Garibaldi

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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. 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.

Interval chain

In the following table, prime harmonics are in bold.

# Cents* Approximate Ratios
7-limit 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, 42/25 22/13 22/13 22/13
4 408.24 63/50, 80/63 14/11
5 1110.29 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 20/11
11 522.65 27/20 15/11
12 24.71 50/49, 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
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
19 139.12 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 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 15/13
27 955.59 216/125, 256/147 26/15
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
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 39/28 18/13
36 74.12 256/245 27/26
37 776.18 384/245
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

Tuning spectra

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]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
16/15 701.676
5/4 701.711
[0 -10 17⟩ 701.728 5-odd-limit least squares
6/5 701.738 5-odd-limit minimax
10/9 701.760
15/13 701.9355
13/10 701.9362
4/3 701.955
16/13 702.026
13/12 702.030
18/13 702.034
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
[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
7/6 702.209 7-odd-limit minimax
8/7 702.227
14/11 702.230
11/8 702.231
12/11 702.244
11/9 702.258
15/14 702.778
7/5 702.915
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]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
16/15 701.676
5/4 701.711
6/5 701.738 5-odd-limit minimax
10/9 701.760
4/3 701.955
9/7 702.193 9-odd-limit minimax
7/6 702.209 7-odd-limit minimax
8/7 702.227
11/9 702.630 11-odd-limit minimax
12/11 702.665
11/8 702.705
18/13 702.756 13 and 15-odd-limit minimax
15/14 702.778
13/12 702.792
16/13 702.832
7/5 702.915
15/11 703.359
15/13 703.410
11/10 703.500
13/10 703.522
13/11 703.597
14/13 704.043
14/11 704.377

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]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
14/11 701.094
14/13 701.489
11/10 701.591
15/11 701.607
11/8 701.623
12/11 701.633
11/9 701.644 11, 13, and 15-odd-limit minimax
16/15 701.676
5/4 701.711
6/5 701.738 5-odd-limit minimax
10/9 701.760
16/13 701.802
13/12 701.807
18/13 701.811
13/10 701.831
15/13 701.836
4/3 701.955
9/7 702.193 9-odd-limit minimax
7/6 702.209 7-odd-limit minimax
8/7 702.227
15/14 702.778
7/5 702.915
13/11 703.597

Scales

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