Garibaldi: Difference between revisions

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{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! #
! rowspan="2" | #
! Cents*
! rowspan="2" | Cents*
! Approximate Ratios
! colspan="4" | Approximate Ratios
|-
!7-limit
!13-limit Extension<br>Cassandra Mapping
!13-limit Extension<br>Andromeda Mapping
!13-limit Extension<br>Helenus Mapping
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| 1/1
|
|
|
|-
|-
| 1
| 1
| 702.1
| 702.1
| '''3/2'''
| '''3/2'''
|
|
|
|-
|-
| 2
| 2
| 204.2
| 204.2
| 9/8
| 9/8
|
|
|
|-
|-
| 3
| 3
| 906.3
| 906.3
| 22/13, 27/16
| 27/16, 42/25
| 22/13
| 22/13
| 22/13
|-
|-
| 4
| 4
| 408.5
| 408.3
| 33/26, 63/50
| 63/50, 80/63
|
| 14/11
|
|-
|-
| 5
| 5
| 1110.6
| 1110.4
| 40/21
| 40/21
|
| 21/11
|
|-
|-
| 6
| 6
| 612.7
| 612.5
| 10/7
| 10/7
|
|
|
|-
|-
| 7
| 7
| 114.8
| 114.6
| 15/14, 16/15
| 15/14, 16/15
|
| 14/13
|
|-
|-
| 8
| 8
| 816.9
| 816.7
| '''8/5'''
| '''8/5'''
|
| 21/13
|
|-
|-
| 9
| 9
| 319.0
| 318.8
| 6/5
| 6/5
|
| 40/33
|
|-
|-
| 10
| 10
| 1021.1
| 1020.9
| 9/5
| 9/5
|
| 20/11
|
|-
|-
| 11
| 11
| 523.2
| 522.9
| 27/20
| 27/20
|
| 15/11
|
|-
|-
| 12
| 12
| 25.4
| 25.0
| 50/49, 64/63, 81/80
| 50/49, 64/63, 81/80
|
| 40/39, 45/44
|
|-
|-
| 13
| 13
| 727.5
| 727.1
| 32/21
| 32/21
|
| 20/13
|
|-
|-
| 14
| 14
| 229.6
| 229.2
| '''8/7'''
| '''8/7'''
|
| 15/13
|
|-
|-
| 15
| 15
| 931.7
| 931.3
| 12/7
| 12/7
|
|
|
|-
|-
| 16
| 16
| 433.8
| 433.4
| 9/7
| 9/7
|
|
| 14/11
|-
|-
| 17
| 17
| 1135.9
| 1135.4
| 27/14, 48/25
| 27/14, 48/25
| 52/27
| 64/33
| 21/11
|-
|-
| 18
| 18
| 638.0
| 637.5
| 36/25, 81/56
| 13/9
| 13/9
| '''16/11'''
|
|-
|-
| 19
| 19
| 140.1
| 139.6
| 27/25
| 13/12
| 13/12
| 12/11
| 14/13
|-
|-
| 20
| 20
| 842.3
| 841.7
| '''13/8'''
| 80/49, 81/50
| '''13/8''', 44/27
| 18/11, 64/39
| 21/13
|-
|-
| 21
| 21
| 344.4
| 343.8
| 11/9
| 60/49
| 11/9, 39/32
| '''16/13''', 27/22
| 40/33
|-
|-
| 22
| 22
| 1046.5
| 1045.9
| 64/35
| 11/6
| 11/6
| 24/13
| 20/11
|-
|-
| 23
| 23
| 548.6
| 548.0
| 48/35
| '''11/8'''
| '''11/8'''
| 18/13
| 15/11
|-
|-
| 24
| 24
| 50.7
| 50.0
| 33/32, 36/35
| 36/35
| 33/32
| 27/26
| 40/39, 45/44
|-
|-
| 25
| 25
| 752.8
| 752.1
| 54/35
| 54/35
|
|
| 20/13
|-
|-
| 26
| 26
| 254.9
| 254.2
| 81/70
| 81/70, 144/125
|
|
| 15/13
|-
|-
| 27
| 27
| 957.0
| 956.3
| 15/13
| 216/125, 256/147
| 26/15
|
|
|-
|-
| 28
| 28
| 459.2
| 458.4
| 64/49
| 13/10
| 13/10
|
|
|-
|-
| 29
| 29
| 1161.3
| 1160.5
| 39/20, 88/45, 96/49
| 96/49
| 39/20, 88/45
|
| 64/33
|}
|}
<nowiki>*</nowiki> in 13-limit POTE tuning
<nowiki>*</nowiki> in 7-limit POTE tuning


== Scales ==
== Scales ==

Revision as of 02:47, 3 June 2021

Garibaldi temperament is a 7-limit (and higher) temperament of the schismatic family. It is an extension of helmholtz temperament beyond the 5-limit but with the same simple chain-of-fifths structure (so that standard notation may be used). As in helmholtz temperament, 5/4 is mapped to the diminished fourth (e.g. A-Db), and the new mapping specific to garibaldi is that 7/4 is mapped to the double diminished octave (e.g. A-Abb). This makes garibaldi a marvel temperament.

Immediate 11-limit extensions include cassandra (41&53), mapping 11/8 to +23 steps, andromeda (29&41), mapping 11/8 to -18 steps, and helenus (53&65d), mapping 11/8 to -30 steps.

Interval chain

In the following table, prime harmonics are in bold.

# Cents* Approximate Ratios
7-limit 13-limit Extension
Cassandra Mapping 		13-limit Extension
Andromeda Mapping 		13-limit Extension
Helenus Mapping
0 0.0 1/1
1 702.1 3/2
2 204.2 9/8
3 906.3 27/16, 42/25 22/13 22/13 22/13
4 408.3 63/50, 80/63 14/11
5 1110.4 40/21 21/11
6 612.5 10/7
7 114.6 15/14, 16/15 14/13
8 816.7 8/5 21/13
9 318.8 6/5 40/33
10 1020.9 9/5 20/11
11 522.9 27/20 15/11
12 25.0 50/49, 64/63, 81/80 40/39, 45/44
13 727.1 32/21 20/13
14 229.2 8/7 15/13
15 931.3 12/7
16 433.4 9/7 14/11
17 1135.4 27/14, 48/25 52/27 64/33 21/11
18 637.5 36/25, 81/56 13/9 16/11
19 139.6 27/25 13/12 12/11 14/13
20 841.7 80/49, 81/50 13/8, 44/27 18/11, 64/39 21/13
21 343.8 60/49 11/9, 39/32 16/13, 27/22 40/33
22 1045.9 64/35 11/6 24/13 20/11
23 548.0 48/35 11/8 18/13 15/11
24 50.0 36/35 33/32 27/26 40/39, 45/44
25 752.1 54/35 20/13
26 254.2 81/70, 144/125 15/13
27 956.3 216/125, 256/147 26/15
28 458.4 64/49 13/10
29 1160.5 96/49 39/20, 88/45 64/33

* in 7-limit POTE tuning

Scales

Spectrum of garibaldi tunings by eigenmonzos

Eigenmonzo Fifth Comments
16/15 701.676
(69\118) 701.695
5/4 701.711
[0 -10 17⟩ 701.728 5-odd-limit least squares
6/5 701.738 5-odd-limit minimax
100\171 701.754
10/9 701.760
(31\53) 701.887
15/13 701.9355
13/10 701.9362
4/3 701.955
16/13 702.026
13/12 702.030
18/13 702.034
86\147 702.041
11/10 702.097
15/11 702.102
14/13 702.109 13- and 15-odd-limit minimax
[0 -95 -137 -129 167 143⟩ 702.112 15-odd-limit least squares
[0 -27 7 17⟩ 702.114 9-odd-limit least squares
(55\94) 702.12766
[0 -38 -80 -122 137 116⟩ 702.12770 13-odd-limit least squares
[0 -25 11 35⟩ 702.140 7-odd-limit least squares
[0 17 -52 -88 134⟩ 702.183 11-odd-limit least squares
9/7 702.193 9- and 11-odd-limit minimax
7/6 702.209 7-odd-limit minimax
(79\135) 702.222
8/7 702.227
14/11 702.230
11/8 702.231
12/11 702.244
11/9 702.258
(24\41) 702.439
15/14 702.778
7/5 702.915
(17\29) 703.448
13/11 703.597

Template:IoT

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