Xenharmonic Wiki

Augene: Difference between revisions

← Older edit
VisualWikitext
Frostburn (talk | contribs)
autopatrolled, emailconfirmed
322 edits
Fix typo MOS sequence -> optimal ET sequence. Add link.
→Tunings: + prime-optimized tunings
 
(11 intermediate revisions by 4 users not shown)
Line 1: Line 1:
Augene is a regular temperament of the [[Augmented_family#x-Augene|augmented family]], which means that 128/125 is tempered out and three 5/4s make a 2/1. Augene is distinguished from its relative [[august|august]] by tempering out 64/63, which means that tunings in which the fifth is larger than 7\12 (700 cents) are optimal. Therefore the [[optimal ET sequence]] of augene goes 12, 15, 27... in constrast to august's which goes 9, 12, 21...
'''Augene''' is a [[regular temperament]] of the [[augmented family]], which means that [[128/125]] is [[tempering out|tempered out]] and three [[5/4]]'s make a [[2/1]]. Augene is distinguished from its relative [[august]] by tempering out [[64/63]], which means that tunings in which the fifth is larger than 7\12 (700 cents) are optimal. Therefore the [[optimal ET sequence]] of augene goes 12, 15, 27, … in constrast to august's which goes 9, 12, 21, ….


The first few augene MOSes are [[3L_3s|3L 3s]], [[3L_6s|3L 6s]], [[3L_9s|3L 9s]], [[12L_3s|12L 3s]]... and the first EDOs that reasonably support augene are [[12edo|12edo]], [[15edo|15edo]], and especially [[27edo|27edo]].
It is also the unique [[7-limit]] [[regular temperament]] that tempers out the [[essential tempering commas]] of the [[5-odd-limit|5-]] ([[128/125]]) and [[7-odd-limit]] ([[64/63]], [[126/125]]).


== 11-limit interval chain ==
The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]].
{| class="wikitable"
 
|+Generator based off of 11-limit minimax
See [[Augmented family #Augene]] for technical data. See [[Augene extensions]] for a discussion on 13-limit extensions.
!Gen.
 
!Period 1
== Interval chain ==
!Period 2
In the following table, odd harmonics 1–11 are in '''bold'''.
!Period 3 (0)
 
{| class="wikitable center-1 right-2 right-4 right-6"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
|-
|0
! Cents*
|400.000
! Approx. ratios
|800.000
! Cents*
|1200.000
! Approx. ratios
! Cents*
! Approx. ratios
|-
|-
|1
| 0
|313.148
| 0.0
|713.148
| 1/1
|1113.148
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
|-
|2
| 1
|226.296
| 1113.1
|626.296
| 15/8, 40/21
|1026.296
| 313.1
| 6/5
| 713.1
| '''3/2'''
|-
| 2
| 1026.3
| 9/5, 20/11
| 226.3
| '''8/7''', '''9/8'''
| 626.3
| 10/7, '''16/11'''
|-
|-
|3
| 3
|139.444
| 939.4
|539.444
| 12/7
|939.444
| 139.4
| 12/11, 15/14
| 539.4
| 15/11
|-
|-
|4
| 4
|52.592
| 852.6
|452.592
| 18/11
|852.592
| 52.6
| 45/44
| 452.6
| 9/7
|-
|-
|5
| 5
|1165.740
| 765.7
|365.740
| 54/35
|765.740
| 1165.7
| 27/14, 96/49
| 365.7
| 27/22
|}
|}
<nowiki/>* In 11-odd-limit minimax tuning, octave reduced


== Tuning spectrum ==
== Tunings ==
 
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable center-all"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|-
|-
! | ET<br>generator
! rowspan="2" |  
! | [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! colspan="3" | Euclidean
! | fifth<br>(¢)
! | comments
|-
|-
| |
! Constrained
| | 14/13
! Constrained & skewed
| | 671.702
! Destretched
| |
|-
|-
| | 12\28
! Tenney
| |  
| CTE: ~3/2 = 709.5949{{c}}
| | 685.714
| CWE: ~3/2 = 709.3249{{c}}
| |  
| POTE: ~3/2 = 709.2568{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings
|-
|-
| |
! rowspan="2" |  
| | 16/15
! colspan="3" | Euclidean
| | 688.269
| |
|-
|-
| |
! Constrained
| | 13/11
! Constrained & skewed
| | 689.210
! Destretched
| |
|-
|-
| | 7\12
! Tenney
| |
| CTE: ~3/2 = 713.5701{{c}}
| | 700.000
| CWE: ~3/2 = 711.6031{{c}}
| |
| POTE: ~3/2 = 711.1766{{c}}
|-
|}
| | 31\53
 
| |
=== Tuning spectrum ===
| | 701.887
{| class="wikitable center-all left-4"
| |
|-
| |
| | 4/3
| | 701.955
| |
|-
| | 30\51
| |
| | 705.882
| |
|-
| |
| | 15/14
| | 706.481
| |
|-
|-
| | 23\39
! Edo<br>generator
| |
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
| | 707.692
! Generator (¢)
| |
! Comments
|-
|-
| |  
|  
| | 7/5
| 15/8
| | 708.744
| 688.269
| |
| -1/3 comma
|-
|-
| |
| 7\12
| | 9/7
|  
| | 708.771
| 700.000
| |  
|  
|-
|-
| |  
|  
| | 10/9
| 3/2
| | 708.798
| 701.955
| | 9 limit minimax
| Untempered
|-
|-
| | 39\66
| 30\51
| |  
|  
| | 709.091
| 705.882
| |
| 51cdeee val
|-
|-
| |  
|  
| | 7/6
| 15/14
| | 711.043
| 706.481
| |  
|  
|-
|-
| | 16\27
| 23\39
| |  
|  
| | 711.111
| 707.692
| |
| 39dee val
|-
|-
| |  
|  
| | 15/11
| 7/5
| | 712.317
| 708.744
| |  
|  
|-
|-
| | 57\96
|  
| |
| 9/7
| | 712.5
| 708.771
| |  
|  
|-
|-
| |  
|  
| | 11/9
| 9/5
| | 713.148
| 708.798
| | 11 limit minimax
| 1/6 comma, 9-odd-limit minimax
|-
|-
| | 25\42
| 39\66
| |  
|  
| | 714.286
| 709.091
| |
| 66cdeee val
|-
|-
| |  
|  
| | 8/7
| 7/6
| | 715.587
| 711.043
| | 7, 13 and 15 limit minimax
|  
|-
|-
| |
| 16\27
| | 6/5
|  
| | 715.641
| 711.111
| | 5 limit minimax
| 27e val
|-
|-
| | 34\57
|  
| |
| 15/11
| | 715.789
| 712.317
| |  
|  
|-
|-
| | 25\72
| 41\69
| |  
|  
| | 716.667
| 713.043
| |
| 69bcee val
|-
|-
| |  
|  
| | 12/11
| 11/9
| | 716.879
| 713.148
| |
| 11-odd-limit minimax
|-
|-
| |
| 25\42
| | 11/10
|  
| | 717.498
| 714.286
| |
| 42e val
|-
|-
| | 9\15
|  
| |
| 7/4
| | 720.000
| 715.587
| |
| 7-odd-limit minimax
|-
|-
| |  
|  
| | 18/13
| 5/3
| | 721.127
| 715.641
| |
| 1/3 comma, 5-odd-limit minimax
|-
|-
| |
| 34\57
| | 15/13
|  
| | 723.871
| 715.789
| |
| 53bce val
|-
|-
| |  
|  
| | 11/8
| 11/6
| | 724.341
| 716.879
| |  
|  
|-
|-
| |  
|  
| | 13/12
| 11/10
| | 730.714
| 717.498
| |  
|  
|-
|-
| |
| 9\15
| | 13/10
|  
| | 745.786
| 720.000
| |  
|  
|-
|-
| |  
|  
| | 16/13
| 11/8
| | 759.472
| 724.341
| |  
|  
|}
|}
<nowiki/>* Besides the octave


=Music=
== Music ==
''[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves]'' by [https://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones] in Augene[12] tuned to 27edo
; [[Igliashon Jones]]
* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 ''Sad Like Winter Leaves''] &ndash; in Augene[12] tuned to 27edo


''[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina]'' by [https://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor] in Augene[12] tuned to 27edo
; [[Joel Grant Taylor]]
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] &ndash; in Augene[12] tuned to 27edo


[[Category:Temperaments]]
[[Category:Augene| ]] <!-- Main article -->
[[Category:Augene| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Archytas clan]]
[[Category:Starling temperaments]]
Retrieved from "https://en.xen.wiki/w/Augene"