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Scales and Tuning
Due whenever
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Compute the ratio and the value in cents corresponding to each
of the following:
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The Pythagorean comma is the tiny interval by which
twelve just perfect fifths differ
from seven octaves.
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The Syntonic comma, also called simply the comma,
is the tiny interval by which the just
major third differs from four perfect fifths less two octaves.
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The Septimal comma is the tiny interval by which the
septimal minor seventh (having a
ratio 7:4) differs from the Pythagorean minor seventh.
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Assuming 12-tone equal temperament (12-TET), find a set whose
interval vector is <2,2,4,2,2,3>.
Consider the 17-TET scale, whose notes are numbered 0-16.
Consider also the 12-TET scale,
whose notes are labelled C, C#, D, D#, etc. We can embed 12-TET
into 17-TET by assigning
C in 12-TET to 0 in 17-TET, and then considering a fifth to be the ratio 10/17,
and assigning
each of the other notes in a "circle of fifths" manner.
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Complete this embedding -- i.e. give the other note assignments.
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How many non-trivial (i.e. non-zero) interval classes are in
17-TET?
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Write the above embedding as a (singly) generated set, if
possible.
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What is the interval vector of the embedded scale?
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Which of the following properties does this embedded scale have:
G, WF, MP, DE, ME, DP, DT.
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