Diaschisma: Difference between revisions

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[http://micro.soonlabel.com/petr_parizek/pp_pump_examples/pump_1.ogg parizek1] A [[comma pump]] progression that assumes that the diaschisma is tempered out (i.e. equates two notes that are separated by a diaschisma).

In the progression, the bassline moves as follows: D-(up 5/4)-F#-(down 4/3)-C#-(down 4/3)-G#-(up 5/4)-C-(up 4/3)-G-(up 3/2)-D (*). If we ignore octaves, the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]], and the last three steps (G#-D) are the same moves as the first three, thus it moves us 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. The fact that the D we come back to is exactly the same as the first D, indicates that the 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, which is (2/1) / (45/32)^2 = 2048/2025 is tempered out.

In the progression, the bassline moves as follows: D-(up 5/4)-F#-(down 4/3)-C#-(down 4/3)-G#-(up 5/4)-C-(up 4/3)-G-(up 3/2)-D (*). If we ignore octaves, the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]], and the last three steps (G#-D) are the same moves as the first three, thus it moves us 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. 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, which is (2/1) / (45/32)^2 = 2048/2025 is tempered out.

== See also ==

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 [http://micro.soonlabel.com/petr_parizek/pp_pump_examples/pump_1.ogg parizek1] A [[comma pump]] progression that assumes that the diaschisma is tempered out (i.e. equates two notes that are separated by a diaschisma).
-In the progression, the bassline moves as follows: D-(up 5/4)-F#-(down 4/3)-C#-(down 4/3)-G#-(up 5/4)-C-(up 4/3)-G-(up 3/2)-D (*). If we ignore octaves, the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]], and the last three steps (G#-D) are the same moves as the first three, thus it moves us 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. The fact that the D we come back to is exactly the same as the first D, indicates that the 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, which is (2/1) / (45/32)^2 = 2048/2025 is tempered out.
+In the progression, the bassline moves as follows: D-(up 5/4)-F#-(down 4/3)-C#-(down 4/3)-G#-(up 5/4)-C-(up 4/3)-G-(up 3/2)-D (*). If we ignore octaves, the first three steps (cumulatively D to G#) moves us up by the tritone [[45/32]], and the last three steps (G#-D) are the same moves as the first three, thus it moves us 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. 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, which is (2/1) / (45/32)^2 = 2048/2025 is tempered out.
 == See also ==