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 [[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.
'''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 ==
Line 40: Line 40:
| 386.5
| 386.5
| '''5/4'''
| '''5/4'''
| 96/77
|  
|-
|-
| 3
| 3
Line 50: Line 50:
| 773.0
| 773.0
| '''25/16'''
| '''25/16'''
| 39/25, 120/77
| 39/25
|-
|-
| 5
| 5
Line 74: Line 74:
| 9
| 9
| 539.2
| 539.2
| 175/128
|  
| 15/11
| 15/11
|-
|-
Line 95: Line 95:
| 112.1
| 112.1
| '''16/15'''
| '''16/15'''
| 77/72
|  
|-
|-
| 14
| 14
Line 125: Line 125:
| 71.5
| 71.5
| 25/24
| 25/24
| 26/25, 80/77
| 26/25
|-
|-
| 20
| 20
Line 149: Line 149:
| 24
| 24
| 1037.7
| 1037.7
| 175/96
|  
| 20/11
| 20/11
|-
|-
Line 162: Line 162:
{{Main| Chords of hemithirds }}
{{Main| Chords of hemithirds }}


== Tuning spectrum ==
== 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"
{| class="wikitable center-all left-4"
! EDO<br />generator
! EDO<br />generator
Line 171: Line 177:
| [[25edo|4\25]]
| [[25edo|4\25]]
|  
|  
| 192.000
| 192.0000
| '''Lower bound of 7- and 9-odd-limit diamond monotone'''
| 25ef val, '''lower bound of 7- and 9-odd-limit diamond monotone'''
|-
|-
| [[81edo|13\81]]
| [[81edo|13\81]]
|  
|  
| 192.593
| 192.5926
| 81bef val
|-
| [[56edo|9\56]]
|
| 192.8571
| '''Lower bound of 11- to 15-odd-limit diamond monotone'''
|-
|
| [[32/21]]
| 192.9219
|  
|  
|-
|-
| [[56edo|9\56]]
|  
|  
| 192.857
| [[40/21]]
| 192.9601
|  
|  
|-
|-
| [[143edo|23\143]]
| [[143edo|23\143]]
|  
|  
| 193.007
| 193.0070
|
|-
|
| [[64/63]]
| 193.0906
|  
|  
|-
|-
| [[87edo|14\87]]
| [[87edo|14\87]]
|  
|  
| 193.103
| 193.1034
|  
|  
|-
|-
|  
|  
| [[5/4]]
| [[5/4]]
| 193.157
| 193.1569
| 1/2-comma
| 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)
|-
|-
| [[205edo|33/205]]
|  
|  
| 193.171
| [[16/15]]
| 205d val (hemithirds) <br /> ↓ Lunatic
| 193.2101
|  
|-  
|-  
| [[118edo|19\118]]
| [[118edo|19\118]]
|  
|  
| 193.220
| 193.2203
|  ↑ Lunatic
|  ↑ 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]]
| [[31edo|5\31]]
|  
|  
| 193.548
| 193.5484
| '''Upper bound of 9-odd-limit diamond monotone'''
| '''Upper bound of 9- to 15-odd-limit diamond monotone'''
|-
|
| [[7/4]]
| 193.7652
| 2/5-comma didacus
|-
|-
| [[37edo|6\37]]
| [[37edo|6\37]]
|  
|  
| 194.595
| 194.5946
| 37b val, '''upper bound of 7-odd-limit diamond monotone'''
| 37beff val, '''upper bound of 7-odd-limit diamond monotone'''
|-
|
| [[28/25]]
| 196.1985
| Untempered didacus
|}
|}
<nowiki />* Besides the octave
<nowiki />* Besides the octave


Gencom: [2 28/25; 196/195 352/351 385/384 625/624]
=== 15-odd-limit eigenmonzos ===
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
Gencom mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
|+ style="font-size: 105%; white-space: nowrap;" | 15-odd-limit eigenmonzos of 13-limit hemithirds
 
{| class="wikitable center-all left-3"
|-
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
Line 324: 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]]

Latest revision as of 09:21, 9 February 2026

Hemithirds
Subgroups 2.3.5.7
Comma basis 1029/1024, 3136/3125 (2.3.5.7)
Reduced mapping ⟨1; -15 2 5]
ET join 31 & 87
Generators (CWE) ~28/25 = 193.2 ¢
MOS scales 1L 5s, 6L 1s, 6L 7s, 6L 13s, 6L 19s, 25L 6s
Ploidacot iota-beta-15-cot
Pergen (P8, ccP4/15)
Color name Latrizo & Zozoquinguti
Minimax error 7-odd-limit: 2.5 ¢;
7-limit 25-odd-limit: 3.6 ¢
Target scale size 7-odd-limit: 56 notes;
7-limit 25-odd-limit: 87 notes

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 cents) 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.

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).

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.

# Cents* 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
4 773.0 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

* In CWE 7-limit hemithirds tuning

Chords

Main article: 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.

EDO
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Comments
4\25 192.0000 25ef val, lower bound of 7- and 9-odd-limit diamond monotone
13\81 192.5926 81bef val
9\56 192.8571 Lower bound of 11- to 15-odd-limit diamond monotone
32/21 192.9219
40/21 192.9601
23\143 193.0070
64/63 193.0906
14\87 193.1034
5/4 193.1569 1/2-comma didacus
33\205 193.1707 205d val (hemithirds)
↑ Hemithirds
↓ Lunatic
52\323 193.1889
25/24 193.1933
71\441 193.1973
5/3 193.1976 5-odd-limit minimax tuning
10/9 193.2001
90\559 193.2021
4/3 193.2030 (2.3.7) 21- and 27-odd-limit minimax tuning (hemithirds)
16/15 193.2101
19\118 193.2203 ↑ Lunatic
↓ Hemithirds
28/27 193.2592 2.3.7 CEE tuning
14/9 193.2833 9-odd-limit minimax tuning
24\149 193.2886 149f val
29\180 193.3333 180ef val
7/6 193.3435 7-odd-limit minimax tuning
28/15 193.3643
34\211 193.3649 211eff val
49/48 193.4279
5\31 193.5484 Upper bound of 9- to 15-odd-limit diamond monotone
7/4 193.7652 2/5-comma didacus
6\37 194.5946 37beff val, upper bound of 7-odd-limit diamond monotone
28/25 196.1985 Untempered didacus

* Besides the octave

15-odd-limit eigenmonzos

15-odd-limit eigenmonzos of 13-limit hemithirds
Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
14/13 192.872
12/11 192.948
15/11 192.995
13/10 193.058
16/13 193.066
13/11 193.094
15/13 193.118
13/12 193.120
11/8 193.122
11/10 193.125
18/13 193.144
5/4 193.157
6/5 193.198 5-odd-limit minimax
10/9 193.200
4/3 193.203
16/15 193.210
14/11 193.241 11-odd-limit minimax
9/7 193.283 9-odd-limit minimax
7/6 193.344 7-odd-limit minimax
15/14 193.364
11/9 193.426
8/7 193.765
7/5 194.171
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