FKH Extended-diatonic Interval Names: Difference between revisions

Line 1,123:

There are edos whose best fifth is flatter even than 4\7. In such edos major intervals are smaller than minor intervals, augmented smaller than major and diminished larger than minor. We expand our definition of well-ordered intervals to include that within each degree... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where sk _ ≤ s/k_ ≤ _ ≤ S/K_ ≤ SK _ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...). In order to obtain well-ordered interval-name sets, we use enharmonic equivalences, replacing diatonic intervals with altered intervals.

In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence ~~cla M~~ = m, where meantone can be defined by cM = M, and schismatic by cM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:

In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence kM = m, where meantone can be defined by kM = M, and schismatic by kM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:

P1 M2 M3 m3 P4 P5 M6 m6 m7 P8.

Line 1,139:

P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.

Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are:

Even though major is now below minor, we may still give the name 'neural' to an interval half way between major and minor with an interval class. Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are:

P1 Km2 N2 N3 kM3 P4 P5 Km6 N6 N7 kM7 P8, in which we can see Neutral[7] 3|3.

Where 11edo's best fifth is 47c flat, most would not really call it a P5. Accordingly we may wish instead to label 11edo as every second step of 22edo, where we still get major seconds and thirds, and minor sixths and sevenths:

P1 Km2 M2 Km3 M3 K4 k5 m6 kM6 m7 kM7 P8.

Since this labeling is based on a much more accurate fifth, the interval names mach their size much more closely, e.g. using 11edo's fifth, 2\11 was labelled a N2, but at 218c, is much closer to a M2, as it is labeled using 22edo's fifth. Since 11edo is abnormally flat even for Mavila, I suggest labeling using 22edo's fifth.

16edo has diatonic interval names:

@@ Line 1,123: / Line 1,123: @@
 There are edos whose best fifth is flatter even than 4\7. In such edos major intervals are smaller than minor intervals, augmented smaller than major and diminished larger than minor. We expand our definition of well-ordered intervals to include that within each degree... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where sk _ ≤ s/k_ ≤ _ ≤ S/K_ ≤ SK _ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...). In order to obtain well-ordered interval-name sets, we use enharmonic equivalences, replacing diatonic intervals with altered intervals.
-In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cla M = m, where meantone can be defined by cM = M, and schismatic by cM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:
+In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence kM = m, where meantone can be defined by kM = M, and schismatic by kM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names:
 P1 M2 M3 m3 P4 P5 M6 m6 m7 P8.
@@ Line 1,139: / Line 1,139: @@
 P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.
-Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are:
+Even though major is now below minor, we may still give the name 'neural' to an interval half way between major and minor with an interval class. Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are:
 P1 Km2 N2 N3 kM3 P4 P5 Km6 N6 N7 kM7 P8, in which we can see Neutral[7] 3|3.
+Where 11edo's best fifth is 47c flat, most would not really call it a P5. Accordingly we may wish instead to label 11edo as every second step of 22edo, where we still get major seconds and thirds, and minor sixths and sevenths:
+P1 Km2 M2 Km3 M3 K4 k5 m6 kM6 m7 kM7 P8.
+Since this labeling is based on a much more accurate fifth, the interval names mach their size much more closely, e.g. using 11edo's fifth, 2\11 was labelled a N2, but at 218c, is much closer to a M2, as it is labeled using 22edo's fifth. Since 11edo is abnormally flat even for Mavila, I suggest labeling using 22edo's fifth.
 edo has diatonic interval names: