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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 as a 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]].
 
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>.  


See [[Breedsmic temperaments #Hemififths]] and [[No-fives_subgroup_temperaments#Hemif]] for more technical data.
See [[Breedsmic temperaments #Hemififths]] and [[No-fives subgroup temperaments #Hemif]] for more technical data.


== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 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" | &#35;
|-
! rowspan="2" | #
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! colspan="2" | Approximate Ratios
! colspan="2" | Approximate ratios
! rowspan="2" |ups and downs
notation **
|-
|-
! 7-limit
! 7-limit
! 13-limit Extension
! 13-limit extension
|-
|-
| 0
| 0
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|  
|P1
|-
|-
| 1
| 1
| 351.4
| 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
|~3 = ^m3 = vM3
|-
|-
| 2
| 2
Line 33: Line 47:
| '''3/2'''
| '''3/2'''
|
|
|P5
|-
|-
| 3
| 3
| 1054.3
| 1054.4
| 90/49
| 90/49
| 11/6, 24/13
| 11/6, 24/13
|~7 = ^m7 = vM7
|-
|-
| 4
| 4
| 205.8
| 205.9
| '''9/8'''
| '''9/8'''
|  
|  
|M2
|-
|-
| 5
| 5
| 557.2
| 557.3
| 112/81
| 112/81
| '''11/8''', 18/13
| '''11/8''', 18/13
|~4 = ^4 = vA4
|-
|-
| 6
| 6
| 908.7
| 908.8
| 27/16
| 27/16
| 22/13
| 22/13
|M6
|-
|-
| 7
| 7
| 60.1
| 60.3
| 28/27
| 28/27
| 33/32, 27/26
| 33/32, 27/26
|^1 = \m2
|-
|-
| 8
| 8
| 411.6
| 411.7
| 80/63, 81/64
| 80/63, 81/64
| 14/11, 33/26
| 14/11, 33/26
|M3
|-
|-
| 9
| 9
| 763.0
| 763.2
| 14/9
| 14/9
|  
|  
|^5 = \m6
|-
|-
| 10
| 10
| 1114.5
| 1114.7
| 40/21
| 40/21
| 21/11
| 21/11
|M7
|-
|-
| 11
| 11
| 265.9
| 266.1
| 7/6
| 7/6
|  
|  
|^M2 = \m3
|-
|-
| 12
| 12
| 617.4
| 617.6
| 10/7
| 10/7
|  
|  
|A4 = \~5
|-
|-
| 13
| 13
| 968.8
| 969.1
| '''7/4'''
| '''7/4'''
|  
|  
|^M6 = \m7
|-
|-
| 14
| 14
| 120.2
| 120.5
| 15/14
| 15/14
| 14/13
| 14/13
|A1 = \~2
|-
|-
| 15
| 15
| 471.7
| 472.0
| '''21/16'''
| '''21/16'''
|  
|  
|^M3 = \4
|-
|-
| 16
| 16
| 823.1
| 823.5
| 45/28
| 45/28
| 21/13
| 21/13
|A5 = \~6
|-
|-
| 17
| 17
| 1174.6
| 1174.9
| 63/32, 160/81
| 63/32, 160/81
|
| 55/28, 65/33, 77/39
|^M7 = \8
|-
|-
| 18
| 18
| 326.0
| 326.4
| 98/81, 135/112
| 98/81, 135/112
| 40/33
| 40/33
|A2 = \~3
|-
|-
| 19
| 19
| 677.5
| 677.9
| 40/27
| 40/27
|  
|  
|^A4 = \5
|-
|-
| 20
| 20
| 1028.9
| 1029.3
| 49/27
| 49/27
| 20/11
| 20/11
|A6 = \~7
|-
|-
| 21
| 21
| 180.4
| 180.8
| 10/9
| 10/9
|  
|  
|^A1 = \M2
|-
|-
| 22
| 22
| 531.8
| 532.3
| 49/36
| 49/36
| 15/11
| 15/11
|A3 = \~4
|-
|-
| 23
| 23
| 883.3
| 883.7
| 5/3
| 5/3
|  
|  
|^A5 = \M6
|-
|-
| 24
| 24
| 34.7
| 35.2
| 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
| 386.2
| 386.7
| '''5/4'''
| '''5/4'''
|  
|  
|^A2 = \M3
|-
|-
| 26
| 26
| 737.6
| 738.1
| 49/32
| 49/32
| 20/13
| 20/13
|AA4 = ^\5
|-
|-
| 27
| 27
| 1089.1
| 1089.6
| '''15/8'''
| '''15/8'''
|  
|  
|^A6 = \M7
|-
|-
| 28
| 28
| 240.5
| 241.1
| 147/128
| 147/128
| 15/13
| 15/13
|AA1= ^\2
|-
|-
| 29
| 29
| 591.9
| 592.5
| 45/32
| 45/32
|  
|  
|^A3 = \A4
|}
|}
<nowiki>*</nowiki> In 7-limit CTE tuning, generator = 351.445¢, P5 = 702.89¢, c = 2.89¢
<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.  


<nowiki>**</nowiki> Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ - 8.5c
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 208: Line 199:
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).  
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"
{| class="wikitable center-1 center-3"
|+ style="font-size: 105%;" | Hemififths nomenclature<br />for selected intervals
|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
|-
|-
! Ratio
! Ratio
Line 237: Line 228:
Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).  
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"
{| class="wikitable center-1 center-3"
|+ style="font-size: 105%;" | Hemififths nomenclature<br />for selected intervals
|+ style="font-size: 105%;" | Hemififths nomenclature<br>for selected intervals
|-
|-
! Ratio
! Ratio
Line 264: 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 273: 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 310: 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 350: Line 554:
| 25/24
| 25/24
| 351.472
| 351.472
| Very close to [[Logarithmic approximants#Argent temperament|argent temperament]] with neutral intervals (351.47186 cents)
| Very close to [[Argent tuning|argent tuning]] with neutral intervals (351.47186 cents)
|-
|-
|  
|  
Line 395: 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 450: 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 462: Line 666:
|  
|  
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


== Notes ==
== 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:Hemifamity temperaments]]
[[Category:Hemimage temperaments]]
[[Category:Hemimage temperaments]]
Retrieved from "https://en.xen.wiki/w/Hemififths"