Hemififths: Difference between revisions

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== Notation ==
== Notation ==
Hemififths can be notated in [[neutral circle-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. The just major triad on C is {{dash|C, D{{sesquisharp2}}, G|med}}, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step. There are two solutions:  
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. The just major triad on C is {{dash|C, D{{sesquisharp2}}, G|med}}, 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:  
# let an arrow represent a bend by the syntonic~septimal comma (17 gensteps, semidiminished second);  
# The syntonic~septimal comma (-17 gensteps, semidiminished second);  
# let an arrow represent a bend by the Pythagorean comma (24 gensteps, negative diminished 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.  
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma.  
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| 3/2
| 3/2
| Perfect fifth
| Perfect fifth
| C-G
| C–G
|-
|-
| 5/4
| 5/4
| Down major third
| Down major third
| C-vE
| C–vE
|-
|-
| 7/4
| 7/4
| Down minor seventh
| Down minor seventh
| C-vBb
| C–vBb
|-
|-
| 11/8
| 11/8
| Semi-augmented fourth
| Semi-augmented fourth
| C-F+
| C–Ft
|-
|-
| 13/8
| 13/8
| Neutral sixth
| Neutral sixth
| C-Ad
| C–Ad
|}
|}


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| 3/2
| 3/2
| Perfect fifth
| Perfect fifth
| C-G
| C–G
|-
|-
| 5/4
| 5/4
| Up neutral third
| Up neutral third
| C-^Ed
| C–^Ed
|-
|-
| 7/4
| 7/4
| Up semidiminished seventh
| Up semidiminished seventh
| C-^Bdb
| C–^Bdb
|-
|-
| 11/8
| 11/8
| Semi-augmented fourth
| Semi-augmented fourth
| C-F+
| C–Ft
|-
|-
| 13/8
| 13/8
| Neutral sixth
| Neutral sixth
| C-Ad
| C–Ad
|}
|}


Revision as of 14:26, 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 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:

  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.

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.

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 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

Notes

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