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 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 doubly-diminished octave (e.g. C–C𝄫). This makes garibaldi a [[marvel temperaments|marvel temperament]].

'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|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). 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 doubly-diminished octave (e.g. C–C𝄫; a comma-flat Pythagorean minor seventh). This makes garibaldi a [[marvel temperaments|marvel 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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== Notation ==

Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[chain-of-fifths notation]]. The just major triad on C is C–Fb–G, for example. ~~One may want~~ to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.

Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[chain-of-fifths notation]]. The just major triad on C is C–Fb–G, for example. Due to the generalized comma of garibaldi, a natural choice is to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.

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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 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 doubly-diminished octave (e.g. C–C&#x1D12B;). This makes garibaldi a [[marvel temperaments|marvel temperament]].
+'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|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). 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 doubly-diminished octave (e.g. C–C&#x1D12B;; a comma-flat Pythagorean minor seventh). This makes garibaldi a [[marvel temperaments|marvel temperament]].
 Immediate 11-limit extensions include ''cassandra'' {{nowrap|(41 &amp; 53)}}, mapping 11/8 to +23 fifths, ''andromeda'' {{nowrap|(29 &amp; 41)}}, mapping 11/8 to &minus;18 fifths, and ''helenus'' {{nowrap|(53 &amp; 65d)}}, mapping 11/8 to &minus;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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 == Notation ==
-Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[chain-of-fifths notation]]. The just major triad on C is C–Fb–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.
+Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[chain-of-fifths notation]]. The just major triad on C is C–Fb–G, for example. Due to the generalized comma of garibaldi, a natural choice is to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.
 {| class="wikitable center-1 center-3"