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Hemififths: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{About|the regular temperament|the irrational interval of a hemififth|Sqrt(3/2)}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{Infobox regtemp
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-28 23:58:40 UTC</tt>.<br>
| Title = Hemififths
: The original revision id was <tt>473144602</tt>.<br>
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
: The revision comment was: <tt></tt><br>
| 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)
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Edo join 1 = 41 | Edo join 2 = 58
<h4>Original Wikitext content:</h4>
| Mapping = 1; 2 25 13 5 -1
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">See [[Breedsmic temperaments#Hemififths]].
| 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]].  


Gencom: [2 11/9; 144/143 196/195 243/242 364/363]
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).
Gencom map: [&lt;1 1 -5 -1 2 4|, &lt;0 2 25 13 5 -1|]


=Spectrum of Hemififths Tunings by Eigenmonzos=
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>.  
||~ Eigenmonzo ||~ Neutral Third ||
|| 11/9 || 347.408 ||
|| 12/11 || 349.788 ||
|| 7\24 || 350.000 ||
|| 11/8 || 350.264 ||
|| 4/3 || 350.978 ||
|| 12\41 || 351.220 ||
|| 15/14 || 351.389 ||
|| 16/15 || 351.417 ||
|| 41\140 || 351.429 ||
|| 8/7 || 351.448 (7, 9 and 11 limit hemif minimax) ||
|| 5/4 || 351.453 (5, 7, 9 and 11 limit minimax) ||
|| 7/5 || 351.457 ||
|| 6/5 || 351.494 ||
|| 29\99 || 351.515 ||
|| 7/6 || 351.534 ||
|| 10/9 || 351.543 ||
|| 9/7 || 351.657 ||
|| 15/11 || 351.680 ||
|| 15/13 || 351.705 (15 limit minimax) ||
|| 17\58 || 351.724 ||
|| 11/10 || 351.750 ||
|| 13/10 || 351.761 (13 limit minimax) ||
|| 13/11 || 351.798 (13 and 15 limit hemif minimax) ||
|| 22\75 || 352.000 ||
|| 14/13 || 352.021 ||
|| 14/11 || 352.188 ||
|| 18/13 || 352.676 ||
|| 13/12 || 353.809 ||
|| 16/13 || 359.472 ||
</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Hemififths&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Breedsmic%20temperaments#Hemififths"&gt;Breedsmic temperaments&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
Gencom: [2 11/9; 144/143 196/195 243/242 364/363]&lt;br /&gt;
Gencom map: [&amp;lt;1 1 -5 -1 2 4|, &amp;lt;0 2 25 13 5 -1|]&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Spectrum of Hemififths Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Spectrum of Hemififths Tunings by Eigenmonzos&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
See [[Breedsmic temperaments #Hemififths]] and [[No-fives subgroup temperaments #Hemif]] for more technical data.
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Neutral Third&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;347.408&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;349.788&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7\24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;350.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;350.264&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;350.978&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12\41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.220&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.389&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.417&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;41\140&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.429&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.448 (7, 9 and 11 limit hemif minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.453 (5, 7, 9 and 11 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.457&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.494&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;29\99&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.515&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.534&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.543&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.657&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.680&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.705 (15 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17\58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.724&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.750&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.761 (13 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.798 (13 and 15 limit hemif minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22\75&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;352.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;352.021&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;352.188&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;352.676&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;353.809&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;359.472&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
== Interval chain ==
In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents*
! colspan="2" | 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
|
|}
<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 ==
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).
 
Below is tabulated how to notate each prime harmonic with an arrow representing a commatic step (thus ↑C = D{{sesquiflat2}}).
 
{| class="wikitable center-1 center-3"
|+ style="font-size: 105%;" | Hemififths nomenclature<br>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{{demisharp2}}
|-
| 13/8
| Neutral sixth
| C–A{{demiflat2}}
|}
 
=== 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 {{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 ==
* [[Hemif7]]
* [[Hemif10]]
* [[Hemif17]]
 
== 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 ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 11/9
| 347.408
|
|-
|
| 11/6
| 349.788
|
|-
| [[24edo|7\24]]
|
| 350.000
| 24c val, lower bound of 7- and 9-odd-limit diamond monotone
|-
|
| 11/8
| 350.264
|
|-
|
| 3/2
| 350.978
|
|-
| [[41edo|12\41]]
|
| 351.220
| Lower bound of 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
|-
|
| 21/16
| 351.385
|
|-
|
| 15/14
| 351.389
|
|-
|
| 15/8
| 351.417
|
|-
| [[140edo|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
|
|-
| [[99edo|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
|-
| [[58edo|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
|
|-
| [[75edo|22\75]]
|
| 352.000
| 75ce val
|-
|
| 13/7
| 352.021
|
|-
|
| 11/7
| 352.188
|
|-
|
| 13/9
| 352.676
|
|-
| [[17edo|5\17]]
|
| 352.941
| 17c val, upper bound of 7- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
|-
|
| 13/12
| 353.809
|
|-
|
| 13/8
| 359.472
|
|}
<nowiki/>* Besides the octave
 
== References ==
<references/>
 
[[Category:Hemififths| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Aberschismic temperaments]]
[[Category:Hemimage temperaments]]
Retrieved from "https://en.xen.wiki/w/Hemififths"