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 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''' is a [[regular temperament|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 [[41edo|41-]], [[58edo|58-]], and [[99edo]].  


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>.  


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


== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 are labeled in '''bold'''.  
In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.  
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2" | &#35;
! rowspan="2" | #
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! colspan="2" | Approximate Ratios
! colspan="2" | Approximate ratios
! rowspan="2" | [[Ups and downs notation|Ups and downs<br />notation]]**
! rowspan="2" | [[Ups and downs notation|Ups and downs<br>notation]]**
|-
|-
! 7-limit
! 7-limit
! 13-limit Extension
! 13-limit extension
|-
|-
| 0
| 0
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| ^A3 = \A4
| ^A3 = \A4
|}
|}
<nowiki />* In 7-limit CTE tuning, {{nowrap|generator {{=}} 351.445¢,|P5 {{=}} 702.89¢|and c {{=}} 2.89¢}}
<nowiki/>* In 7-limit CTE tuning, {{nowrap|generator {{=}} 351.445¢ }}, {{nowrap|P5 {{=}} 702.89¢}} and {{nowrap|c {{=}} 2.89¢}}


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


== Notation ==
== Notation ==
Line 208: Line 208:
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).  
Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).  
{| 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
|-
|-
! Ratio
! Ratio
Line 237: Line 237:
Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).  
Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).  
{| 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
|-
|-
! Ratio
! Ratio
Line 274: Line 274:
== Tunings ==
== Tunings ==
=== Tuning spectrum ===
=== Tuning spectrum ===
{| 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 310: Line 309:
|  
|  
| 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 350: Line 349:
| 25/24
| 25/24
| 351.472
| 351.472
| Very close to [[Logarithmic approximants#Argent temperament|argent temperament]] with neutral intervals (351.47186 cents)
| Very close to [[Logarithmic approximants #Argent temperament|argent tuning]] with neutral intervals (351.47186 cents)
|-
|-
|  
|  
Line 450: Line 449:
|  
|  
| 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 462: Line 461:
|  
|  
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


== References ==
== References ==
<references />
<references/>


[[Category:Temperaments]]
[[Category:Temperaments]]

Revision as of 14:28, 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
1 351.4 49/40, 60/49 11/9, 16/13, 27/22, 39/32 ~3 = ^m3 = vM3
2 702.9 3/2 P5
3 1054.3 90/49 11/6, 24/13 ~7 = ^m7 = vM7
4 205.8 9/8 M2
5 557.2 112/81 11/8, 18/13 ~4 = ^4 = vA4
6 908.7 27/16 22/13 M6
7 60.1 28/27 33/32, 27/26 ^1 = \m2
8 411.6 80/63, 81/64 14/11, 33/26 M3
9 763.0 14/9 ^5 = \m6
10 1114.5 40/21 21/11 M7
11 265.9 7/6 ^M2 = \m3
12 617.4 10/7 A4 = \~5
13 968.8 7/4 ^M6 = \m7
14 120.2 15/14 14/13 A1 = \~2
15 471.7 21/16 ^M3 = \4
16 823.1 45/28 21/13 A5 = \~6
17 1174.6 63/32, 160/81 ^M7 = \8
18 326.0 98/81, 135/112 40/33 A2 = \~3
19 677.5 40/27 ^A4 = \5
20 1028.9 49/27 20/11 A6 = \~7
21 180.4 10/9 ^A1 = \M2
22 531.8 49/36 15/11 A3 = \~4
23 883.3 5/3 ^A5 = \M6
24 34.7 49/48, 50/49 40/39, 45/44, 55/54, 65/64 A7 - P8 = -d2 = ^\1
25 386.2 5/4 ^A2 = \M3
26 737.6 49/32 20/13 AA4 = ^\5
27 1089.1 15/8 ^A6 = \M7
28 240.5 147/128 15/13 AA1= ^\2
29 591.9 45/32 ^A3 = \A4

* In 7-limit CTE tuning, generator = 351.445¢, P5 = 702.89¢ and c = 2.89¢

** 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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