Garibaldi

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Garibaldi 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. C-F♭), and the new mapping specific to garibaldi is that 7/4 is mapped to the double diminished octave (e.g. C-Cbb). This makes garibaldi a marvel temperament.

Immediate 11-limit extensions include cassandra (41 & 53), mapping 11/8 to +23 fifths, andromeda (29 & 41), mapping 11/8 to -18 fifths, and helenus (53 & 65d), mapping 11/8 to -30 fifths. Garibaldi is most naturally a 2.3.5.7.19 subgroup temperament due to its immediate availability of 19/16 at the minor third (C-Eb). This is sometimes known as garibaldi nestoria.

Garibaldi was named in honor of Eduardo Sábat-Garibaldi, who developed the dinarra, a 53-tone microtonal guitar in the 1/9-schisma tuning.

Interval chain

In the following table, odd harmonics 1–21 are in bold.

# Cents* Approximate Ratios
2.3.5.7.19 Subgroup 13-limit Extension
Cassandra Andromeda Helenus
0 0.00 1/1
1 702.06 3/2
2 204.12 9/8
3 906.18 27/16, 32/19, 42/25 22/13 22/13 22/13
4 408.24 19/15, 24/19, 63/50, 80/63 14/11
5 1110.29 19/10, 36/19, 40/21 21/11
6 612.35 10/7
7 114.41 15/14, 16/15 14/13
8 816.47 8/5 21/13
9 318.53 6/5 40/33
10 1020.59 9/5, 38/21 20/11
11 522.65 19/14, 27/20 15/11
12 24.71 50/49, 57/56, 64/63, 81/80 40/39, 45/44
13 726.77 32/21 20/13
14 228.82 8/7 15/13
15 930.88 12/7 19/11
16 432.94 9/7 14/11
17 1135.00 27/14, 48/25 52/27 64/33 21/11
18 637.06 36/25, 81/56 13/9 16/11, 19/13
19 139.12 27/25 13/12 12/11 14/13
20 841.18 80/49, 81/50 13/8, 44/27 18/11, 64/39 21/13
21 343.24 60/49 11/9, 39/32 16/13, 27/22 40/33
22 1045.30 64/35 11/6 24/13 20/11
23 547.35 48/35 11/8, 26/19 18/13 15/11
24 49.41 36/35 33/32 27/26 40/39, 45/44
25 751.47 54/35 20/13
26 253.53 81/70, 144/125 22/19 15/13
27 955.59 216/125, 256/147 26/15 19/11
28 457.65 64/49 13/10
29 1159.71 96/49 39/20, 88/45 64/33
30 661.77 72/49 22/15 16/11, 19/13
31 163.83 54/49 11/10 12/11
32 865.88 81/49 33/20 18/11, 64/39
33 367.94 216/175 26/21 16/13, 27/22
34 1070.00 324/175 13/7 24/13
35 572.06 243/175 18/13
36 74.12 256/245 22/21 27/26
37 776.18 384/245 11/7
38 278.24 288/245
39 980.30 432/245
40 482.36 324/245
41 1184.41 486/245

* in 7-limit CTE tuning

Notation

Using garibaldi can be a challenge because it defies the tradition of tertian harmony in circle-of-fifths notation. The just major triad on C is C-Fb-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C-vE-G.

Cassandra 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 Double-up fourth C-^^F
13/8 Double-up minor sixth C-^^Ab
19/16 Minor third C-Eb
Andromeda nomenclature for selected intervals
Ratio Nominal Example
11/8 Down diminished fifth
Double-down augmented fourth 		C-vGb
C-vvF#
13/8 Double down major sixth C-vvA
Helenus nomenclature for selected intervals
Ratio Nominal Example
11/8 Double-down diminished fifth
Triple-down augmented fourth 		C-vvGb
C-v3F#
13/8 Triple-down major sixth C-v3A

Chords and harmony

Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:

  • 1-19/15-3/2 (C-E-G)
  • 1-19/16-3/2 (C-Eb-G)

Note that the major third also represents 24/19, and the minor third, 13/11. These chords are typically associated with a sort of coldness and metalness, like those in 12edo if not more so.

If a warm, sweet, laid-back sound is desired, the thirds can be inflected inwards by a comma to yield

  • 1-5/4-3/2 (C-vE-G)
  • 1-6/5-3/2 (C-^Eb-G)

Contrarily, for a more sour and active sound, they can be inflected outwards by a comma to yield

  • 1-9/7-3/2 (C-^E-G)
  • 1-7/6-3/2 (C-vEb-G)

Scales

Tunings

Tuning spectra

Garibaldi

Tuning spectrum for garibaldi
Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9-odd-limit,
2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
19/12, 24/19 701.1105 1/4 undevicesimal schisma
38\65 701.5385
15/8, 16/15 701.676 1/7 schisma
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
5/3, 5/4 701.738 5-odd-limit minimax, 1/9 schisma
9/5, 10/9 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
31\53 701.8868
3/2 701.9550 Pythagorean tuning
36/35 702.0321
9/7 702.193 9-odd-limit minimax, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4, 8/7 702.227 1/14 septimal schisma
19/10, 20/19 702.2399
21/16 702.2476 1/13 septimal schisma
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390
19/14 702.6079
21/19 702.6732
15/14 702.778
7/5, 10/7, 50/49 702.915
21/20 703.1066
17\29 703.4483 Upper bound of 9-odd-limit,
2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone
13/11 703.597

Cassandra

Gencom: [2 4/3; 225/224 275/273 325/324 385/384]

Gencom mapping: [⟨1 2 -1 -3 13 12], ⟨0 -1 8 14 -23 -20]]

Tuning spectrum for cassandra
Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
19/12 701.1105 1/4 undevicesimal schisma
38\65 701.5385
15/8 701.676 1/7 schisma
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
[0 -10 17⟩ 701.728 5-odd-limit least squares
5/3 701.738 5-odd-limit minimax, 1/9 schisma
9/5 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
19/13 701.8702
31\53 701.8868 Lower bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
15/13 701.9355
13/10 701.9362
3/2 701.9550 Pythagorean tuning
13/8 702.026
13/12 702.030
36/35 702.0321
13/9 702.034
19/11 702.0694
11/10 702.097
15/11 702.102
13/7 702.109 13- and 15-odd-limit minimax
[0 -95 -137 -129 167 143⟩ 702.112 15-odd-limit least squares
21/13 702.1135
[0 -27 7 17⟩ 702.114 9-odd-limit least squares
[0 -38 -80 -122 137 116⟩ 702.128 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, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4 702.227 1/14 septimal schisma
11/7 702.230
11/8 702.231
21/11 702.2371
19/10 702.2399
11/6 702.244
21/16 702.2476 1/13 septimal schisma
11/9 702.258
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390 Upper bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/14 702.6079
21/19 702.6732
15/14 702.778
7/5 702.915
21/20 703.1066
17\29 703.4483 Upper bound of 9-odd-limit diamond monotone
13/11 703.597

Andromeda

Gencom: [2 4/3; 100/99 105/104 196/195 245/242]

Gencom mapping: [⟨1 2 -1 -3 -4 -5], ⟨0 -1 8 14 18 21]]

Tuning spectrum for andromeda
Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9- and 11-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
19/12 701.1105 1/4 undevicesimal schisma
38\65 701.5385
15/8 701.676 1/7 schisma
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
5/3 701.738 5-odd-limit minimax, 1/9 schisma
9/5 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
31\53 701.8868
3/2 701.9550 Pythagorean tuning
36/35 702.0321
9/7 702.193 9-odd-limit minimax, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4 702.227 1/14 septimal schisma
21/16 702.2476 1/13 septimal schisma
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390 Lower bound of 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/14 702.6079
11/9 702.630 11-odd-limit minimax
11/6 702.665
21/19 702.6732
11/8 702.705
13/9 702.756 13- and 15-odd-limit minimax
15/14 702.778
13/12 702.792
13/8 702.832
7/5 702.915
19/11 703.0797
21/20 703.1066
19/13 703.1659
15/11 703.359
15/13 703.410
17\29 703.4483 Upper bound of 9-, 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
11/10 703.500
13/10 703.522
13/11 703.597
21/13 701.7817
19/10 702.2399
21/11 703.8926
13/7 704.043
11/7 704.377

Helenus

Gencom: [2 4/3; 99/98 176/175 275/273 847/845]

Gencom mapping: [⟨1 2 -1 -3 -9 -10], ⟨0 -1 8 14 30 33]]

Tuning spectrum for helenus
Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9- and 11-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
11/7 701.094
19/12 701.1105 1/4 undevicesimal schisma
21/11 701.1149
13/7 701.489
21/13 701.5127
38\65 701.5385 Lower bound of 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
11/10 701.591
15/11 701.607
11/8 701.623
11/6 701.633
11/9 701.644 11-, 13-, and 15-odd-limit minimax
15/8 701.676 1/7 schisma
19/11 701.7109
5/4 701.711 1/8 schisma
25/24 701.7252 2/17 schisma
5/3 701.738 5-odd-limit minimax, 1/9 schisma
9/5 701.760 1/10 schisma
81/80 701.7922 1/12 schisma
13/8 701.802
13/12 701.807
13/9 701.811
13/10 701.831
15/13 701.836
31\53 701.8868 Upper bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/13 701.8995
3/2 701.9550 Pythagorean tuning
36/35 702.0321
9/7 702.193 9-odd-limit minimax, 1/16 septimal schisma
7/6 702.209 7-odd-limit minimax, 1/15 septimal schisma
49/48 702.2174 2/29 septimal schisma
7/4 702.227 1/14 septimal schisma
19/10 702.2399
21/16 702.2476 1/13 septimal schisma
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390
19/14 702.6079
21/19 702.6732
15/14 702.778
7/5 702.915
21/20 703.1066
17\29 703.4483 Upper bound of 9-odd-limit diamond monotone
13/11 703.597
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