Xenharmonic Wiki

Hemififths: Difference between revisions

← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,171 edits
→Tuning spectrum: improve and standardize tuning spectra tables
→Tunings: note the vals
 
(49 intermediate revisions by 9 users not shown)
Line 1: Line 1:
The '''hemififths''' temperament tempers out the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]], and as the name suggests, uses a neutral-third generator. '''Hemif''' is the no-5 subgroup version of hemififths.  
{{About|the regular temperament|the irrational interval of a hemififth|Sqrt(3/2)}}
{{Infobox regtemp
| Title = Hemififths
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[2401/2400]], [[5120/5103]] (7-limit); <br> [[243/242]], [[441/440]], [[896/891]] (11-limit); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (13-limit)
| Edo join 1 = 41 | Edo join 2 = 58
| Mapping = 1; 2 25 13 5 -1
| Generators = 49/40
| Generators tuning = 351.5
| Optimization method = CWE
| Pergen = (P8, P5/2)
| MOS scales = [[3L&nbsp;4s]], [[7L&nbsp;3s]], [[7L&nbsp;10s]], [[17L&nbsp;7s]], [[17L 24s]]
| Odd limit 1 = 9 | Mistuning 1 = 1.90 | Complexity 1 = 41
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 7.77 | Complexity 2 = 41
}}
'''Hemififths''' is a [[regular temperament|temperament]] that uses a neutral third as a [[generator]], just as the name suggests. A stack of 13 generators represents [[7/4]] and a stack of 25 generators represents [[5/4]], [[tempering out]] the breedsma, [[2401/2400]], and the argent comma, [[5120/5103]].  


See [[Breedsmic temperaments #Hemififths]] for more technical data.
It extends fairly naturally to the [[11-limit|11-]] and [[13-limit]] by treating the generator as [[11/9]][[~]][[16/13]]. This lowers the overall accuracy, but supplies more harmonic resources. The no-5 subgroup [[restriction]], called '''hemif''', is also notable. Possible tunings include [[41edo|41-]], [[58edo|58-]], and [[99edo]] (using the 99ef val in the 13-limit).
 
Hemififths was named by [[Gene Ward Smith]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html Yahoo! Tuning Group (Archive) | ''Names for important high-complexity temperaments'']</ref>.
 
See [[Breedsmic temperaments #Hemififths]] and [[No-fives subgroup temperaments #Hemif]] for more technical data.


== Interval chain ==
== Interval chain ==
In the following table, prime harmonics are labeled in '''bold'''.  
In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.  
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | #
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! colspan="2" | Approximate Ratios
! colspan="2" | Approximate ratios
|-
|-
! 7-limit
! 7-limit
! 13-limit Extension
! 13-limit extension
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| '''1/1'''
|
|  
|-
|-
| 1
| 1
| 351.477
| 351.5
| 49/40, 60/49
| 49/40, 60/49
| 11/9, '''16/13''', 27/22, 39/32
| 11/9, '''16/13''', 27/22, 39/32
|-
|-
| 2
| 2
| 702.955
| 702.9
| '''3/2'''
| '''3/2'''
|
|
|-
|-
| 3
| 3
| 1054.432
| 1054.4
| 90/49
| 90/49
| 11/6, 24/13
| 11/6, 24/13
|-
|-
| 4
| 4
| 205.910
| 205.9
| 9/8
| '''9/8'''
|  
|  
|-
|-
| 5
| 5
| 557.387
| 557.3
| 112/81
| 112/81
| '''11/8''', 18/13
| '''11/8''', 18/13
|-
|-
| 6
| 6
| 908.865
| 908.8
| 27/16
| 27/16
| 22/13
| 22/13
|-
|-
| 7
| 7
| 60.342
| 60.3
| 28/27
| 28/27
| 33/32, 27/26
| 33/32, 27/26
|-
|-
| 8
| 8
| 411.819
| 411.7
| 81/64, 80/63
| 80/63, 81/64
| 14/11, 33/26
| 14/11, 33/26
|-
|-
| 9
| 9
| 763.297
| 763.2
| 14/9
| 14/9
|  
|  
|-
|-
| 10
| 10
| 1114.774
| 1114.7
| 40/21
| 40/21
| 21/11
| 21/11
|-
|-
| 11
| 11
| 266.252
| 266.1
| 7/6
| 7/6
|  
|  
|-
|-
| 12
| 12
| 617.729
| 617.6
| 10/7
| 10/7
|  
|  
|-
|-
| 13
| 13
| 969.206
| 969.1
| '''7/4'''
| '''7/4'''
|  
|  
|-
|-
| 14
| 14
| 120.684
| 120.5
| 15/14
| 15/14
| 14/13
| 14/13
|-
|-
| 15
| 15
| 472.161
| 472.0
| 21/16
| '''21/16'''
|  
|  
|-
|-
| 16
| 16
| 823.639
| 823.5
| 45/28
| 45/28
| 21/13
| 21/13
|-
|-
| 17
| 17
| 1175.116
| 1174.9
| 63/32, 160/81
| 63/32, 160/81
|
| 55/28, 65/33, 77/39
|-
|-
| 18
| 18
| 326.594
| 326.4
| 98/81, 135/112
| 98/81, 135/112
| 40/33
| 40/33
|-
|-
| 19
| 19
| 678.071
| 677.9
| 40/27
| 40/27
|  
|  
|-
|-
| 20
| 20
| 1029.549
| 1029.3
| 49/27
| 49/27
| 20/11
| 20/11
|-
|-
| 21
| 21
| 181.026
| 180.8
| 10/9
| 10/9
|  
|  
|-
|-
| 22
| 22
| 532.503
| 532.3
| 49/36
| 49/36
| 15/11
| 15/11
|-
|-
| 23
| 23
| 883.981
| 883.7
| 5/3
| 5/3
|  
|  
|-
|-
| 24
| 24
| 35.458
| 35.2
| 49/48, 50/49
| 49/48, 50/49
| 45/44, 55/54
| 40/39, 45/44, 55/54, 65/64
|-
|-
| 25
| 25
| 386.936
| 386.7
| '''5/4'''
| '''5/4'''
|  
|  
|-
| 26
| 738.1
| 49/32
| 20/13
|-
| 27
| 1089.6
| '''15/8'''
|
|-
| 28
| 241.1
| 147/128
| 15/13
|-
| 29
| 592.5
| 45/32
|
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced
=== As a detemperament of 17et ===
[[File: Hemififths 17et Detempering.png|thumb|Hemififths as a 58-tone 17et detempering]]
Hemififths is very naturally considered as a [[detemperament]] of the [[17edo|17 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -28 to +29. 58 is the largest number of tones for a mos where intervals in the 17 categories do not overlap. Each category may be further divided into "sub", "plain" and "super" qualities, separated by the commatic step of -17 generator steps, which represents [[56/55]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], [[91/90]], [[99/98]], [[121/120]], and [[169/168]]. Combining this division with the minor, neutral, and major qualities of the 17 equal temperament, hemififths gives us at least ''nine'' qualities for each diatonic category: subminor, minor, supraminor, subneutral, neutral, supraneutral, submajor, major, and supermajor.
Notice also the little interval between the largest of a category and the smallest of the next. This interval separates supraminor from subneutral and supraneutral from submajor, and spans 41 generator steps. 41edo tempers it out so that it conflates supraminor with subneutral and supraneutral with submajor, whereas 58edo exaggerates it to the size of the comma. 99edo tunes it to one half the size of the commatic step, which can be seen as a good compromise.
== Notation ==
Hemififths can be notated in [[neutral chain-of-fifths notation]], in which case 5/4 is represented by a sesqui-augmented second (C–D{{sesquisharp2}}), and 7/4 by a semi-augmented sixth (C–A{{demisharp2}}). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C–F{{demisharp2}}), and 13/8 by the neutral sixth (C–A{{demiflat2}}). This, of course, defies the tradition of tertian harmony, as the [[just major triad]] on C is C–D{{sesquisharp2}}–G, for example, so one may want to adopt one or more additional modules of accidentals such as arrows or +/- signs to represent the commatic steps (-17 generator steps, a semidiminished second).
Below is tabulated how to notate each prime harmonic with an arrow representing a commatic step (thus ↑C = D{{sesquiflat2}}).
{| class="wikitable center-1 center-3"
|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
|-
! Ratio
! Nominal
! Example
|-
| 3/2
| Perfect fifth
| C–G
|-
| 5/4
| Down major third
| C–↓E
|-
| 7/4
| Down minor seventh
| C–↓B♭
|-
| 11/8
| Semi-augmented fourth
| C–F{{demisharp2}}
|-
| 13/8
| Neutral sixth
| C–A{{demiflat2}}
|}
|}
<nowiki>*</nowiki> in 7-limit POTE tuning


== Chords ==
=== Ups and downs notation ===
{{Main| Chords of hemififths }}
In [[Kite's ups and downs notation]], the equivalences are vvA1 and v\m2. Let ''c'' be the amount by which the fifth exceeds 7\12, then {{nowrap| ^1 {{=}} 50{{c}} + 3.5''c'' }} and {{nowrap| /1 {{=}} 50{{c}} − 8.5''c'' }}. For 7-limit CWE tuning, {{nowrap| ''c'' {{=}} 2.934{{c}} }}.
 
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Ups and downs<br>notation
! Associated ratios
|-
| 0
| 0.0
| P1
| 1/1
|-
| 1
| 351.5
| ~3 = ^m3 = vM3
| 11/9~16/13
|-
| 2
| 702.9
| P5
| 3/2
|-
| 3
| 1054.4
| ~7 = ^m7 = vM7
| 11/6~24/13
|-
| 4
| 205.9
| M2
| 9/8
|-
| 5
| 557.3
| ~4 = ^4 = vA4
| 11/8~18/13
|-
| 6
| 908.8
| M6
| 22/13~27/16
|-
| 7
| 60.3
| ^1 = \m2
| 27/26~33/32
|-
| 8
| 411.7
| M3
| 14/11~33/26
|-
| 9
| 763.2
| ^5 = \m6
| 14/9
|-
| 10
| 1114.7
| M7
| 21/11~40/21
|-
| 11
| 266.1
| ^M2 = \m3
| 7/6
|-
| 12
| 617.6
| A4 = \~5
| 10/7
|-
| 13
| 969.1
| ^M6 = \m7
| 7/4
|-
| 14
| 120.5
| A1 = \~2
| 14/13~15/14
|-
| 15
| 472.0
| ^M3 = \4
| 21/16
|-
| 16
| 823.5
| A5 = \~6
| 21/13
|-
| 17
| 1174.9
| ^M7 = \8
| 63/32~160/81
|-
| 18
| 326.4
| A2 = \~3
| 40/33
|-
| 19
| 677.9
| ^A4 = \5
| 40/27
|-
| 20
| 1029.3
| A6 = \~7
| 20/11
|-
| 21
| 180.8
| ^A1 = \M2
| 10/9
|-
| 22
| 532.3
| A3 = \~4
| 15/11
|-
| 23
| 883.7
| ^A5 = \M6
| 5/3
|-
| 24
| 35.2
| A7 - P8 = -d2 = ^\1
| 49/48~50/49
|-
| 25
| 386.7
| ^A2 = \M3
| 5/4
|-
| 26
| 738.1
| AA4 = ^\5
| 20/13
|-
| 27
| 1089.6
| ^A6 = \M7
| 15/8
|-
| 28
| 241.1
| AA1= ^\2
| 15/13
|-
| 29
| 592.5
| ^A3 = \A4
| 45/32
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced
 
== Chords and harmony ==
{{See also| Chords of hemififths }}


== Scales ==
== Scales ==
Line 153: Line 395:
* [[Hemif17]]
* [[Hemif17]]


== Tuning spectrum ==
== 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
|-
! Equilateral
| CEE: ~49/40 = 351.4464{{c}}
| CSEE: ~49/40 = 351.4671{{c}}
| POEE: ~49/40 = 351.4774{{c}}
|-
! Tenney
| CTE: ~49/40 = 351.4492{{c}}
| CWE: ~49/40 = 351.4639{{c}}
| POTE: ~49/40 = 351.4834{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~49/40 = 351.4447{{c}}
| CSBE: ~49/40 = 351.4675{{c}}
| POBE: ~49/40 = 351.4787{{c}}
|}


Gencom: [2 11/9; 144/143 196/195 243/242 364/363]
{| 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
|-
! Equilateral
| CEE: ~11/9 = 351.4230{{c}}
| CSEE: ~11/9 = 351.5800{{c}}
| POEE: ~11/9 = 351.6627{{c}}
|-
! Tenney
| CTE: ~11/9 = 351.4331{{c}}
| CWE: ~11/9 = 351.5438{{c}}
| POTE: ~11/9 = 351.5734{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~11/9 = 351.4380{{c}}
| CSBE: ~11/9 = 351.5144{{c}}
| POBE: ~11/9 = 351.5243{{c}}
|}


Gencom map: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]
=== Tuning spectrum ===
 
{| class="wikitable center-all left-4"
{| class="wikitable center-all"
|-
|-
! ET<br>generator
! Edo<br>generator
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! generator<br>(¢)
! Generator (¢)
! comments
! Comments
|-
|-
|  
|  
Line 172: Line 462:
|-
|-
|  
|  
| 12/11
| 11/6
| 349.788
| 349.788
|  
|  
|-
|-
| 7\24
| [[24edo|7\24]]
|  
|  
| 350.000
| 350.000
|  
| 24c val, lower bound of 7- and 9-odd-limit diamond monotone
|-
|-
|  
|  
Line 187: Line 477:
|-
|-
|  
|  
| 4/3
| 3/2
| 350.978
| 350.978
|  
|  
|-
|-
| 12\41
| [[41edo|12\41]]
|  
|  
| 351.220
| 351.220
| Lower bound of 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
|-
|
| 21/16
| 351.385
|  
|  
|-
|-
Line 202: Line 497:
|-
|-
|  
|  
| 16/15
| 15/8
| 351.417
| 351.417
|  
|  
|-
|-
| 41\140
| [[140edo|41\140]]
|  
|  
| 351.429
| 351.429
|  
| 140ef val
|-
|-
|  
|  
| 8/7
| 7/4
| 351.448
| 351.448
| 7-, 9- and 11-odd-limit hemif minimax
| 7-, 9- and 11-odd-limit hemif minimax
Line 227: Line 522:
|-
|-
|  
|  
| 6/5
| 25/24
| 351.472
| Very close to [[argent tuning]] with neutral intervals (351.47186 cents)
|-
|
| 49/48
| 351.487
|
|-
|
| 5/3
| 351.494
| 351.494
|  
|  
|-
|-
| 29\99
| [[99edo|29\99]]
|  
|  
| 351.515
| 351.515
|  
| 99ef val
|-
|-
|  
|  
Line 242: Line 547:
|-
|-
|  
|  
| 10/9
| 9/5
| 351.543
| 351.543
|
|-
|
| 21/20
| 351.553
|  
|  
|-
|-
Line 259: Line 569:
| 15/13
| 15/13
| 351.705
| 351.705
| 15-odd-limit minimax
| 15-odd-limit and 13-limit 21-odd-limit minimax
|-
|-
| 17\58
| [[58edo|17\58]]
|  
|  
| 351.724
| 351.724
Line 281: Line 591:
| 13- and 15-odd-limit hemif minimax
| 13- and 15-odd-limit hemif minimax
|-
|-
| 22\75
|
| 21/13
| 351.891
|
|-
|
| 21/11
| 351.946
|
|-
| [[75edo|22\75]]
|  
|  
| 352.000
| 352.000
|  
| 75ce val
|-
|-
|  
|  
| 14/13
| 13/7
| 352.021
| 352.021
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 352.188
| 352.188
|  
|  
|-
|-
|  
|  
| 18/13
| 13/9
| 352.676
| 352.676
|  
|  
|-
| [[17edo|5\17]]
|
| 352.941
| 17c val, upper bound of 7- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 307: Line 632:
|-
|-
|  
|  
| 16/13
| 13/8
| 359.472
| 359.472
|  
|  
|}
|}
<nowiki/>* Besides the octave
== References ==
<references/>


{{IoT}}
[[Category:Hemififths| ]] <!-- Main article -->
[[Category:Hemififths| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Aberschismic temperaments]]
[[Category:Hemimage temperaments]]
[[Category:Hemimage temperaments]]
Retrieved from "https://en.xen.wiki/w/Hemififths"