Schismic–Pythagorean equivalence continuum: Difference between revisions
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All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| -19 12 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[12edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is. | All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| -19 12 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[12edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is. | ||
The [[Pythagorean comma]] is the characteristic 3-limit comma tempered out in 12edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For an ''n'' that is a divisor of 12, however, the corresponding temperament splits the ''[[octave]]'' into ''n'' parts instead. For example: | |||
* [[Meantone]] (''n'' = 1) does not split the octave; | |||
* [[Diaschismic]] (''n'' = 2) splits the octave in two; | |||
* [[Misty]] (''n'' = 3) splits the octave in three; | |||
* [[Undim]] (''n'' = 4) splits the octave in four; | |||
* [[Quindromeda]] (''n'' = 5) does not split the octave but splits the fourth in five, as 5 does not divide 12. | |||
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We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''syntonic-Pythagorean equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588… | We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''syntonic-Pythagorean equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588… The [[syntonic comma]] is way larger but way simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments. | ||
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