View source for 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]]. 

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 ==
In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''. 
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
|-
! 7-limit
! 13-limit extension
|-
| 0
| 0.0
| '''1/1'''
| 
|-
| 1
| 351.5
| 49/40, 60/49
| 11/9, '''16/13''', 27/22, 39/32
|-
| 2
| 702.9
| '''3/2'''
|
|-
| 3
| 1054.4
| 90/49
| 11/6, 24/13
|-
| 4
| 205.9
| '''9/8'''
| 
|-
| 5
| 557.3
| 112/81
| '''11/8''', 18/13
|-
| 6
| 908.8
| 27/16
| 22/13
|-
| 7
| 60.3
| 28/27
| 33/32, 27/26
|-
| 8
| 411.7
| 80/63, 81/64
| 14/11, 33/26
|-
| 9
| 763.2
| 14/9
| 
|-
| 10
| 1114.7
| 40/21
| 21/11
|-
| 11
| 266.1
| 7/6
| 
|-
| 12
| 617.6
| 10/7
| 
|-
| 13
| 969.1
| '''7/4'''
| 
|-
| 14
| 120.5
| 15/14
| 14/13
|-
| 15
| 472.0
| '''21/16'''
| 
|-
| 16
| 823.5
| 45/28
| 21/13
|-
| 17
| 1174.9
| 63/32, 160/81
| 55/28, 65/33, 77/39
|-
| 18
| 326.4
| 98/81, 135/112
| 40/33
|-
| 19
| 677.9
| 40/27
| 
|-
| 20
| 1029.3
| 49/27
| 20/11
|-
| 21
| 180.8
| 10/9
| 
|-
| 22
| 532.3
| 49/36
| 15/11
|-
| 23
| 883.7
| 5/3
| 
|-
| 24
| 35.2
| 49/48, 50/49
| 40/39, 45/44, 55/54, 65/64
|-
| 25
| 386.7
| '''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 -17 generator steps, which represents the syntonic~septimal comma. 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 syntonic~septimal comma. 99edo tunes it to one half the size of the syntonic~septimal comma, 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. 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"
|+ 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–vE
|-
| 7/4
| Down minor seventh
| C–vBb
|-
| 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
| Semi-augmented fourth
| C–Ft
|-
| 13/8
| Neutral sixth
| C–Ad
|}

=== Ups and downs notation ===
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 ==
* [[Hemif7]]
* [[Hemif10]]
* [[Hemif17]]

== 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}}
|}

{| 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}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| 
| 11/9
| 347.408
| 
|-
| 
| 11/6
| 349.788
| 
|-
| [[24edo|7\24]]
| 
| 350.000
| Lower bound of 7- and 9-odd-limit diamond monotone
|-
| 
| 11/8
| 350.264
| 
|-
| 
| 3/2
| 350.978
| 
|-
| [[41edo|12\41]]
| 
| 351.220
| Lower bound of 11- to 15-odd-limit<br>and (13-limit) 21-odd-limit diamond monotone
|-
| 
| 21/16
| 351.385
| 
|-
| 
| 15/14
| 351.389
| 
|-
| 
| 15/8
| 351.417
| 
|-
| [[140edo|41\140]]
| 
| 351.429
| 
|-
| 
| 7/4
| 351.448
| 7-, 9- and 11-odd-limit hemif minimax
|-
| 
| 5/4
| 351.453
| 5-, 7-, 9- and 11-odd-limit minimax
|-
| 
| 7/5
| 351.457
| 
|-
| 
| 25/24
| 351.472
| Very close to [[Argent tuning|argent tuning]] with neutral intervals (351.47186 cents)
|-
| 
| 49/48
| 351.487
| 
|-
| 
| 5/3
| 351.494
| 
|-
| [[99edo|29\99]]
| 
| 351.515
| 
|-
| 
| 7/6
| 351.534
| 
|-
| 
| 9/5
| 351.543
| 
|-
| 
| 21/20
| 351.553
| 
|-
| 
| 9/7
| 351.657
| 
|-
| 
| 15/11
| 351.680
| 
|-
| 
| 15/13
| 351.705
| 15-odd-limit and (13-limit) 21-odd-limit minimax
|-
| [[58edo|17\58]]
| 
| 351.724
| 
|-
| 
| 11/10
| 351.750
| 
|-
| 
| 13/10
| 351.761
| 13-odd-limit minimax
|-
| 
| 13/11
| 351.798
| 13- and 15-odd-limit hemif minimax
|-
| 
| 21/13
| 351.891
| 
|-
| 
| 21/11
| 351.946
| 
|-
| [[75edo|22\75]]
| 
| 352.000
| 
|-
| 
| 13/7
| 352.021
| 
|-
| 
| 11/7
| 352.188
| 
|-
| 
| 13/9
| 352.676
| 
|-
| [[17edo|5\17]]
| 
| 352.941
| Upper bound of 7- to 15-odd-limit<br>and (13-limit) 21-odd-limit diamond monotone
|-
| 
| 13/12
| 353.809
| 
|-
| 
| 13/8
| 359.472
| 
|}
<nowiki/>* Besides the octave

== References ==
<references/>

[[Category:Hemififths| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Hemimage temperaments]]