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Luna and hemithirds: Difference between revisions

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{{URWTC}}
{{Infobox regtemp
{{Infobox Regtemp
| Title = Hemithirds
| Title = Hemithirds
| Subgroups = 2.3.5.7
| Subgroups = 2.3.5.7
| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)
| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7)
| Edo join 1 = 31 | Edo join 2 = 56
| Edo join 1 = 31 | Edo join 2 = 87
| Generator = 28/25 | Generator tuning = 193.239 | Optimization method = CWE
| 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]]
| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]
| Mapping = 1; -15 2 5
| Pergen = (P8, ccP4/15)
| Odd limit 1 = 7 | Mistuning 1 = ??? | Complexity 1 = ???
| Color name = Latrizo & Zozoquinguti
| Odd limit 2 = (7-limit) 25 | Mistuning 2 = ??? | Complexity 2 = ???
| 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.


The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, in the range between [[25edo]] and [[31edo]] tuning, 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 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 [[1001/1000]] to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.
'''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).
 
'''Luna''' is a restriction of hemithirds to the [[5-limit]] that is a [[microtemperament]], supported by such high-precision tuning systems as [[118edo]] 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).


See [[Hemimean clan #Hemithirds]] and [[Luna family #Luna]] for more information.  
See [[Hemimean clan #Hemithirds]] and [[Luna family #Luna]] for more information.


== Intervals ==
== Intervals ==
In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.  
In the following table, odd harmonics and subharmonics 1–39 are labeled in '''bold'''.  


{| class="wikitable sortable center-all right-2"
{| class="wikitable sortable center-all right-2"
Line 35: Line 35:
| 193.2
| 193.2
| 28/25, 125/112
| 28/25, 125/112
|
| 39/35
|-
|-
| 2
| 2
| 386.5
| 386.5
| '''5/4'''
| '''5/4'''
|
|  
|-
|-
| 3
| 3
| 579.7
| 579.7
| 7/5
| 7/5
|
| 39/28, 88/63
|-
|-
| 4
| 4
| 773.0
| 773.0
| '''25/16'''
| '''25/16'''
|
| 39/25
|-
|-
| 5
| 5
| 966.2
| 966.2
| '''7/4'''
| '''7/4'''
|
| 96/55, 110/63
|-
|-
| 6
| 6
| 1159.4
| 1159.4
| 49/25, 125/64
| 49/25, 125/64
|
| 39/20, 88/45
|-
|-
| 7
| 7
| 152.7
| 152.7
| '''35/32'''
| '''35/32'''
|
| 12/11
|-
|-
| 8
| 8
| 345.9
| 345.9
| 49/40, 128/105
| 49/40, 128/105
|
| 11/9, '''39/32'''
|-
|-
| 9
| 9
| 539.2
| 539.2
| 175/128
|  
|
| 15/11
|-
|-
| 10
| 10
| 732.4
| 732.4
| '''32/21''', 49/32
| '''32/21''', 49/32
|
| 55/36, 84/55
|-
|-
| 11
| 11
| 925.6
| 925.6
| 128/75
| 128/75
|
| 77/45
|-
|-
| 12
| 12
| 1118.9
| 1118.9
| 40/21
| 40/21
|
| 21/11
|-
|-
| 13
| 13
| 112.1
| 112.1
| '''16/15'''
| '''16/15'''
|
|  
|-
|-
| 14
| 14
Line 110: Line 110:
| 691.8
| 691.8
| 112/75
| 112/75
|
| 52/35
|-
|-
| 17
| 17
| 885.1
| 885.1
| 5/3
| 5/3
|
| 128/77
|-
|-
| 18
| 18
| 1078.3
| 1078.3
| 28/15
| 28/15
|
| 13/7
|-
|-
| 19
| 19
| 71.5
| 71.5
| 25/24
| 25/24
|
| 26/25
|-
|-
| 20
| 20
| 264.8
| 264.8
| 7/6
| 7/6
|
| 64/55
|-
|-
| 21
| 21
| 458.0
| 458.0
| 98/75, 125/96
| 98/75, 125/96
|
| 13/10
|-
|-
| 22
| 22
| 651.3
| 651.3
| 35/24
| 35/24
|
| '''16/11'''
|-
|-
| 23
| 23
| 844.5
| 844.5
| 49/30
| 49/30
|
| '''13/8''', 44/27
|-
|-
| 24
| 24
| 1037.7
| 1037.7
| 175/96
|  
|
| 20/11
|-
|-
| 25
| 25
| 31.0
| 31.0
| 64/63, 49/48
| 64/63, 49/48
|
| 55/54, 56/55, 65/64
|}
|}
<nowiki />* In [[CWE]] 7-limit hemithirds tuning
<nowiki />* In [[CWE]] 7-limit hemithirds tuning
Line 162: Line 162:
{{Main| Chords of hemithirds }}
{{Main| Chords of hemithirds }}


== Tuning spectrum ==
== Tunings ==
Gencom: [2 28/25; 196/195 352/351 385/384 625/624]
 
=== Tuning spectrum ===
{{see also|Didacus #Tuning spectrum}}
 
Vals are for 13-limit hemithirds and 7-limit lunatic in their respective ranges.


Gencom mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
{| 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


{| class="wikitable center-all left-3"
=== 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)]]
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
Line 266: Line 433:
|}
|}


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