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== Interval chain ==
== Interval chain ==
{| class="wikitable right-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! #
! #
Line 28: Line 28:
|-
|-
| 0
| 0
| 0.00
| 0.0
| '''1/1'''
| '''1/1'''
|-
|-
| 1
| 1
| 176.95
| 176.9
| 10/9, 11/10
| 10/9, 11/10
|-
|-
| 2
| 2
| 353.91
| 353.7
| 11/9, '''16/13'''
| 11/9, '''16/13'''
|-
|-
| 3
| 3
| 530.86
| 530.6
| 15/11
| 15/11
|-
|-
| 4
| 4
| 707.81
| 707.5
| '''3/2'''
| '''3/2'''
|-
|-
| 5
| 5
| 884.77
| 884.4
| 5/3
| 5/3
|-
|-
| 6
| 6
| 1061.72
| 1061.2
| 11/6, 13/7, 24/13
| 11/6, 13/7, 24/13
|-
|-
| 7
| 7
| 38.67
| 38.1
| 36/35, 40/39, 45/44, 55/54
| 36/35, 40/39, 45/44, 55/54
|-
|-
| 8
| 8
| 215.63
| 215.0
| '''8/7''', '''9/8'''
| '''8/7''', '''9/8'''
|-
|-
| 9
| 9
| 392.58
| 391.9
| '''5/4'''
| '''5/4'''
|-
|-
| 10
| 10
| 569.53
| 568.7
| '''11/8''', 18/13
| '''11/8''', 18/13
|-
|-
| 11
| 11
| 746.49
| 745.6
| 20/13
| 20/13
|-
|-
| 12
| 12
| 923.44
| 922.5
| 12/7, 22/13
| 12/7, 22/13
|-
|-
| 13
| 13
| 1100.39
| 1099.4
| 15/8, 40/21
| 15/8, 40/21
|-
|-
| 14
| 14
| 77.35
| 76.2
| 22/21, 25/24, 27/26
| 22/21, 25/24, 27/26
|-
|-
| 15
| 15
| 254.30
| 253.1
| 15/13
| 15/13
|-
|-
| 16
| 16
| 431.25
| 430.0
| 9/7
| 9/7
|-
|-
| 17
| 17
| 608.20
| 606.8
| 10/7
| 10/7
|-
|-
| 18
| 18
| 785.16
| 783.7
| 11/7
| 11/7
|-
|-
| 19
| 19
| 962.11
| 960.6
|  
| 45/26
|-
|-
| 20
| 20
| 1139.06
| 1137.5
| 27/14
| 27/14
|-
| 21
| 116.02
| 15/14
|}
|}
<nowiki/>* in 13-limit POTE tuning
<nowiki/>* in 13-limit CWE tuning


== Tunings ==
== Tunings ==

Revision as of 12:12, 15 April 2026

Modus
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis 64/63, 4375/4374 (7-limit);
64/63, 100/99, 243/242 (11-limit)
64/63, 78/77, 100/99, 144/143
(13-limit)
Reduced mapping ⟨1; 4 9 -8 10 -2]
ET join 27e & 34d
Generators (CWE) ~10/9 = 176.8 ¢
MOS scales 6L 1s, 7L 6s, 7L 13s, 7L 20s
Ploidacot tetracot
Minimax error 9-odd-limit: 13.6 ¢;
13-odd-limit: 16.7 ¢
Target scale size 9-odd-limit: 20 notes;
13-odd-limit: 20 notes

The modus temperament is one of the 7-limit extensions of tetracot, the 5-limit temperament tempering out the tetracot comma (20000/19683), and is naturally a full 13-limit temperament.

In addition to the tetracot comma, modus tempers out 64/63, making it a member of the archytas clan. As such, septimal intervals are tempered together with Pythagorean intervals; in particular, a stack of two perfect fifths octave reduced represents 8/7~9/8 at 8 generator steps. Modus also tempers out 4375/4374, making it a ragismic temperament. In the 11- and 13-limit it can be viewed as a weak extension of suhajira as well.

Additionally, the generator can be taken to represent 21/19, which gives us an extension for prime 19 at -5 generator steps.

See Tetracot family #Modus for technical data.

Interval chain

# Cents* Approximate ratios
0 0.0 1/1
1 176.9 10/9, 11/10
2 353.7 11/9, 16/13
3 530.6 15/11
4 707.5 3/2
5 884.4 5/3
6 1061.2 11/6, 13/7, 24/13
7 38.1 36/35, 40/39, 45/44, 55/54
8 215.0 8/7, 9/8
9 391.9 5/4
10 568.7 11/8, 18/13
11 745.6 20/13
12 922.5 12/7, 22/13
13 1099.4 15/8, 40/21
14 76.2 22/21, 25/24, 27/26
15 253.1 15/13
16 430.0 9/7
17 606.8 10/7
18 783.7 11/7
19 960.6 45/26
20 1137.5 27/14

* in 13-limit CWE tuning

Tunings

Tuning spectrum

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

Music

See Tetracot #Music.

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