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'''Hemififths''' is the [[temperament]] [[tempering 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. It is supported by [[41edo|41-]], [[58edo|58-]], and [[99edo|99et]].  
{{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>*</nowiki> in 7-limit POTE tuning
<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 ==
== Notation ==
Hemififths can be notated in [[neutral circle-of-fifths notation]], in which case 5/4 is represented by a sesqui-augmented second (C-D#+), and 7/4 by a semi-augmented sixth (C-A+). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C-F+), and 13/8 by the neutral sixth (C-Ad). This, of course, defies the tradition of tertian harmony. The just major triad on C is C-D#+-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step. There are two solutions:
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).
# let an arrow represent a bend by the syntonic~septimal comma (17 gensteps, semidiminished second);
 
# let an arrow represent a bend by the Pythagorean comma (24 gensteps, negative diminished second).  
Below is tabulated how to notate each prime harmonic with an arrow representing a commatic step (thus ↑C = D{{sesquiflat2}}).  


Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma.
{| class="wikitable center-1 center-3"
{| class="wikitable center-1 center-3"
|+Hemififths nomenclature<br>for selected intervals
|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
|-
! Ratio
! Ratio
! Nominal
! Nominal
Line 159: Line 206:
| 3/2
| 3/2
| Perfect fifth
| Perfect fifth
| C-G
| C–G
|-
|-
| 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
| 11/8
| Semi-augmented fourth
| Semi-augmented fourth
| C-F+
| C–F{{demisharp2}}
|-
|-
| 13/8
| 13/8
| Neutral sixth
| Neutral sixth
| C-Ad
| C–A{{demiflat2}}
|}
|}


Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma.  
=== Ups and downs notation ===
{| class="wikitable center-1 center-3"
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}} }}.  
|+Hemififths nomenclature<br>for selected intervals
 
! Ratio
{| class="wikitable center-1 right-2"
! Nominal
|-
! Example
! #
! 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
| 3/2
| Perfect fifth
| C-G
|-
|-
| 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
| 5/4
| Up neutral third
| C-^Ed
|-
|-
| 7/4
| 26
| Up semidiminished seventh
| 738.1
| C-^Bdb
| AA4 = ^\5
| 20/13
|-
| 27
| 1089.6
| ^A6 = \M7
| 15/8
|-
|-
| 11/8
| 28
| Semi-augmented fourth
| 241.1
| C-F+
| AA1= ^\2
| 15/13
|-
|-
| 13/8
| 29
| Neutral sixth
| 592.5
| C-Ad
| ^A3 = \A4
| 45/32
|}
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced


== Chords ==
== Chords and harmony ==
{{Main| Chords of hemififths }}
{{See also| Chords of hemififths }}


== Scales ==
== Scales ==
Line 214: 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
Gencom mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
|-
! 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"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 233: Line 462:
|-
|-
|  
|  
| 12/11
| 11/6
| 349.788
| 349.788
|  
|  
|-
|-
| 7\24
| [[24edo|7\24]]
|  
|  
| 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
|-
|-
|  
|  
Line 248: 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
| Lower bound of 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
|-
|
| 21/16
| 351.385
|
|-
|-
|  
|  
Line 263: 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 288: 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 303: Line 547:
|-
|-
|  
|  
| 10/9
| 9/5
| 351.543
| 351.543
|
|-
|
| 21/20
| 351.553
|  
|  
|-
|-
Line 320: 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 342: 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
|  
|  
|-
|-
| 5\17
| [[17edo|5\17]]
|  
|  
| 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
|-
|-
|  
|  
Line 373: Line 632:
|-
|-
|  
|  
| 16/13
| 13/8
| 359.472
| 359.472
|  
|  
|}
|}
<nowiki>*</nowiki> besides the octave
<nowiki/>* Besides the octave
 
== References ==
<references/>


[[Category:Temperaments]]
[[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]]

Latest revision as of 15:02, 13 September 2026

This page is about the regular temperament. For the irrational interval of a hemififth, see Sqrt(3/2).
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);
243/242, 441/440, 896/891 (11-limit);
144/143, 196/195, 243/242, 364/363 (13-limit)
Reduced mapping ⟨1; 2 25 13 5 -1]
ET join 41 & 58
Generators (CWE) ~49/40 = 351.5 ¢
MOS scales 3L 4s, 7L 3s, 7L 10s, 17L 7s, 17L 24s
Ploidacot dicot
Pergen (P8, P5/2)
Minimax error 9-odd-limit: 1.90 ¢;
13-limit 21-odd-limit: 7.77 ¢
Target scale size 9-odd-limit: 41 notes;
13-limit 21-odd-limit: 41 notes

Hemififths is a 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- 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 41-, 58-, and 99edo (using the 99ef val in the 13-limit).

Hemififths was named by Gene Ward Smith in 2004[1].

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.

# Cents* 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

* In 7-limit CWE tuning, octave reduced

As a detemperament of 17et

Hemififths as a 58-tone 17et detempering

Hemififths is very naturally considered as a detemperament of the 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 ⁠), and 7/4 by a semi-augmented sixth (C–A⁠ ⁠). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C–F⁠ ⁠), and 13/8 by the neutral sixth (C–A⁠ ⁠). This, of course, defies the tradition of tertian harmony, as the just major triad on C is C–D⁠ ⁠–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).

Hemififths nomenclature
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 ⁠
13/8 Neutral sixth C–A⁠ ⁠

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 ^1 = 50 ¢ + 3.5c and /1 = 50 ¢ − 8.5c. For 7-limit CWE tuning, c = 2.934 ¢.

# Cents* Ups and downs
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

* In 7-limit CWE tuning, octave reduced

Chords and harmony

See also: Chords of hemififths

Scales

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~49/40 = 351.4464 ¢ CSEE: ~49/40 = 351.4671 ¢ POEE: ~49/40 = 351.4774 ¢
Tenney CTE: ~49/40 = 351.4492 ¢ CWE: ~49/40 = 351.4639 ¢ POTE: ~49/40 = 351.4834 ¢
Benedetti,
Wilson 		CBE: ~49/40 = 351.4447 ¢ CSBE: ~49/40 = 351.4675 ¢ POBE: ~49/40 = 351.4787 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~11/9 = 351.4230 ¢ CSEE: ~11/9 = 351.5800 ¢ POEE: ~11/9 = 351.6627 ¢
Tenney CTE: ~11/9 = 351.4331 ¢ CWE: ~11/9 = 351.5438 ¢ POTE: ~11/9 = 351.5734 ¢
Benedetti,
Wilson 		CBE: ~11/9 = 351.4380 ¢ CSBE: ~11/9 = 351.5144 ¢ POBE: ~11/9 = 351.5243 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
11/9 347.408
11/6 349.788
7\24 350.000 24c val, lower bound of 7- and 9-odd-limit diamond monotone
11/8 350.264
3/2 350.978
12\41 351.220 Lower bound of 11- to 15-odd-limit,
and 13-limit 21-odd-limit diamond monotone
21/16 351.385
15/14 351.389
15/8 351.417
41\140 351.429 140ef val
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 with neutral intervals (351.47186 cents)
49/48 351.487
5/3 351.494
29\99 351.515 99ef val
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
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
22\75 352.000 75ce val
13/7 352.021
11/7 352.188
13/9 352.676
5\17 352.941 17c val, upper bound of 7- to 15-odd-limit,
and 13-limit 21-odd-limit diamond monotone
13/12 353.809
13/8 359.472

* Besides the octave

References

  1. ↑ Yahoo! Tuning Group (Archive) | Names for important high-complexity temperaments
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