Hemififths: Difference between revisions

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→Interval chain: added ups and downs notation
Line 11: Line 11:
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! colspan="2" | Approximate Ratios
! colspan="2" | Approximate Ratios
! rowspan="2" |ups and downs
notation **
|-
|-
! 7-limit
! 7-limit
Line 19: Line 21:
| '''1/1'''
| '''1/1'''
|
|
|P1
|-
|-
| 1
| 1
Line 24: Line 27:
| 49/40, 60/49
| 49/40, 60/49
| 11/9, '''16/13''', 27/22, 39/32
| 11/9, '''16/13''', 27/22, 39/32
|~3 = ^m3 = vM3
|-
|-
| 2
| 2
Line 29: Line 33:
| '''3/2'''
| '''3/2'''
|
|
|P5
|-
|-
| 3
| 3
Line 34: Line 39:
| 90/49
| 90/49
| 11/6, 24/13
| 11/6, 24/13
|~7 = ^m7 = vM7
|-
|-
| 4
| 4
Line 39: Line 45:
| '''9/8'''
| '''9/8'''
|  
|  
|M2
|-
|-
| 5
| 5
Line 44: Line 51:
| 112/81
| 112/81
| '''11/8''', 18/13
| '''11/8''', 18/13
|~4 = ^4 = vA4
|-
|-
| 6
| 6
Line 49: Line 57:
| 27/16
| 27/16
| 22/13
| 22/13
|M6
|-
|-
| 7
| 7
Line 54: Line 63:
| 28/27
| 28/27
| 33/32, 27/26
| 33/32, 27/26
|^1 = \m2
|-
|-
| 8
| 8
Line 59: Line 69:
| 80/63, 81/64
| 80/63, 81/64
| 14/11, 33/26
| 14/11, 33/26
|M3
|-
|-
| 9
| 9
Line 64: Line 75:
| 14/9
| 14/9
|  
|  
|^5 = \m6
|-
|-
| 10
| 10
Line 69: Line 81:
| 40/21
| 40/21
| 21/11
| 21/11
|M7
|-
|-
| 11
| 11
Line 74: Line 87:
| 7/6
| 7/6
|  
|  
|^M2 = \m3
|-
|-
| 12
| 12
Line 79: Line 93:
| 10/7
| 10/7
|  
|  
|A4 = \~5
|-
|-
| 13
| 13
Line 84: Line 99:
| '''7/4'''
| '''7/4'''
|  
|  
|^M6 = \m7
|-
|-
| 14
| 14
Line 89: Line 105:
| 15/14
| 15/14
| 14/13
| 14/13
|A1 = \~2
|-
|-
| 15
| 15
Line 94: Line 111:
| '''21/16'''
| '''21/16'''
|  
|  
|^M3 = \4
|-
|-
| 16
| 16
Line 99: Line 117:
| 45/28
| 45/28
| 21/13
| 21/13
|A5 = \~6
|-
|-
| 17
| 17
Line 104: Line 123:
| 63/32, 160/81
| 63/32, 160/81
|
|
|^M7 = \8
|-
|-
| 18
| 18
Line 109: Line 129:
| 98/81, 135/112
| 98/81, 135/112
| 40/33
| 40/33
|A2 = \~3
|-
|-
| 19
| 19
Line 114: Line 135:
| 40/27
| 40/27
|  
|  
|^A4 = \5
|-
|-
| 20
| 20
Line 119: Line 141:
| 49/27
| 49/27
| 20/11
| 20/11
|A6 = \~7
|-
|-
| 21
| 21
Line 124: Line 147:
| 10/9
| 10/9
|  
|  
|^A1 = \M2
|-
|-
| 22
| 22
Line 129: Line 153:
| 49/36
| 49/36
| 15/11
| 15/11
|A3 = \~4
|-
|-
| 23
| 23
Line 134: Line 159:
| 5/3
| 5/3
|  
|  
|^A5 = \M6
|-
|-
| 24
| 24
Line 139: Line 165:
| 49/48, 50/49
| 49/48, 50/49
| 40/39, 45/44, 55/54, 65/64
| 40/39, 45/44, 55/54, 65/64
|A7 - P8 = -d2 = ^\1
|-
|-
| 25
| 25
Line 144: Line 171:
| '''5/4'''
| '''5/4'''
|  
|  
|^A2 = \M3
|-
|-
| 26
| 26
Line 149: Line 177:
| 49/32
| 49/32
| 20/13
| 20/13
|AA4 = ^\5
|-
|-
| 27
| 27
Line 154: Line 183:
| '''15/8'''
| '''15/8'''
|  
|  
|^A6 = \M7
|-
|-
| 28
| 28
Line 159: Line 189:
| 147/128
| 147/128
| 15/13
| 15/13
|AA1= ^\2
|-
|-
| 29
| 29
Line 164: Line 195:
| 45/32
| 45/32
|  
|  
|^A3 = \A4
|}
|}
<nowiki />* In 7-limit CTE tuning
<nowiki>*</nowiki> In 7-limit CTE tuning, generator = 351.445¢, P5 = 702.89¢, c = 2.89¢
 
<nowiki>**</nowiki> Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ - 8.5c


== Notation ==
== Notation ==
Line 172: Line 206:
# The Pythagorean comma (+24 gensteps, inverse 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 (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
Line 201: Line 235:
|}
|}


Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma.  
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

Revision as of 19:43, 15 November 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 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¢, 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 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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