Xenharmonic Wiki

Modus: Difference between revisions

← Older edit
VisualWikitext
Complete interval table
ArrowHead294 (talk | contribs)
15,213 edits
mNo edit summary
 
(4 intermediate revisions by one other user not shown)
Line 8: Line 8:
| Generators tuning = 176.8
| Generators tuning = 176.8
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[6L 1s]], [[7L 6s]], [[7L 13s]], [[7L 20s]]
| MOS scales = [[6L 1s]], [[7L 6s]], [[7L 13s]], [[7L 20s]]
| Odd limit 1 = 9 | Mistuning 1 = 13.6 | Complexity 1 = 20
| Odd limit 1 = 9 | Mistuning 1 = 13.6 | Complexity 1 = 20
| Odd limit 2 = 13 | Mistuning 2 = 16.7 | Complexity 2 = 20
| Odd limit 2 = 13 | Mistuning 2 = 16.7 | Complexity 2 = 20
Line 14: Line 14:
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.  
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.  
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.  
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.  
See [[Tetracot family #Modus]] for technical data.  


== Interval chain ==
== Interval chain ==
In the following tables, odd harmonics 1–13 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
Line 114: Line 118:


== 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-3"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|
| 11/10
| 11/10
| 165.004
| 165.004
|  
|  
|-
|-
| 1\7
|
| 171.429
|
|-
|
| 11/9
| 11/9
| 173.704
| 173.704
|  
|  
|-
|-
| 12/11
|  
| 11/6
| 174.894
| 174.894
|  
|  
|-
|-
|
| 11/8
| 11/8
| 175.132
| 175.132
|  
|  
|-
|-
| 4/3
|  
| 3/2
| 175.489
| 175.489
|  
|  
|-
|-
|
| 13/11
| 13/11
| 175.899
| 175.899
|  
|  
|-
|-
| 16/15
|  
| 15/8
| 176.021
| 176.021
|  
|  
|-
|-
|
| 5/4
| 5/4
| 176.257
| 176.257
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 18/13
|  
| 13/9
| 176.338
| 176.338
|  
|  
|-
|-
| 5\34
|
| 176.471
| 34d val, lower bound of 7- to 15-odd-limit diamond monotone
|-
|
| 15/13
| 15/13
| 176.516
| 176.516
|  
|  
|-
|-
| 14/11
|  
| 11/7
| 176.805
| 176.805
| 11, 13 and 15-odd-limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|-
| 6/5
|  
| 5/3
| 176.872
| 176.872
|  
|  
|-
|-
|
| 13/10
| 13/10
| 176.890
| 176.890
|  
|  
|-
|-
|
| 13/12
| 13/12
| 176.905
| 176.905
|  
|  
|-
|-
| 9\61
|
| 177.049
| 61de val
|-
|
| 15/14
| 15/14
| 177.116
| 177.116
|  
|  
|-
|-
|
| 9/7
| 9/7
| 177.193
| 177.193
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
|
| 7/5
| 7/5
| 177.499
| 177.499
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| 7/6
| 7/6
| 177.761
| 177.761
|  
|  
|-
|-
| 14/13
| 4\27
|
| 177.778
| 27e val, upper bound of 11- to 15-odd-limit diamond monotone
|-
|
| 13/7
| 178.617
| 178.617
|  
|  
|-
|-
| 8/7
|  
| 7/4
| 178.897
| 178.897
|  
|  
|-
|-
|
| 15/11
| 15/11
| 178.984
| 178.984
|  
|  
|-
|-
| 16/13
|  
| 13/8
| 179.736
| 179.736
|  
|  
|-
|-
| 10/9
| 3\20
|
| 180.000
| 20ce val, upper bound of 7- and 9-odd-limit diamond monotone
|-
|
| 9/5
| 182.404
| 182.404
|  
|  
|}
|}
<nowiki/>* Besides the octave


== Music ==
== Music ==
See [[Tetracot #Music]].  
See [[Tetracot #Music]].  
== References ==


[[Category:Modus| ]] <!-- main article -->
[[Category:Modus| ]] <!-- main article -->
Retrieved from "https://en.xen.wiki/w/Modus"