Schismic–commatic equivalence continuum: Difference between revisions

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The '''schismic-Pythagorean equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments ~~supported~~ by [[12edo]].

The '''schismic-Pythagorean equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[12edo]].

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 [[tempering out|tempers out]] both commas and thus tempers out 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 not coprime with 12, however, the corresponding temperament splits the ''[[octave]]'' into gcd (''n'', 12) parts, and splits the interval class of 3 into ''n''/gcd (''n'', 12). For example:

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 not coprime with 12, however, the corresponding temperament splits the [[octave]] into gcd (''n'', 12) parts, and splits the interval class of 3 into ''n''/gcd (''n'', 12). For example:

* [[Meantone]] (''n'' = 1) is generated by a fifth with an unsplit octave;

* [[Diaschismic]] (''n'' = 2) splits the octave in two;

* [[Diaschismic]] (''n'' = 2) splits the octave in two, as 2 divides 12;

* [[Misty]] (''n'' = 3) splits the octave in three;

* [[Misty]] (''n'' = 3) splits the octave in three, as 3 divides 12;

* [[Undim]] (''n'' = 4) splits the octave in four;

* [[Undim]] (''n'' = 4) splits the octave in four, as 4 divides 12;

* [[Quindromeda]] (''n'' = 5) does not split the octave but splits the fourth in five, as 5 is coprime with 12.

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-The '''schismic-Pythagorean equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[12edo]].
+The '''schismic-Pythagorean equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[12edo]].
-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 [[tempering out|tempers out]] both commas and thus tempers out 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 not coprime with 12, however, the corresponding temperament splits the ''[[octave]]'' into gcd (''n'', 12) parts, and splits the interval class of 3 into ''n''/gcd (''n'', 12). For example:
+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 not coprime with 12, however, the corresponding temperament splits the [[octave]] into gcd (''n'', 12) parts, and splits the interval class of 3 into ''n''/gcd (''n'', 12). For example:
 * [[Meantone]] (''n'' = 1) is generated by a fifth with an unsplit octave;
-* [[Diaschismic]] (''n'' = 2) splits the octave in two;
+* [[Diaschismic]] (''n'' = 2) splits the octave in two, as 2 divides 12;
-* [[Misty]] (''n'' = 3) splits the octave in three;
+* [[Misty]] (''n'' = 3) splits the octave in three, as 3 divides 12;
-* [[Undim]]  (''n'' = 4) splits the octave in four;
+* [[Undim]]  (''n'' = 4) splits the octave in four, as 4 divides 12;
 * [[Quindromeda]]  (''n'' = 5) does not split the octave but splits the fourth in five, as 5 is coprime with 12.