Blackdye: Difference between revisions

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"Blackdye" is a name given by [[User:ks26|ks26]]. The name is a pun on "blackwood" and "diatonic" since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[blackwood]][10] ([[5L 5s]]).

== Properties ==

The fifth (3L + M + 2S) of blackdye must be a diatonic fifth.

# The fifth (3L + M + 2S) of blackdye must be a diatonic fifth.

#* (To see this: Consider the chain of fifths generated by the fifth. The diatonic chroma = 7 fifths - 4 octaves = 21L + 7M + 14S - (20L + 8M + 12S) = L - M + 2S. Since L - M >= 0, this must always be nonnegative, so the fifth must be >= 4\7. The fifth cannot be sharper than 3\5, since it generates the [[2L 3s|2L' 3s']] subset of blackdye with L' = ML and s' = SL.)

(To see this: Consider the chain of fifths generated by the fifth. The diatonic chroma = 7 fifths - 4 octaves = 21L + 7M + 14S - (20L + 8M + 12S) = L - M + 2S. Since L - M >= 0, this must always be nonnegative, so the fifth must be >= 4\7. The fifth cannot be sharper than 3\5, since it generates the [[2L 3s|2L' 3s']] subset of blackdye with L' = ML and s' = SL.)

# The fifth is:

#* sharper than 700¢, if L + 3S > 2M

#* equal to 700¢, if L + 3S = 2M

#* flatter than 700¢, if L + 3S < 2M

== Tunings ==

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 "Blackdye" is a name given by [[User:ks26|ks26]]. The name is a pun on "blackwood" and "diatonic" since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[blackwood]][10] ([[5L 5s]]).
 == Properties ==
-The fifth (3L + M + 2S) of blackdye must be a diatonic fifth.
+# The fifth (3L + M + 2S) of blackdye must be a diatonic fifth.
+#* (To see this: Consider the chain of fifths generated by the fifth. The diatonic chroma = 7 fifths - 4 octaves = 21L + 7M + 14S - (20L + 8M + 12S) = L - M + 2S. Since L - M >= 0, this must always be nonnegative, so the fifth must be >= 4\7. The fifth cannot be sharper than 3\5, since it generates the [[2L 3s|2L' 3s']] subset of blackdye with L' = ML and s' = SL.)
-(To see this: Consider the chain of fifths generated by the fifth. The diatonic chroma = 7 fifths - 4 octaves = 21L + 7M + 14S - (20L + 8M + 12S) = L - M + 2S. Since L - M >= 0, this must always be nonnegative, so the fifth must be >= 4\7. The fifth cannot be sharper than 3\5, since it generates the [[2L 3s|2L' 3s']] subset of blackdye with L' = ML and s' = SL.)
+# The fifth is:
+#* sharper than 700¢, if L + 3S > 2M
+#* equal to 700¢, if L + 3S = 2M
+#* flatter than 700¢, if L + 3S < 2M
 == Tunings ==