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Luna and hemithirds: Difference between revisions

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See [[Gamelismic clan #Hemithirds]] or [[Luna family #Hemithirds]].  
{{Infobox regtemp
| Title = Hemithirds
| Subgroups = 2.3.5.7
| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)
| Edo join 1 = 31 | Edo join 2 = 87
| Mapping = 1; -15 2 5
| Generators = 28/25 | Generators tuning = 193.2 | Optimization method = CWE
| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]
| Pergen = (P8, ccP4/15)
| Color name = Latrizo & Zozoquinguti
| Odd limit 1 = 7 | Mistuning 1 = 2.5 | Complexity 1 = 56
| Odd limit 2 = 7-limit 25 | Mistuning 2 = 3.6 | Complexity 2 = 87
}}
The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, which is defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus in the range between [[25edo]] and [[31edo]] tuning, by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.


==Spectrum of Hemithirds Tunings by Eigenmonzos ==
'''Luna''' is a restriction of hemithirds to the [[5-limit]] that is a [[microtemperament]], supported by such high-precision tuning systems as [[323edo]] and [[441edo]]; another notable tuning of luna is [[1000edo]]. It can further be re-extended to the 7-limit in the form of [[lunatic]] by adding [[4375/4374]] to the comma list, but that extension is extremely complex (finding the 7th harmonic at 113 generators down).


{| class="wikitable center-all"
See [[Hemimean clan #Hemithirds]] and [[Luna family #Luna]] for more information.
 
== Intervals ==
In the following table, odd harmonics and subharmonics 1–39 are labeled in '''bold'''.
 
{| class="wikitable sortable center-all right-2"
|-
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
! 7-limit hemithirds !! 13-limit extension
|-
| 0
| 0.0
| '''1/1'''
|
|-
| 1
| 193.2
| 28/25, 125/112
| 39/35
|-
| 2
| 386.5
| '''5/4'''
|
|-
| 3
| 579.7
| 7/5
| 39/28, 88/63
|-
|-
! Eigenmonzo
| 4
! Hemithird
| 773.0
! Comment
| '''25/16'''
| 39/25
|-
| 5
| 966.2
| '''7/4'''
| 96/55, 110/63
|-
| 6
| 1159.4
| 49/25, 125/64
| 39/20, 88/45
|-
| 7
| 152.7
| '''35/32'''
| 12/11
|-
| 8
| 345.9
| 49/40, 128/105
| 11/9, '''39/32'''
|-
| 9
| 539.2
|
| 15/11
|-
| 10
| 732.4
| '''32/21''', 49/32
| 55/36, 84/55
|-
| 11
| 925.6
| 128/75
| 77/45
|-
| 12
| 1118.9
| 40/21
| 21/11
|-
| 13
| 112.1
| '''16/15'''
|
|-
| 14
| 305.3
| 25/21
|
|-
| 15
| 498.6
| '''4/3'''
|
|-
| 16
| 691.8
| 112/75
| 52/35
|-
| 17
| 885.1
| 5/3
| 128/77
|-
| 18
| 1078.3
| 28/15
| 13/7
|-
| 19
| 71.5
| 25/24
| 26/25
|-
| 20
| 264.8
| 7/6
| 64/55
|-
| 21
| 458.0
| 98/75, 125/96
| 13/10
|-
| 22
| 651.3
| 35/24
| '''16/11'''
|-
| 23
| 844.5
| 49/30
| '''13/8''', 44/27
|-
| 24
| 1037.7
|
| 20/11
|-
| 25
| 31.0
| 64/63, 49/48
| 55/54, 56/55, 65/64
|}
<nowiki />* In [[CWE]] 7-limit hemithirds tuning
 
== Chords ==
{{Main| Chords of hemithirds }}
 
== Tunings ==
 
=== Tuning spectrum ===
{{see also|Didacus #Tuning spectrum}}
 
Vals are for 13-limit hemithirds and 7-limit lunatic in their respective ranges.
 
{| class="wikitable center-all left-4"
! EDO<br />generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
! Generator (¢)
! Comments
|-
| [[25edo|4\25]]
|
| 192.0000
| 25ef val, '''lower bound of 7- and 9-odd-limit diamond monotone'''
|-
| [[81edo|13\81]]
|
| 192.5926
| 81bef val
|-
| [[56edo|9\56]]
|
| 192.8571
| '''Lower bound of 11- to 15-odd-limit diamond monotone'''
|-
|
| [[32/21]]
| 192.9219
|
|-
|
| [[40/21]]
| 192.9601
|
|-
| [[143edo|23\143]]
|
| 193.0070
|
|-
|
| [[64/63]]
| 193.0906
|
|-
| [[87edo|14\87]]
|
| 193.1034
|
|-
|
| [[5/4]]
| 193.1569
| 1/2-comma didacus
|-
| [[205edo|33\205]]
|
| 193.1707
| 205d val (hemithirds) <br /> ↑ Hemithirds <br /> ↓ Lunatic
|-
| [[323edo|52\323]]
|
| 193.1889
|
|-
|
| [[25/24]]
| 193.1933
|
|-
| [[441edo|71\441]]
|
| 193.1973
|
|-
|
| [[5/3]]
| 193.1976
| 5-odd-limit minimax tuning
|-
|
| [[10/9]]
| 193.2001
|
|-
| [[559edo|90\559]]
|
| 193.2021
|
|-
|
| [[4/3]]
| 193.2030
| (2.3.7) 21- and 27-odd-limit minimax tuning (hemithirds)
|-
|
| [[16/15]]
| 193.2101
|
|-
| [[118edo|19\118]]
|
| 193.2203
|  ↑ Lunatic <br /> ↓ Hemithirds
|-
|
| [[28/27]]
| 193.2592
| 2.3.7 [[CEE]] tuning
|-
|
| [[14/9]]
| 193.2833
| 9-odd-limit minimax tuning
|-
| [[149edo|24\149]]
|
| 193.2886
| 149f val
|-
| [[180edo|29\180]]
|
| 193.3333
| 180ef val
|-
|
| [[7/6]]
| 193.3435
| 7-odd-limit minimax tuning
|-
|
| [[28/15]]
| 193.3643
|
|-
| [[211edo|34\211]]
|
| 193.3649
| 211eff val
|-
|
| [[49/48]]
| 193.4279
|
|-
| [[31edo|5\31]]
|
| 193.5484
| '''Upper bound of 9- to 15-odd-limit diamond monotone'''
|-
|
| [[7/4]]
| 193.7652
| 2/5-comma didacus
|-
| [[37edo|6\37]]
|
| 194.5946
| 37beff val, '''upper bound of 7-odd-limit diamond monotone'''
|-
|
| [[28/25]]
| 196.1985
| Untempered didacus
|}
<nowiki />* Besides the octave
 
=== 15-odd-limit eigenmonzos ===
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 15-odd-limit eigenmonzos of 13-limit hemithirds
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! Comments
|-
| 14/13
| 192.872
|
|-
|-
| 12/11
| 12/11
| 192.948
| 192.948
|
|-
| 15/11
| 192.995
|
|-
| 13/10
| 193.058
|
|-
| 16/13
| 193.066
|
|-
| 13/11
| 193.094
|
|-
| 15/13
| 193.118
|
|-
| 13/12
| 193.120
|  
|  
|-
|-
Line 19: Line 378:
| 11/10
| 11/10
| 193.125
| 193.125
|
|-
| 18/13
| 193.144
|  
|  
|-
|-
Line 27: Line 390:
| 6/5
| 6/5
| 193.198
| 193.198
| 5 limit minimax
| 5-odd-limit minimax
|-
|-
| 10/9
| 10/9
Line 35: Line 398:
| 4/3
| 4/3
| 193.203
| 193.203
|
|-
| 16/15
| 193.210
|  
|  
|-
|-
| 14/11
| 14/11
| 193.241
| 193.241
| 11 limit minimax
| 11-odd-limit minimax
|-
|-
| 9/7
| 9/7
| 193.283
| 193.283
| 9 limit minimax
| 9-odd-limit minimax
|-
|-
| 7/6
| 7/6
| 193.344
| 193.344
| 7 limit minimax
| 7-odd-limit minimax
|-
| 15/14
| 193.364
|
|-
|-
| 11/9
| 11/9
Line 62: Line 433:
|}
|}


[[Category:Temperament]]
[[Category:Hemithirds| ]] <!-- main article -->
[[Category:Hemithirds]]
[[Category:Rank-2 temperaments]]
[[Category:Luna]]
[[Category:Luna family]]
[[Category:Hemimean clan]]
[[Category:Gamelismic clan]]
Retrieved from "https://en.xen.wiki/w/Luna_and_hemithirds"