Hemififths: Difference between revisions

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| ^A3 = \A4
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<nowiki/>* In 7-limit CTE tuning, {{nowrap|generator {{=}} 351.445¢ }}, {{nowrap|P5 {{=}} 702.89¢}} and {{nowrap|c {{=}} 2.89¢}}
<nowiki/>* In 7-limit CWE tuning, {{nowrap|generator {{=}} 351.467{{c}} }}, {{nowrap| P5 {{=}} 702.934{{c}} }} and {{nowrap| c {{=}} 2.934{{c}} }}


<nowiki/>** Enharmonic equivalences: vvA1 and v\m2. Cents: {{nowrap|^1 {{=}} 50¢ + 3.5c}} and {{nowrap|/1 {{=}} 50¢ − 8.5c}}
<nowiki/>** Enharmonic equivalences: vvA1 and v\m2. Cents: {{nowrap| ^1 {{=}} 50¢ + 3.5c }} and {{nowrap| /1 {{=}} 50¢ − 8.5c }}


== Notation ==
== Notation ==

Revision as of 14:40, 18 March 2025

Hemififths is a 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 hemifamity comma, 5120/5103. It extends fairly naturally to the 11- and 13-limit by treating the generator as 11/9~16/13. The no-5 subgroup restriction, called hemif, is also notable. Possible tunings include 41-, 58-, and 99edo.

Hemififths was named by Gene Ward Smith in 2004[1].

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

Interval chain

In the following table, odd harmonics 1–21 and their inversions are labeled in bold.

# Cents* Approximate ratios Ups and downs
notation**
7-limit 13-limit extension
0 0.0 1/1 P1
1 351.5 49/40, 60/49 11/9, 16/13, 27/22, 39/32 ~3 = ^m3 = vM3
2 702.9 3/2 P5
3 1054.4 90/49 11/6, 24/13 ~7 = ^m7 = vM7
4 205.9 9/8 M2
5 557.3 112/81 11/8, 18/13 ~4 = ^4 = vA4
6 908.8 27/16 22/13 M6
7 60.3 28/27 33/32, 27/26 ^1 = \m2
8 411.7 80/63, 81/64 14/11, 33/26 M3
9 763.2 14/9 ^5 = \m6
10 1114.7 40/21 21/11 M7
11 266.1 7/6 ^M2 = \m3
12 617.6 10/7 A4 = \~5
13 969.1 7/4 ^M6 = \m7
14 120.5 15/14 14/13 A1 = \~2
15 472.0 21/16 ^M3 = \4
16 823.5 45/28 21/13 A5 = \~6
17 1174.9 63/32, 160/81 55/28, 65/33, 77/39 ^M7 = \8
18 326.4 98/81, 135/112 40/33 A2 = \~3
19 677.9 40/27 ^A4 = \5
20 1029.3 49/27 20/11 A6 = \~7
21 180.8 10/9 ^A1 = \M2
22 532.3 49/36 15/11 A3 = \~4
23 883.7 5/3 ^A5 = \M6
24 35.2 49/48, 50/49 40/39, 45/44, 55/54, 65/64 A7 - P8 = -d2 = ^\1
25 386.7 5/4 ^A2 = \M3
26 738.1 49/32 20/13 AA4 = ^\5
27 1089.6 15/8 ^A6 = \M7
28 241.1 147/128 15/13 AA1= ^\2
29 592.5 45/32 ^A3 = \A4

* In 7-limit CWE tuning, generator = 351.467 ¢, P5 = 702.934 ¢ and c = 2.934 ¢

** Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ − 8.5c

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:

  1. The syntonic~septimal comma (-17 gensteps, semidiminished second);
  2. 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 (thus ^C = Ddb).

Hemififths nomenclature
for selected intervals
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 (thus ^C = B#).

Hemififths nomenclature
for selected intervals
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

Main article: Chords of hemififths

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 tuning 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

References

  1. ↑ Yahoo! Tuning Group (Archive) | Names for important high-complexity temperaments
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