Garibaldi: Difference between revisions

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'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family #Garibaldi|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|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). The garibaldi temperament tempers together the Pythagorean, syntonic, and Archytas commas into a singular generalized "comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat Pythagorean major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat Pythagorean minor seventh). This makes garibaldi a [[marvel temperaments|marvel temperament]]~~. Garibaldi can also be seen as~~ a ~~temperament of~~ [[hemifamity]]~~, which further identifies hemifamity's 81/80~64/63 with 531441/524288, the Pythagorean comma, collapsing it to a rank-2 structure~~.

'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family #Garibaldi|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|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). The garibaldi temperament tempers together the Pythagorean, syntonic, and Archytas commas into a singular generalized "comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat Pythagorean major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat Pythagorean minor seventh). This makes garibaldi a [[marvel temperaments|marvel temperament]] and a [[hemifamity temperaments|hemifamity temperament]].

Immediate 11-limit extensions include ''cassandra'' ({{nowrap|41 & 53}}), mapping 11/8 to +23 fifths, ''andromeda'' ({{nowrap|29 & 41}}), mapping 11/8 to −18 fifths, and ''helenus'' ({{nowrap|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–E♭). This is sometimes known as ''garibaldi nestoria''.

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	'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament\|temperament]] of the [[schismatic family #Garibaldi\|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)\|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). The garibaldi temperament tempers together the Pythagorean, syntonic, and Archytas commas into a singular generalized "comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat Pythagorean major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat Pythagorean minor seventh). This makes garibaldi a [[marvel temperaments\|marvel temperament]]~~. Garibaldi can also be seen as~~ a ~~temperament of~~ [[hemifamity]]~~, which further identifies hemifamity's 81/80~64/63 with 531441/524288, the Pythagorean comma, collapsing it to a rank-2 structure~~.		'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament\|temperament]] of the [[schismatic family #Garibaldi\|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)\|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). The garibaldi temperament tempers together the Pythagorean, syntonic, and Archytas commas into a singular generalized "comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat Pythagorean major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat Pythagorean minor seventh). This makes garibaldi a [[marvel temperaments\|marvel temperament]] and a [[hemifamity temperaments\|hemifamity temperament]].

	Immediate 11-limit extensions include ''cassandra'' ({{nowrap\|41 & 53}}), mapping 11/8 to +23 fifths, ''andromeda'' ({{nowrap\|29 & 41}}), mapping 11/8 to −18 fifths, and ''helenus'' ({{nowrap\|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–E♭). This is sometimes known as ''garibaldi nestoria''.		Immediate 11-limit extensions include ''cassandra'' ({{nowrap\|41 & 53}}), mapping 11/8 to +23 fifths, ''andromeda'' ({{nowrap\|29 & 41}}), mapping 11/8 to −18 fifths, and ''helenus'' ({{nowrap\|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–E♭). This is sometimes known as ''garibaldi nestoria''.