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'''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- and 13-limit by treating the generator as [[11/9]][[~]][[16/13]]. The no-5 subgroup [[restriction]], called '''hemif''', is also notable. Possible tunings include [[41edo|41-]], [[58edo|58-]], and [[99edo]].  
{{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>.  
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>.  
Line 12: 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 20: Line 36:
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|  
| P1
|-
|-
| 1
| 1
Line 27: 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 33: Line 47:
| '''3/2'''
| '''3/2'''
|
|
| P5
|-
|-
| 3
| 3
Line 39: Line 52:
| 90/49
| 90/49
| 11/6, 24/13
| 11/6, 24/13
| ~7 = ^m7 = vM7
|-
|-
| 4
| 4
Line 45: Line 57:
| '''9/8'''
| '''9/8'''
|  
|  
| M2
|-
|-
| 5
| 5
Line 51: Line 62:
| 112/81
| 112/81
| '''11/8''', 18/13
| '''11/8''', 18/13
| ~4 = ^4 = vA4
|-
|-
| 6
| 6
Line 57: Line 67:
| 27/16
| 27/16
| 22/13
| 22/13
| M6
|-
|-
| 7
| 7
Line 63: Line 72:
| 28/27
| 28/27
| 33/32, 27/26
| 33/32, 27/26
| ^1 = \m2
|-
|-
| 8
| 8
Line 69: Line 77:
| 80/63, 81/64
| 80/63, 81/64
| 14/11, 33/26
| 14/11, 33/26
| M3
|-
|-
| 9
| 9
Line 75: Line 82:
| 14/9
| 14/9
|  
|  
| ^5 = \m6
|-
|-
| 10
| 10
Line 81: Line 87:
| 40/21
| 40/21
| 21/11
| 21/11
| M7
|-
|-
| 11
| 11
Line 87: Line 92:
| 7/6
| 7/6
|  
|  
| ^M2 = \m3
|-
|-
| 12
| 12
Line 93: Line 97:
| 10/7
| 10/7
|  
|  
| A4 = \~5
|-
|-
| 13
| 13
Line 99: Line 102:
| '''7/4'''
| '''7/4'''
|  
|  
| ^M6 = \m7
|-
|-
| 14
| 14
Line 105: Line 107:
| 15/14
| 15/14
| 14/13
| 14/13
| A1 = \~2
|-
|-
| 15
| 15
Line 111: Line 112:
| '''21/16'''
| '''21/16'''
|  
|  
| ^M3 = \4
|-
|-
| 16
| 16
Line 117: Line 117:
| 45/28
| 45/28
| 21/13
| 21/13
| A5 = \~6
|-
|-
| 17
| 17
Line 123: 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 129: Line 127:
| 98/81, 135/112
| 98/81, 135/112
| 40/33
| 40/33
| A2 = \~3
|-
|-
| 19
| 19
Line 135: Line 132:
| 40/27
| 40/27
|  
|  
| ^A4 = \5
|-
|-
| 20
| 20
Line 141: Line 137:
| 49/27
| 49/27
| 20/11
| 20/11
| A6 = \~7
|-
|-
| 21
| 21
Line 147: Line 142:
| 10/9
| 10/9
|  
|  
| ^A1 = \M2
|-
|-
| 22
| 22
Line 153: Line 147:
| 49/36
| 49/36
| 15/11
| 15/11
| A3 = \~4
|-
|-
| 23
| 23
Line 159: Line 152:
| 5/3
| 5/3
|  
|  
| ^A5 = \M6
|-
|-
| 24
| 24
Line 165: 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 171: Line 162:
| '''5/4'''
| '''5/4'''
|  
|  
| ^A2 = \M3
|-
|-
| 26
| 26
Line 177: Line 167:
| 49/32
| 49/32
| 20/13
| 20/13
| AA4 = ^\5
|-
|-
| 27
| 27
Line 183: Line 172:
| '''15/8'''
| '''15/8'''
|  
|  
| ^A6 = \M7
|-
|-
| 28
| 28
Line 189: Line 177:
| 147/128
| 147/128
| 15/13
| 15/13
| AA1= ^\2
|-
|-
| 29
| 29
Line 195: 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 ===
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.
[[File: Hemififths 17et Detempering.png|thumb|Hemififths as a 58-tone 17et detempering]]


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.  
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.  


{| class="wikitable center-all mw-collapsible mw-collapsed"
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.
|-
! 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
| 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
| 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;" |  ||  ||
|}


== Notation ==
== Notation ==
Line 430: 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 439: Line 426:


== Tunings ==
== 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 ===
=== 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 475: Line 514:
|  
|  
| 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 515: Line 554:
| 25/24
| 25/24
| 351.472
| 351.472
| Very close to [[Logarithmic approximants #Argent temperament|argent tuning]] with neutral intervals (351.47186 cents)
| Very close to [[Argent tuning|argent tuning]] with neutral intervals (351.47186 cents)
|-
|-
|  
|  
Line 560: Line 599:
| 15/13
| 15/13
| 351.705
| 351.705
| 15-odd-limit minimax
| 15-odd-limit and (13-limit) 21-odd-limit minimax
|-
|-
| [[58edo|17\58]]
| [[58edo|17\58]]
Line 615: Line 654:
|  
|  
| 352.941
| 352.941
| Upper bound of 7- to 15-odd-limit<br>and 13-limit 21-odd-limit diamond monotone
| Upper bound of 7- to 15-odd-limit<br>and (13-limit) 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 632: Line 671:
<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]]
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