Hemififths: Difference between revisions
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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 | '''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]]. | ||
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>. | ||
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In the following table, odd harmonics 1–21 are labeled in '''bold'''. | In the following table, odd harmonics 1–21 are labeled in '''bold'''. | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
! rowspan="2" | # | ! rowspan="2" | # | ||
! rowspan="2" | Cents* | ! rowspan="2" | Cents* | ||
! colspan="2" | Approximate Ratios | ! colspan="2" | Approximate Ratios | ||
| Line 165: | Line 165: | ||
| | | | ||
|} | |} | ||
<nowiki>* | <nowiki />* In 7-limit CTE tuning | ||
== Notation == | == Notation == | ||
| Line 243: | Line 243: | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! Edo<br>generator | ! Edo<br />generator | ||
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]* | ! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
| Line 276: | Line 276: | ||
| | | | ||
| 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 416: | Line 416: | ||
| | | | ||
| 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 428: | Line 428: | ||
| | | | ||
|} | |} | ||
<nowiki>* | <nowiki />* Besides the octave | ||
== Notes == | == Notes == | ||
[[Category:Temperaments]] | [[Category:Temperaments]] | ||
[[Category:Hemififths| ]] <!-- | [[Category:Hemififths| ]] <!-- Main article --> | ||
[[Category:Breedsmic temperaments]] | [[Category:Breedsmic temperaments]] | ||
[[Category:Hemifamity temperaments]] | [[Category:Hemifamity temperaments]] | ||
[[Category:Hemimage temperaments]] | [[Category:Hemimage temperaments]] | ||
Revision as of 15:06, 20 October 2024
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 41-, 58-, and 99et.
Hemififths was named by Gene Ward Smith in 2004[1].
See Breedsmic temperaments #Hemififths for more technical data.
Interval chain
In the following table, odd harmonics 1–21 are labeled in bold.
| # | Cents* | Approximate Ratios | |
|---|---|---|---|
| 7-limit | 13-limit Extension | ||
| 0 | 0.0 | 1/1 | |
| 1 | 351.4 | 49/40, 60/49 | 11/9, 16/13, 27/22, 39/32 |
| 2 | 702.9 | 3/2 | |
| 3 | 1054.3 | 90/49 | 11/6, 24/13 |
| 4 | 205.8 | 9/8 | |
| 5 | 557.2 | 112/81 | 11/8, 18/13 |
| 6 | 908.7 | 27/16 | 22/13 |
| 7 | 60.1 | 28/27 | 33/32, 27/26 |
| 8 | 411.6 | 80/63, 81/64 | 14/11, 33/26 |
| 9 | 763.0 | 14/9 | |
| 10 | 1114.5 | 40/21 | 21/11 |
| 11 | 265.9 | 7/6 | |
| 12 | 617.4 | 10/7 | |
| 13 | 968.8 | 7/4 | |
| 14 | 120.2 | 15/14 | 14/13 |
| 15 | 471.7 | 21/16 | |
| 16 | 823.1 | 45/28 | 21/13 |
| 17 | 1174.6 | 63/32, 160/81 | |
| 18 | 326.0 | 98/81, 135/112 | 40/33 |
| 19 | 677.5 | 40/27 | |
| 20 | 1028.9 | 49/27 | 20/11 |
| 21 | 180.4 | 10/9 | |
| 22 | 531.8 | 49/36 | 15/11 |
| 23 | 883.3 | 5/3 | |
| 24 | 34.7 | 49/48, 50/49 | 40/39, 45/44, 55/54, 65/64 |
| 25 | 386.2 | 5/4 | |
| 26 | 737.6 | 49/32 | 20/13 |
| 27 | 1089.1 | 15/8 | |
| 28 | 240.5 | 147/128 | 15/13 |
| 29 | 591.9 | 45/32 | |
* In 7-limit CTE tuning
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 
), and 7/4 by a semi-augmented sixth (C–A 
). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C–F ), and 13/8 by the neutral sixth (C–A 
). This, of course, defies the tradition of tertian harmony. The just major triad on C is C – D – G, for example. One may want to adopt one or more additional modules of accidentals such as arrows or +/- signs to represent the comma steps. There are two notable comma steps:
- The syntonic~septimal comma (-17 gensteps, semidiminished second);
- The Pythagorean comma (+24 gensteps, inverse diminished second).
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma.
| Ratio | Nominal | Example |
|---|---|---|
| 3/2 | Perfect fifth | C–G |
| 5/4 | Down major third | C–vE |
| 7/4 | Down minor seventh | C–vBb |
| 11/8 | Semi-augmented fourth | C–Ft |
| 13/8 | Neutral sixth | C–Ad |
Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma.
| Ratio | Nominal | Example |
|---|---|---|
| 3/2 | Perfect fifth | C–G |
| 5/4 | Up neutral third | C–^Ed |
| 7/4 | Up semidiminished seventh | C–^Bdb |
| 11/8 | Semi-augmented fourth | C–Ft |
| 13/8 | Neutral sixth | C–Ad |
Chords
Scales
Tunings
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 11/9 | 347.408 | ||
| 11/6 | 349.788 | ||
| 7\24 | 350.000 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 11/8 | 350.264 | ||
| 3/2 | 350.978 | ||
| 12\41 | 351.220 | Lower bound of 11- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone | |
| 21/16 | 351.385 | ||
| 15/14 | 351.389 | ||
| 15/8 | 351.417 | ||
| 41\140 | 351.429 | ||
| 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 temperament with neutral intervals (351.47186 cents) | |
| 49/48 | 351.487 | ||
| 5/3 | 351.494 | ||
| 29\99 | 351.515 | ||
| 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 minimax | |
| 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 | ||
| 22\75 | 352.000 | ||
| 13/7 | 352.021 | ||
| 11/7 | 352.188 | ||
| 13/9 | 352.676 | ||
| 5\17 | 352.941 | Upper bound of 7- to 15-odd-limit and 13-limit 21-odd-limit diamond monotone | |
| 13/12 | 353.809 | ||
| 13/8 | 359.472 |
* Besides the octave