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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&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]], [[7L&nbsp;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.


[[Category:Temperaments]]
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 {{nowrap|[[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.
 
Modus was named by [[Mike Battaglia]] in 2012 for its fantastic [[modmos]] structures<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_102416.html#102467 Yahoo! Tuning Group | ''Guaranteed meantone successor'']</ref>.
 
See [[Tetracot family #Modus]] for technical data.
 
== Interval chain ==
In the following tables, odd harmonics 1–13 and their inverses are in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
! #
! 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
|}
<nowiki/>* in 13-limit CWE tuning
 
== 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 ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(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
|
|}
<nowiki/>* Besides the octave
 
== Music ==
See [[Tetracot #Music]].
 
== References ==
 
[[Category:Modus| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Tetracot family]]
[[Category:Tetracot family]]
[[Category:Archytas clan]]
[[Category:Ragismic microtemperaments]]

Latest revision as of 21:00, 16 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.

Modus was named by Mike Battaglia in 2012 for its fantastic modmos structures[1].

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

* Besides the octave

Music

See Tetracot #Music.

References

  1. ↑ Yahoo! Tuning Group | Guaranteed meantone successor
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