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#redirect [[Tetracot family #Modus]]
{{Infobox regtemp
| Title = 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);<br>[[64/63]], [[100/99]], [[243/242]] (11-limit)<br>[[64/63]], [[78/77]], [[100/99]], [[144/143]]<br>(13-limit)
| Edo join 1 = 27e | Edo join 2 = 34d
| Mapping = 1; 4 9 -8 10 -2
| Generators = 10/9
| Generators tuning = 176.8
| Optimization method = CWE
| MOS scales = [[6L 1s]], [[7L 6s]], [[7L 13s]], [[7L 20s]]
| Odd limit 1 = 9 | Mistuning 1 = 13.6 | Complexity 1 = 20
| Odd limit 2 = 13 | Mistuning 2 = 16.7 | Complexity 2 = 20
}}
The '''modus''' [[regular temperament|temperament]] is one of the [[7-limit]] [[extension]]s 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 reduction|octave reduced]] represents [[8/7]][[~]][[9/8]] at 8 generator steps. Modus also tempers out [[4375/4374]], making it a [[ragismic microtemperaments|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 ==
{| class="wikitable right-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.00
| '''1/1'''
|-
| 1
| 176.95
| 10/9, 11/10
|-
| 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, 13/7, 24/13
|-
| 7
| 38.67
| 36/35, 40/39, 45/44, 55/54
|-
| 8
| 215.63
| '''8/7''', '''9/8'''
|-
| 9
| 392.58
| '''5/4'''
|-
| 10
| 569.53
| '''11/8''', 18/13
|-
| 11
| 746.49
| 20/13
|-
| 12
| 923.44
| 12/7, 22/13
|-
| 13
| 1100.39
| 15/8, 40/21
|-
| 14
| 77.35
| 22/21, 25/24, 27/26
|-
| 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
|}
<nowiki/>* in 13-limit POTE tuning
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-3"
|-
! [[Eigenmonzo|Eigenmonzo<br>(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]].
[[Category:Modus| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Tetracot family]]
[[Category:Tetracot family]]
[[Category:Archytas clan]]
[[Category:Archytas clan]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]

Revision as of 12:05, 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 0.00 1/1
1 176.95 10/9, 11/10
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, 13/7, 24/13
7 38.67 36/35, 40/39, 45/44, 55/54
8 215.63 8/7, 9/8
9 392.58 5/4
10 569.53 11/8, 18/13
11 746.49 20/13
12 923.44 12/7, 22/13
13 1100.39 15/8, 40/21
14 77.35 22/21, 25/24, 27/26
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

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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