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Hemififths: Difference between revisions

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: ''This page is about the regular temperament. For the irrational interval of a hemififth, see [[sqrt(3/2)]].''
{{About|the regular temperament|the irrational interval of a hemififth|Sqrt(3/2)}}
{{Infobox regtemp
{{Infobox regtemp
| Title = Hemififths
| Title = Hemififths
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[2401/2400]], [[5120/5103]] (L7); <br> [[243/242]], [[441/440]], [[896/891]] (L11); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (L13)
| 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)
| Generator = 49/40
| Edo join 1 = 41 | Edo join 2 = 58
| Mapping = 1; 2 25 13 5 -1
| Mapping = 1; 2 25 13 5 -1
| Generators = 49/40
| Generators tuning = 351.5
| Optimization method = CWE
| Pergen = (P8, P5/2)
| Pergen = (P8, P5/2)
| Edo join 1 = 41 | Edo join 2 = 58
| Optimization method = CTE
| Generator tuning = 351.4
| MOS scales = [[3L&nbsp;4s]], [[7L&nbsp;3s]], [[7L&nbsp;10s]], [[17L&nbsp;7s]], [[17L 24s]]
| 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 1 = 9 | Mistuning 1 = 1.90 | Complexity 1 = 41
| Odd limit 2 = (13-limit) 21 | Mistuning 2 = 7.77 | Complexity 2 = 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]].  
'''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]].  


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[[File: Hemififths 17et Detempering.png|thumb|Hemififths as a 58-tone 17et detempering]]
[[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 table below shows a 58-tone detempered scale, with a generator range of -28 to +29. Each interval category of the 17 equal temperament is further divided into "sub", "plain" and "super" qualities, separated by -17 generator steps, which represents the syntonic~septimal comma; the "plain" type here consists of a [[7L 10s]] scale in 8|8 mode. 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.  
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 comma between supraminor and subneutral, and between supraneutral and submajor. This interval 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 syntonic~septimal comma. 99edo tunes it to one half the size of the syntonic~septimal comma, which can be seen as a good compromise.  
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.


{| class="wikitable center-all mw-collapsible mw-collapsed"
== 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).  
! rowspan="2" | #
! rowspan="2" | Interval<br>category
! colspan="3" style="border-left: double;" | "Double-Sub"
! colspan="3" style="border-left: double;" | "Sub"
! colspan="3" style="border-left: double;" | "Plain"
! colspan="3" style="border-left: double;" | "Super"
! colspan="3" style="border-left: double;" | "Double-super"
|-
! style="border-left: double;" | Gen. || Cents* || Ratios
! style="border-left: double;" | Gen. || Cents* || Ratios
! style="border-left: double;" | Gen. || Cents* || Ratios
! style="border-left: double;" | Gen. || Cents* || Ratios
! style="border-left: double;" | Gen. || Cents* || Ratios
|-
| 0
| P1
| style="border-left: double;" |  ||  ||
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 0 || 0.0 || 1/1
| style="border-left: double;" | -17 || 25.9 || 64/63~81/80
| style="border-left: double;" |  ||  ||
|-
| 1
| m2
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 24 || 35.2 || 49/48~50/49
| style="border-left: double;" | 7 || 60.3 || 28/27
| style="border-left: double;" | -10 || 85.3 || 21/20~22/21
| style="border-left: double;" | -27 || 110.4 || 16/15
|-
| 2
| n2
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 14 || 120.5 || 14/13~15/14
| style="border-left: double;" | -3 || 145.6 || 12/11~13/12
| style="border-left: double;" | -20 || 170.7 || 11/10
| style="border-left: double;" |  ||  ||
|-
| 3
| M2
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 21 || 180.8 || 10/9
| style="border-left: double;" | 4 || 205.9 || 9/8
| style="border-left: double;" | -13 || 230.9 || 8/7
| style="border-left: double;" |  ||  ||
|-
| 4
| m3
| style="border-left: double;" | 28 || 241.1 || 15/13
| style="border-left: double;" | 11 || 266.1 || 7/6
| style="border-left: double;" | -6 || 291.2 || 13/11
| style="border-left: double;" | -23 || 316.3 || 6/5
| style="border-left: double;" |  ||  ||
|-
| 5
| n3
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 18 || 326.4 || 40/33
| style="border-left: double;" | 1 || 351.5 || 11/9~16/13
| style="border-left: double;" | -16 || 376.5 || 26/21
| style="border-left: double;" |  ||  ||
|-
| 6
| M3
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 25 || 386.7 || 5/4
| style="border-left: double;" | 8 || 411.7 || 14/11
| style="border-left: double;" | -9 || 436.8 || 9/7
| style="border-left: double;" | -26 || 461.9 || 13/10
|-
| 7
| P4
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 15 || 472.0 || 21/16
| style="border-left: double;" | -2 || 497.1 || 4/3
| style="border-left: double;" | -19 || 522.1 || 27/20
| style="border-left: double;" |  ||  ||
|-
| 8
| sA4, d5
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 22 || 532.3 || 15/11
| style="border-left: double;" | 5 || 557.3 || 11/8~18/13
| style="border-left: double;" | -12 || 582.4 || 7/5
| style="border-left: double;" |  ||  ||
|-
| 9
| sd5, A4
| style="border-left: double;" | 29 || 592.5 || 45/32
| style="border-left: double;" | 12 || 617.6 || 10/7
| style="border-left: double;" | -5 || 642.7 || 13/9~16/11
| style="border-left: double;" | -22 || 667.7 || 22/15
| style="border-left: double;" |  ||  ||
|-
| 10
| P5
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 19 || 677.9 || 40/27
| style="border-left: double;" | 2 || 702.9 || 3/2
| style="border-left: double;" | -15 || 728.0 || 32/21
| style="border-left: double;" |  ||  ||
|-
| 11
| m6
| style="border-left: double;" | 26 || 738.1 || 20/13
| style="border-left: double;" | 9 || 763.2 || 14/9
| style="border-left: double;" | -8 || 788.3 || 11/7
| style="border-left: double;" | -25 || 813.3 || 8/5
| style="border-left: double;" |  ||  ||
|-
| 12
| n6
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 16 || 823.5 || 21/13
| style="border-left: double;" | -1 || 848.5 || 13/8~18/11
| style="border-left: double;" | -18 || 873.6 || 33/20
| style="border-left: double;" |  ||  ||
|-
| 13
| M6
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 23 || 883.7 || 5/3
| style="border-left: double;" | 6 || 908.8 || 22/13
| style="border-left: double;" | -11 || 933.9 || 12/7
| style="border-left: double;" | -28 || 958.9 || 26/15
|-
| 14
| m7
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 13 || 969.1 || 7/4
| style="border-left: double;" | -4 || 994.1 || 16/9
| style="border-left: double;" | -21 || 1019.2 || 9/5
| style="border-left: double;" |  ||  ||
|-
| 15
| n7
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 20 || 1029.3 || 20/11
| style="border-left: double;" | 3 || 1054.4 || 11/6~24/13
| style="border-left: double;" | -14 || 1079.5 || 13/7~28/15
| style="border-left: double;" |  ||  ||
|-
| 16
| M7
| style="border-left: double;" | 27 || 1089.6 || 15/8
| style="border-left: double;" | 10 || 1114.7 || 21/11~28/15
| style="border-left: double;" | -7 || 1139.7 || 27/14
| style="border-left: double;" | -24 || 1164.8 || 39/20~49/25
| style="border-left: double;" |  ||  ||
|-
| 17
| P8
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 17 || 1174.9 || 55/28~63/32
| style="border-left: double;" | 0 || 1200.0 || 2/1
| style="border-left: double;" |  ||  ||
| style="border-left: double;" |  ||  ||
|}


See the diagram on the right for an isomorphic version.
Below is tabulated how to notate each prime harmonic with an arrow representing a commatic step (thus ↑C = D{{sesquiflat2}}).  


== 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. The just major triad on C is {{dash|C, D{{sesquisharp2}}, G|med}}, for example. One may want to adopt one or more additional modules of accidentals such as arrows or +/- signs to represent the comma steps. There are two notable comma steps:
# The syntonic~septimal comma (-17 gensteps, semidiminished second);
# The Pythagorean comma (+24 gensteps, inverse diminished second).
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).
{| class="wikitable center-1 center-3"
{| class="wikitable center-1 center-3"
|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
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| 5/4
| 5/4
| Down major third
| Down major third
| C–vE
| C–↓E
|-
|-
| 7/4
| 7/4
| Down minor seventh
| Down minor seventh
| C–vBb
| C–↓B♭
|-
| 11/8
| Semi-augmented fourth
| C–Ft
|-
| 13/8
| Neutral sixth
| C–Ad
|}
 
Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).
{| 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
| Up neutral third
| C–^Ed
|-
| 7/4
| Up semidiminished seventh
| C–^Bdb
|-
|-
| 11/8
| 11/8
| Semi-augmented fourth
| Semi-augmented fourth
| C–Ft
| C–F{{demisharp2}}
|-
|-
| 13/8
| 13/8
| Neutral sixth
| Neutral sixth
| C–Ad
| C–A{{demiflat2}}
|}
|}


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|  
|  
| 350.000
| 350.000
| Lower bound of 7- and 9-odd-limit diamond monotone
| 24c val, lower bound of 7- and 9-odd-limit diamond monotone
|-
|-
|  
|  
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|  
|  
| 351.220
| 351.220
| Lower bound of 11- to 15-odd-limit<br>and (13-limit) 21-odd-limit diamond monotone
| Lower bound of 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 698: Line 504:
|  
|  
| 351.429
| 351.429
|  
| 140ef val
|-
|-
|  
|  
Line 718: Line 524:
| 25/24
| 25/24
| 351.472
| 351.472
| Very close to [[Argent tuning|argent tuning]] with neutral intervals (351.47186 cents)
| Very close to [[argent tuning]] with neutral intervals (351.47186 cents)
|-
|-
|  
|  
Line 733: Line 539:
|  
|  
| 351.515
| 351.515
|  
| 99ef val
|-
|-
|  
|  
Line 763: Line 569:
| 15/13
| 15/13
| 351.705
| 351.705
| 15-odd-limit and (13-limit) 21-odd-limit minimax
| 15-odd-limit and 13-limit 21-odd-limit minimax
|-
|-
| [[58edo|17\58]]
| [[58edo|17\58]]
Line 798: Line 604:
|  
|  
| 352.000
| 352.000
|  
| 75ce val
|-
|-
|  
|  
Line 818: Line 624:
|  
|  
| 352.941
| 352.941
| Upper bound of 7- to 15-odd-limit<br>and (13-limit) 21-odd-limit diamond monotone
| 17c val, upper bound of 7- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
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[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Aberschismic temperaments]]
[[Category:Hemimage temperaments]]
[[Category:Hemimage temperaments]]
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