Hemififths: Difference between revisions
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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]] | ||
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 | 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 | 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 == | == 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 | 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" | {| 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 | ||
| Line 211: | Line 210: | ||
| 5/4 | | 5/4 | ||
| Down major third | | Down major third | ||
| | | C–↓E | ||
|- | |- | ||
| 7/4 | | 7/4 | ||
| Down minor seventh | | Down minor seventh | ||
| | | C–↓B♭ | ||
|- | |- | ||
| 11/8 | | 11/8 | ||
| Semi-augmented fourth | | Semi-augmented fourth | ||
| | | C–F{{demisharp2}} | ||
|- | |- | ||
| 13/8 | | 13/8 | ||
| Neutral sixth | | Neutral sixth | ||
| | | C–A{{demiflat2}} | ||
|} | |} | ||
| Line 499: | Line 469: | ||
| | | | ||
| 350.000 | | 350.000 | ||
| | | 24c val, lower bound of 7- and 9-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
| Line 514: | Line 484: | ||
| | | | ||
| 351.220 | | 351.220 | ||
| Lower bound of 11- to 15-odd-limit<br>and | | Lower bound of 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
| Line 534: | Line 504: | ||
| | | | ||
| 351.429 | | 351.429 | ||
| | | 140ef val | ||
|- | |- | ||
| | | | ||
| Line 554: | Line 524: | ||
| 25/24 | | 25/24 | ||
| 351.472 | | 351.472 | ||
| Very close to [[ | | Very close to [[argent tuning]] with neutral intervals (351.47186 cents) | ||
|- | |- | ||
| | | | ||
| Line 569: | Line 539: | ||
| | | | ||
| 351.515 | | 351.515 | ||
| | | 99ef val | ||
|- | |- | ||
| | | | ||
| Line 599: | Line 569: | ||
| 15/13 | | 15/13 | ||
| 351.705 | | 351.705 | ||
| 15-odd-limit and | | 15-odd-limit and 13-limit 21-odd-limit minimax | ||
|- | |- | ||
| [[58edo|17\58]] | | [[58edo|17\58]] | ||
| Line 634: | Line 604: | ||
| | | | ||
| 352.000 | | 352.000 | ||
| | | 75ce val | ||
|- | |- | ||
| | | | ||
| Line 654: | Line 624: | ||
| | | | ||
| 352.941 | | 352.941 | ||
| | | 17c val, upper bound of 7- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
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[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
[[Category:Breedsmic temperaments]] | [[Category:Breedsmic temperaments]] | ||
[[Category: | [[Category:Aberschismic temperaments]] | ||
[[Category:Hemimage temperaments]] | [[Category:Hemimage temperaments]] | ||