Modus: Difference between revisions

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Complete tuning spectrum
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== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 176.8176{{c}}
| CWE: ~10/9 = 177.1188{{c}}
| POTE: ~10/9 = 177.2035{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 176.4456{{c}}
| CWE: ~10/9 = 176.9286{{c}}
| POTE: ~10/9 = 177.0530{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 176.4708{{c}}
| CWE: ~10/9 = 176.8735{{c}}
| POTE: ~10/9 = 176.9532{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"

Revision as of 12:29, 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

In the following tables, odd harmonics 1–13 and their inverses are in bold.

# 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

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 176.8176 ¢ CWE: ~10/9 = 177.1188 ¢ POTE: ~10/9 = 177.2035 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 176.4456 ¢ CWE: ~10/9 = 176.9286 ¢ POTE: ~10/9 = 177.0530 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 176.4708 ¢ CWE: ~10/9 = 176.8735 ¢ POTE: ~10/9 = 176.9532 ¢

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
11/8 175.132
3/2 175.489
13/11 175.899
15/8 176.021
5/4 176.257 5-odd-limit minimax
13/9 176.338
5\34 176.471 34d val, lower bound of 7- to 15-odd-limit diamond monotone
15/13 176.516
11/7 176.805 11-, 13- and 15-odd-limit minimax
5/3 176.872
13/10 176.890
13/12 176.905
9\61 177.049 61de val
15/14 177.116
9/7 177.193 9-odd-limit minimax
7/5 177.499 7-odd-limit minimax
7/6 177.761
4\27 177.778 27e val, upper bound of 11- to 15-odd-limit diamond monotone
13/7 178.617
7/4 178.897
15/11 178.984
13/8 179.736
3\20 180.000 20ce val, upper bound of 7- and 9-odd-limit diamond monotone
9/5 182.404

Music

See Tetracot #Music.

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