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| [[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]] |
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| 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. |
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| 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. | | 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. |
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| {| class="wikitable center-all mw-collapsible mw-collapsed"
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| ! rowspan="2" | #
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| ! rowspan="2" | Interval<br>category
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| ! colspan="3" style="border-left: double;" | "Double-Sub"
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| ! colspan="3" style="border-left: double;" | "Sub"
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| ! colspan="3" style="border-left: double;" | "Plain"
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| ! colspan="3" style="border-left: double;" | "Super"
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| ! colspan="3" style="border-left: double;" | "Double-super"
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| |-
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| ! style="border-left: double;" | Gen. || Cents* || Ratios
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| ! style="border-left: double;" | Gen. || Cents* || Ratios
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| ! style="border-left: double;" | Gen. || Cents* || Ratios
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| ! style="border-left: double;" | Gen. || Cents* || Ratios
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| ! style="border-left: double;" | Gen. || Cents* || Ratios
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| |-
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| | 0
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| | P1
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 0 || 0.0 || 1/1
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| | style="border-left: double;" | -17 || 25.9 || 64/63~81/80
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| | style="border-left: double;" | || ||
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| |-
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| | 1
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| | m2
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 24 || 35.2 || 49/48~50/49
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| | style="border-left: double;" | 7 || 60.3 || 28/27
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| | style="border-left: double;" | -10 || 85.3 || 21/20~22/21
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| | style="border-left: double;" | -27 || 110.4 || 16/15
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| |-
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| | 2
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| | n2
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 14 || 120.5 || 14/13~15/14
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| | style="border-left: double;" | -3 || 145.6 || 12/11~13/12
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| | style="border-left: double;" | -20 || 170.7 || 11/10
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| | style="border-left: double;" | || ||
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| |-
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| | 3
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| | M2
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 21 || 180.8 || 10/9
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| | style="border-left: double;" | 4 || 205.9 || 9/8
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| | style="border-left: double;" | -13 || 230.9 || 8/7
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| | style="border-left: double;" | || ||
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| |-
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| | 4
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| | m3
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| | style="border-left: double;" | 28 || 241.1 || 15/13
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| | style="border-left: double;" | 11 || 266.1 || 7/6
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| | style="border-left: double;" | -6 || 291.2 || 13/11
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| | style="border-left: double;" | -23 || 316.3 || 6/5
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| | style="border-left: double;" | || ||
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| |-
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| | 5
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| | n3
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 18 || 326.4 || 40/33
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| | style="border-left: double;" | 1 || 351.5 || 11/9~16/13
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| | style="border-left: double;" | -16 || 376.5 || 26/21
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| | style="border-left: double;" | || ||
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| |-
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| | 6
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| | M3
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 25 || 386.7 || 5/4
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| | style="border-left: double;" | 8 || 411.7 || 14/11
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| | style="border-left: double;" | -9 || 436.8 || 9/7
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| | style="border-left: double;" | -26 || 461.9 || 13/10
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| |-
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| | 7
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| | P4
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 15 || 472.0 || 21/16
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| | style="border-left: double;" | -2 || 497.1 || 4/3
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| | style="border-left: double;" | -19 || 522.1 || 27/20
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| | style="border-left: double;" | || ||
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| |-
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| | 8
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| | sA4, d5
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 22 || 532.3 || 15/11
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| | style="border-left: double;" | 5 || 557.3 || 11/8~18/13
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| | style="border-left: double;" | -12 || 582.4 || 7/5
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| | style="border-left: double;" | || ||
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| |-
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| | 9
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| | sd5, A4
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| | style="border-left: double;" | 29 || 592.5 || 45/32
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| | style="border-left: double;" | 12 || 617.6 || 10/7
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| | style="border-left: double;" | -5 || 642.7 || 13/9~16/11
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| | style="border-left: double;" | -22 || 667.7 || 22/15
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| | style="border-left: double;" | || ||
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| |-
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| | 10
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| | P5
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 19 || 677.9 || 40/27
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| | style="border-left: double;" | 2 || 702.9 || 3/2
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| | style="border-left: double;" | -15 || 728.0 || 32/21
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| | style="border-left: double;" | || ||
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| |-
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| | 11
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| | m6
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| | style="border-left: double;" | 26 || 738.1 || 20/13
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| | style="border-left: double;" | 9 || 763.2 || 14/9
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| | style="border-left: double;" | -8 || 788.3 || 11/7
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| | style="border-left: double;" | -25 || 813.3 || 8/5
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| | style="border-left: double;" | || ||
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| |-
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| | 12
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| | n6
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 16 || 823.5 || 21/13
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| | style="border-left: double;" | -1 || 848.5 || 13/8~18/11
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| | style="border-left: double;" | -18 || 873.6 || 33/20
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| | style="border-left: double;" | || ||
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| |-
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| | 13
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| | M6
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 23 || 883.7 || 5/3
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| | style="border-left: double;" | 6 || 908.8 || 22/13
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| | style="border-left: double;" | -11 || 933.9 || 12/7
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| | style="border-left: double;" | -28 || 958.9 || 26/15
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| |-
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| | 14
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| | m7
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 13 || 969.1 || 7/4
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| | style="border-left: double;" | -4 || 994.1 || 16/9
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| | style="border-left: double;" | -21 || 1019.2 || 9/5
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| | style="border-left: double;" | || ||
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| |-
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| | 15
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| | n7
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 20 || 1029.3 || 20/11
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| | style="border-left: double;" | 3 || 1054.4 || 11/6~24/13
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| | style="border-left: double;" | -14 || 1079.5 || 13/7~28/15
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| | style="border-left: double;" | || ||
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| |-
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| | 16
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| | M7
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| | style="border-left: double;" | 27 || 1089.6 || 15/8
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| | style="border-left: double;" | 10 || 1114.7 || 21/11~28/15
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| | style="border-left: double;" | -7 || 1139.7 || 27/14
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| | style="border-left: double;" | -24 || 1164.8 || 39/20~49/25
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| | style="border-left: double;" | || ||
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| |-
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| | 17
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| | P8
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | 17 || 1174.9 || 55/28~63/32
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| | style="border-left: double;" | 0 || 1200.0 || 2/1
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| | style="border-left: double;" | || ||
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| | style="border-left: double;" | || ||
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| |}
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| See the diagram on the right for an isomorphic version.
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| == Notation == | | == Notation == |