Garibaldi: Difference between revisions
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'''Garibaldi''' 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. | '''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]]. | ||
Immediate 11-limit extensions include ''cassandra'' (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 ( | Immediate 11-limit extensions include ''cassandra'' (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''. | ||
Garibaldi was named in honor of [[Eduardo Sábat-Garibaldi]], who developed the [[dinarra]], a 53-tone [[microtonal guitar]] in the 1/9-schisma tuning. | Garibaldi was named in honor of [[Eduardo Sábat-Garibaldi]], who developed the [[dinarra]], a 53-tone [[microtonal guitar]] in the 1/9-schisma tuning. | ||
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! rowspan="3" | # | ! rowspan="3" | # | ||
! rowspan="3" | Cents* | ! rowspan="3" | Cents* | ||
! colspan="4" | Approximate | ! colspan="4" | Approximate ratios | ||
|- | |- | ||
! rowspan="2" | 2.3.5.7.19 | ! rowspan="2" | 2.3.5.7.19 subgroup | ||
! colspan="3" | 13-limit | ! colspan="3" | 13-limit extension | ||
|- | |- | ||
! Cassandra | ! Cassandra | ||
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== Notation == | == Notation == | ||
Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[ | 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. | ||
{| class="wikitable center-1 center-3" | {| class="wikitable center-1 center-3" | ||
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| 3/2 | | 3/2 | ||
| Perfect fifth | | Perfect fifth | ||
| | | C–G | ||
|- | |- | ||
| 5/4 | | 5/4 | ||
| Downmajor third | | Downmajor third | ||
| | | C–vE | ||
|- | |- | ||
| 7/4 | | 7/4 | ||
| Downminor seventh | | Downminor seventh | ||
| | | C–vBb | ||
|- | |- | ||
| 11/8 | | 11/8 | ||
| Double-up fourth | | Double-up fourth | ||
| | | C–^^F | ||
|- | |- | ||
| 13/8 | | 13/8 | ||
| Double-up minor sixth | | Double-up minor sixth | ||
| | | C–^^Ab | ||
|- | |- | ||
| 19/16 | | 19/16 | ||
| Minor third | | Minor third | ||
| | | C–Eb | ||
|} | |} | ||
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|- | |- | ||
| 11/8 | | 11/8 | ||
| Down-diminished fifth<br | | Down-diminished fifth<br>Double-down augmented fourth | ||
| | | C–vGb<br>C–vvF# | ||
|- | |- | ||
| 13/8 | | 13/8 | ||
| Double downmajor sixth | | Double downmajor sixth | ||
| | | C–vvA | ||
|} | |} | ||
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|- | |- | ||
| 11/8 | | 11/8 | ||
| Double-down diminished fifth<br | | Double-down diminished fifth<br>Triple-down augmented fourth | ||
| | | C–vvGb<br>C–v<sup>3</sup>F# | ||
|- | |- | ||
| 13/8 | | 13/8 | ||
| Triple-down major sixth | | Triple-down major sixth | ||
| | | C–v<sup>3</sup>A | ||
|} | |} | ||
== Chords and harmony == | == Chords and harmony == | ||
Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality: | 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. | 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 | 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 | 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 == | == Scales == | ||