Garibaldi: Difference between revisions

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'''Schismic''' (or '''helmholtz''') is a 5-limit temperament which takes a roughly justly tuned [[3/2|4/3]] and stacks it eight times to reach [[5/4]], thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭). This can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[Syntonic comma|syntonic commas]] are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a [[microtemperament]], as the 4/3 generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and [[5/4]], the [[schisma]] being tempered out, is approximately 2 cents, which is unnoticeable to most people).

'''Schismic''' (or '''helmholtz''') is a 5-limit temperament which takes a roughly justly tuned [[3/2|4/3]] and stacks it eight times to reach [[5/4]], thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭). This can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[Syntonic comma|syntonic commas]] are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a [[microtemperament]], as the 4/3 generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and [[5/4]], the [[schisma]] being tempered out, is approximately 2 cents, which is unnoticeable to most people). Technically, the best tuning in the 5-limit is to flatten it by a fraction of a cent, though tunings on both sides of the just interval work fine.

To reach intervals of 7, a reasonable choice is to further equate the Pythagorean-syntonic comma with the archytas comma of [[64/63]] (as in [[hemifamity]]), reaching the primary 7-limit extension called '''garibaldi'''. Like with hemifamity, the best tunings involve sharpening the fifth, but in this case only slightly, as the size of the comma is determined by the fifth itself. Thus, tuning the fifth a fraction of a cent sharp gives the best tunings. 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]]. 41edo and 53edo make for good tunings.

Revision as of 07:44, 19 June 2025 view source VectorGraphics (talk \| contribs) 2,399 edits m VectorGraphics moved page Garibaldi to Schismic and garibaldi: Not sure why garibaldi had a page but not schismic ← Older edit		Revision as of 07:52, 19 June 2025 view source VectorGraphics (talk \| contribs) 2,399 edits No edit summary Tag: Visual edit Newer edit →
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	'''Schismic''' (or '''helmholtz''') is a 5-limit temperament which takes a roughly justly tuned [[3/2\|4/3]] and stacks it eight times to reach [[5/4]], thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭). This can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[Syntonic comma\|syntonic commas]] are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a [[microtemperament]], as the 4/3 generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and [[5/4]], the [[schisma]] being tempered out, is approximately 2 cents, which is unnoticeable to most people).		'''Schismic''' (or '''helmholtz''') is a 5-limit temperament which takes a roughly justly tuned [[3/2\|4/3]] and stacks it eight times to reach [[5/4]], thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭). This can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[Syntonic comma\|syntonic commas]] are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a [[microtemperament]], as the 4/3 generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and [[5/4]], the [[schisma]] being tempered out, is approximately 2 cents, which is unnoticeable to most people). Technically, the best tuning in the 5-limit is to flatten it by a fraction of a cent, though tunings on both sides of the just interval work fine.

	To reach intervals of 7, a reasonable choice is to further equate the Pythagorean-syntonic comma with the archytas comma of [[64/63]] (as in [[hemifamity]]), reaching the primary 7-limit extension called '''garibaldi'''. Like with hemifamity, the best tunings involve sharpening the fifth, but in this case only slightly, as the size of the comma is determined by the fifth itself. Thus, tuning the fifth a fraction of a cent sharp gives the best tunings. 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]]. 41edo and 53edo make for good tunings.		To reach intervals of 7, a reasonable choice is to further equate the Pythagorean-syntonic comma with the archytas comma of [[64/63]] (as in [[hemifamity]]), reaching the primary 7-limit extension called '''garibaldi'''. Like with hemifamity, the best tunings involve sharpening the fifth, but in this case only slightly, as the size of the comma is determined by the fifth itself. Thus, tuning the fifth a fraction of a cent sharp gives the best tunings. 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]]. 41edo and 53edo make for good tunings.