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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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! 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]]
Retrieved from "https://en.xen.wiki/w/Hemififths"