Diaschisma: Difference between revisions

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* the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]];

* the last three steps (cumulatively G# to D) are the same moves as the first three, moving up by the tritone 45/32 a second time.

In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)^2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1~~. In temperament contexts, we see this as equivalent to saying that their difference, the diaschisma, is tempered out~~.

In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)^2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1.

This also tells us that if a system tempers out the diaschisma, it has an interval that is equal to exactly half of an octave‚ namely the tempered 45/32 tritone. Thus all edos (such as [[12edo]], [[22edo]], [[34edo]] and [[46edo]]) and MOS scale structures (such as the MOS scales of [[diaschismic family|diaschismic]] and [[pajara]]) that temper out the diaschisma split the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.

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 * the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]];
 * the last three steps (cumulatively G# to D) are the same moves as the first three, moving up by the tritone 45/32 a second time.
-In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)^2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1. In temperament contexts, we see this as equivalent to saying that their difference, the diaschisma, is tempered out.
+In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma, which is (2/1) / (45/32)^2 = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1.
 This also tells us that if a system tempers out the diaschisma, it has an interval that is equal to exactly half of an octave‚ namely the tempered 45/32 tritone. Thus all edos (such as [[12edo]], [[22edo]], [[34edo]] and [[46edo]]) and MOS scale structures (such as the MOS scales of [[diaschismic family|diaschismic]] and [[pajara]]) that temper out the diaschisma split the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.