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.
Interval chain
In the following table, prime harmonics are in bold.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 702.1 | 3/2 |
| 2 | 204.2 | 9/8 |
| 3 | 906.3 | 22/13, 27/16 |
| 4 | 408.5 | 33/26, 63/50 |
| 5 | 1110.6 | 40/21 |
| 6 | 612.7 | 10/7 |
| 7 | 114.8 | 15/14, 16/15 |
| 8 | 816.9 | 8/5 |
| 9 | 319.0 | 6/5 |
| 10 | 1021.1 | 9/5 |
| 11 | 523.2 | 27/20 |
| 12 | 25.4 | 50/49, 64/63, 81/80 |
| 13 | 727.5 | 32/21 |
| 14 | 229.6 | 8/7 |
| 15 | 931.7 | 12/7 |
| 16 | 433.8 | 9/7 |
| 17 | 1135.9 | 27/14, 48/25 |
| 18 | 638.0 | 13/9 |
| 19 | 140.1 | 13/12 |
| 20 | 842.3 | 13/8 |
| 21 | 344.4 | 11/9 |
| 22 | 1046.5 | 11/6 |
| 23 | 548.6 | 11/8 |
| 24 | 50.7 | 33/32, 36/35 |
| 25 | 752.8 | 54/35 |
| 26 | 254.9 | 81/70 |
| 27 | 957.0 | 15/13 |
| 28 | 459.2 | 13/10 |
| 29 | 1161.3 | 39/20, 88/45, 96/49 |
* in 13-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 |