Garibaldi: Difference between revisions
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|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]]. | '''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. Garibaldi is most naturally a 2.3.5.7.19 [[subgroup]] temperament due to its immediate availability of [[19/16]] at the minor third. | 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). | ||
== Interval chain == | == Interval chain == | ||
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== Tuning spectra == | == Chords and harmony == | ||
=== Cassandra === | 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 == | |||
* [[Garibaldi5]] – proper [[2L 3s]] | |||
* [[Garibaldi7]] – improper [[5L 2s]] | |||
* [[Garibaldi12]] – proper [[5L 7s]] | |||
* [[Garibaldi17]] – improper [[12L 5s]] | |||
* [[Garibaldi24opt]] – optimized 24-note scale for 13-limit | |||
== Tunings == | |||
=== Tuning spectra === | |||
==== Cassandra ==== | |||
Gencom: [2 4/3; 225/224 275/273 325/324 385/384] | Gencom: [2 4/3; 225/224 275/273 325/324 385/384] | ||
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=== Andromeda === | ==== Andromeda ==== | ||
Gencom: [2 4/3; 100/99 105/104 196/195 245/242] | Gencom: [2 4/3; 100/99 105/104 196/195 245/242] | ||
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=== Helenus === | ==== Helenus ==== | ||
Gencom: [2 4/3; 99/98 176/175 275/273 847/845] | Gencom: [2 4/3; 99/98 176/175 275/273 847/845] | ||
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[[Category:Garibaldi| ]] <!-- main article --> | [[Category:Garibaldi| ]] <!-- main article --> | ||