Garibaldi: Difference between revisions
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document the 2.3.5.7.19 subgroup extension; linking |
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'''Garibaldi temperament''' is a 7-limit (and higher) temperament of the [[Schismatic family #Garibaldi|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 temperaments|marvel temperament]]. | '''Garibaldi temperament''' is a [[7-limit]] (and higher) temperament of the [[Schismatic family #Garibaldi|schismatic family]]. It is an extension of [[helmholtz]] temperament beyond the 5-limit but with the same simple [[chain of fifths|chain-of-fifths]] structure (so that [[chain-of-fifths notation|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 temperaments|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. | 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. | ||
== Interval chain == | == Interval chain == | ||
In the following table, | In the following table, odd harmonics 1–21 are in '''bold'''. | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
| Line 11: | Line 11: | ||
! colspan="4" | Approximate Ratios | ! colspan="4" | Approximate Ratios | ||
|- | |- | ||
! rowspan="2" | 7 | ! rowspan="2" | 2.3.5.7.19 Subgroup | ||
! colspan="3" | 13-limit Extension | ! colspan="3" | 13-limit Extension | ||
|- | |- | ||
| Line 34: | Line 34: | ||
| 2 | | 2 | ||
| 204.12 | | 204.12 | ||
| 9/8 | | '''9/8''' | ||
| | | | ||
| | | | ||
| Line 41: | Line 41: | ||
| 3 | | 3 | ||
| 906.18 | | 906.18 | ||
| 27/16, 42/25 | | 27/16, '''32/19''', 42/25 | ||
| 22/13 | | 22/13 | ||
| 22/13 | | 22/13 | ||
| Line 48: | Line 48: | ||
| 4 | | 4 | ||
| 408.24 | | 408.24 | ||
| 63/50, 80/63 | | 19/15, 24/19, 63/50, 80/63 | ||
| | | | ||
| 14/11 | | 14/11 | ||
| Line 55: | Line 55: | ||
| 5 | | 5 | ||
| 1110.29 | | 1110.29 | ||
| 40/21 | | 19/10, 36/19, 40/21 | ||
| | | | ||
| 21/11 | | 21/11 | ||
| Line 69: | Line 69: | ||
| 7 | | 7 | ||
| 114.41 | | 114.41 | ||
| 15/14, 16/15 | | 15/14, '''16/15''' | ||
| | | | ||
| 14/13 | | 14/13 | ||
| Line 90: | Line 90: | ||
| 10 | | 10 | ||
| 1020.59 | | 1020.59 | ||
| 9/5 | | 9/5, 38/21 | ||
| | | | ||
| 20/11 | | 20/11 | ||
| Line 97: | Line 97: | ||
| 11 | | 11 | ||
| 522.65 | | 522.65 | ||
| 27/20 | | 19/14, 27/20 | ||
| | | | ||
| 15/11 | | 15/11 | ||
| Line 104: | Line 104: | ||
| 12 | | 12 | ||
| 24.71 | | 24.71 | ||
| 50/49, 64/63, 81/80 | | 50/49, 57/56, 64/63, 81/80 | ||
| | | | ||
| 40/39, 45/44 | | 40/39, 45/44 | ||
| Line 111: | Line 111: | ||
| 13 | | 13 | ||
| 726.77 | | 726.77 | ||
| 32/21 | | '''32/21''' | ||
| | | | ||
| 20/13 | | 20/13 | ||
| Line 127: | Line 127: | ||
| 12/7 | | 12/7 | ||
| | | | ||
| | | 19/11 | ||
| | | | ||
|- | |- | ||
| Line 148: | Line 148: | ||
| 36/25, 81/56 | | 36/25, 81/56 | ||
| 13/9 | | 13/9 | ||
| '''16/11''' | | '''16/11''', 19/13 | ||
| | | | ||
|- | |- | ||
| Line 182: | Line 182: | ||
| 547.35 | | 547.35 | ||
| 48/35 | | 48/35 | ||
| '''11/8''' | | '''11/8''', 26/19 | ||
| 18/13 | | 18/13 | ||
| 15/11 | | 15/11 | ||
| Line 203: | Line 203: | ||
| 253.53 | | 253.53 | ||
| 81/70, 144/125 | | 81/70, 144/125 | ||
| | | 19/11 | ||
| | | | ||
| 15/13 | | 15/13 | ||
| Line 212: | Line 212: | ||
| 26/15 | | 26/15 | ||
| | | | ||
| | | 19/11 | ||
|- | |- | ||
| 28 | | 28 | ||
| Line 233: | Line 233: | ||
| 22/15 | | 22/15 | ||
| | | | ||
| '''16/11''' | | '''16/11''', 19/13 | ||
|- | |- | ||
| 31 | | 31 | ||