Garibaldi
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 |