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, 2.3.5.7.11.13
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[2401/2400]], [[5120/5103]] (2.3.5.7); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (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)
| Generator = 11/9
| 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 = 58
| Odd limit 1 = 9 | Mistuning 1 = 1.90 | Complexity 1 = 41
| Odd limit 2 = (13-limit) 21 | Mistuning 2 = 7.77 | Complexity 2 = 58
| 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 hemifamity 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).  
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).  
Line 30: Line 29:
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
! colspan="2" | Approximate ratios
! rowspan="2" | [[Ups and downs notation|Ups and downs<br>notation]]**
|-
|-
! 7-limit
! 7-limit
Line 38: Line 36:
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|  
| P1
|-
|-
| 1
| 1
Line 45: Line 42:
| 49/40, 60/49
| 49/40, 60/49
| 11/9, '''16/13''', 27/22, 39/32
| 11/9, '''16/13''', 27/22, 39/32
| ~3 = ^m3 = vM3
|-
|-
| 2
| 2
Line 51: Line 47:
| '''3/2'''
| '''3/2'''
|
|
| P5
|-
|-
| 3
| 3
Line 57: Line 52:
| 90/49
| 90/49
| 11/6, 24/13
| 11/6, 24/13
| ~7 = ^m7 = vM7
|-
|-
| 4
| 4
Line 63: Line 57:
| '''9/8'''
| '''9/8'''
|  
|  
| M2
|-
|-
| 5
| 5
Line 69: Line 62:
| 112/81
| 112/81
| '''11/8''', 18/13
| '''11/8''', 18/13
| ~4 = ^4 = vA4
|-
|-
| 6
| 6
Line 75: Line 67:
| 27/16
| 27/16
| 22/13
| 22/13
| M6
|-
|-
| 7
| 7
Line 81: Line 72:
| 28/27
| 28/27
| 33/32, 27/26
| 33/32, 27/26
| ^1 = \m2
|-
|-
| 8
| 8
Line 87: Line 77:
| 80/63, 81/64
| 80/63, 81/64
| 14/11, 33/26
| 14/11, 33/26
| M3
|-
|-
| 9
| 9
Line 93: Line 82:
| 14/9
| 14/9
|  
|  
| ^5 = \m6
|-
|-
| 10
| 10
Line 99: Line 87:
| 40/21
| 40/21
| 21/11
| 21/11
| M7
|-
|-
| 11
| 11
Line 105: Line 92:
| 7/6
| 7/6
|  
|  
| ^M2 = \m3
|-
|-
| 12
| 12
Line 111: Line 97:
| 10/7
| 10/7
|  
|  
| A4 = \~5
|-
|-
| 13
| 13
Line 117: Line 102:
| '''7/4'''
| '''7/4'''
|  
|  
| ^M6 = \m7
|-
|-
| 14
| 14
Line 123: Line 107:
| 15/14
| 15/14
| 14/13
| 14/13
| A1 = \~2
|-
|-
| 15
| 15
Line 129: Line 112:
| '''21/16'''
| '''21/16'''
|  
|  
| ^M3 = \4
|-
|-
| 16
| 16
Line 135: Line 117:
| 45/28
| 45/28
| 21/13
| 21/13
| A5 = \~6
|-
|-
| 17
| 17
Line 141: Line 122:
| 63/32, 160/81
| 63/32, 160/81
| 55/28, 65/33, 77/39
| 55/28, 65/33, 77/39
| ^M7 = \8
|-
|-
| 18
| 18
Line 147: Line 127:
| 98/81, 135/112
| 98/81, 135/112
| 40/33
| 40/33
| A2 = \~3
|-
|-
| 19
| 19
Line 153: Line 132:
| 40/27
| 40/27
|  
|  
| ^A4 = \5
|-
|-
| 20
| 20
Line 159: Line 137:
| 49/27
| 49/27
| 20/11
| 20/11
| A6 = \~7
|-
|-
| 21
| 21
Line 165: Line 142:
| 10/9
| 10/9
|  
|  
| ^A1 = \M2
|-
|-
| 22
| 22
Line 171: Line 147:
| 49/36
| 49/36
| 15/11
| 15/11
| A3 = \~4
|-
|-
| 23
| 23
Line 177: Line 152:
| 5/3
| 5/3
|  
|  
| ^A5 = \M6
|-
|-
| 24
| 24
Line 183: Line 157:
| 49/48, 50/49
| 49/48, 50/49
| 40/39, 45/44, 55/54, 65/64
| 40/39, 45/44, 55/54, 65/64
| A7 - P8 = -d2 = ^\1
|-
|-
| 25
| 25
Line 189: Line 162:
| '''5/4'''
| '''5/4'''
|  
|  
| ^A2 = \M3
|-
|-
| 26
| 26
Line 195: Line 167:
| 49/32
| 49/32
| 20/13
| 20/13
| AA4 = ^\5
|-
|-
| 27
| 27
Line 201: Line 172:
| '''15/8'''
| '''15/8'''
|  
|  
| ^A6 = \M7
|-
|-
| 28
| 28
Line 207: Line 177:
| 147/128
| 147/128
| 15/13
| 15/13
| AA1= ^\2
|-
|-
| 29
| 29
Line 213: Line 182:
| 45/32
| 45/32
|  
|  
| ^A3 = \A4
|}
|}
<nowiki/>* In 7-limit CWE tuning, {{nowrap|generator {{=}} 351.467{{c}} }}, {{nowrap| P5 {{=}} 702.934{{c}} }} and {{nowrap| c {{=}} 2.934{{c}} }}
<nowiki/>* In 7-limit CWE tuning, octave reduced
 
<nowiki/>** Enharmonic equivalences: vvA1 and v\m2. Cents: {{nowrap| ^1 {{=}} 50¢ + 3.5c }} and {{nowrap| /1 {{=}} 50¢ − 8.5c }}


=== As a detemperament of 17et ===
=== As a detemperament of 17et ===
[[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 -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 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.
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|-
! 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.
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 ==
== Notation ==
Line 452: Line 255:
|}
|}


== 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 709: Line 674:
[[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]]

Latest revision as of 12:39, 6 June 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 -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 ⁠), 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. The just major triad on C is C – D⁠ ⁠ – G, 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:

  1. The syntonic~septimal comma (-17 gensteps, semidiminished second);
  2. 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).

Hemififths nomenclature
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#).

Hemififths nomenclature
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 ^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 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
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
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
13/7 352.021
11/7 352.188
13/9 352.676
5\17 352.941 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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