Schismatic family: Difference between revisions
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The arithmetic formula is basically of little value compared to an intro to notation |
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The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields {{multival| 1 -8 -15 }} for the [[wedgie]]. This tells us the generator is a fifth and | The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]]. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields {{multival| 1 -8 -15 }} for the [[wedgie]]. This tells us the generator is a fifth and 5/4 is represented by a diminished fourth. | ||
This defies the tradition of tertian harmony, as 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. | |||
== Schismatic aka helmholtz == | == Schismatic aka helmholtz == | ||
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{{Main| Garibaldi temperament }} | {{Main| Garibaldi temperament }} | ||
Garibaldi tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb). It necessitates a sharper fifth than pure. | Garibaldi tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{See also| Archytas clan #Schism }} | {{See also| Archytas clan #Schism }} | ||
Schism is a low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh. | Schism is a low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Main| Pontiac }} | {{Main| Pontiac }} | ||
Pontiac tempers out the [[nanisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#). | Pontiac tempers out the [[nanisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^<sup>3</sup>A). | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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=== Helenoid === | === Helenoid === | ||
The | The helenoid temperament (53 & 118) is closely related to the helenus temperament, but with the [[4375/4374|ragisma]] rather than the [[225/224|marvel comma]] tempered out. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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=== Bipont === | === Bipont === | ||
The bipont temperament (118 & 224) has a period of half octave and tempers out the [[3025/3024|lehmerisma | The bipont temperament (118 & 224) has a period of half octave and tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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== Grackle == | == Grackle == | ||
Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb). | Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||