A theory for the ranking of consonance and dissonance between simultaneously sounding notes in LucyTuning.
INDEX OF DISSONANCE
Assumptions:
a) Notes played simultaneously, which are closer on the spiral of fourths and fifths, regardless of octave, tend to sound more consonant, than those which are separated by a greater number of steps of fourths and fifths.
b) An Index of Dissonance may be calculated by adding the number of steps between each note and every other simultaneously sounding note, and dividing the result by the number of intervals which have been used in the addition.
c) All notes are LucyTuned; i.e. meantone intervals (5 Large + 2 small per octave (ratio 2))
d) Large interval = 1200 cents/(2*pi) = 190.9858 cents or ratio of 1.116633.
Method:
a) List the notes to be considered.
b) Count the number of steps of fourths or fifths for each pair of notes to be played simultaneously.
c) Add the resulting steps for each pair.
d) Divide the resulting sum by the number of pairs considered.
e) The Index of Dissonance will be constant regardless of key or notenames, and may be considered as an index of dissonance for a type of chord or for a scale.
Examples:
(Using the spiral pattern from fourths to fifths of Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B# for the first 22 notes.)
Chord
Intervals
in
Chord (i)
Number
of steps
of fourths
and fifths (s)
Total
number
of
steps (s) / intervals (i)
Index of
Dissonance (s/i)
C Sixth or A Minor Seventh
C-E-G-A
C-E C-G C-A E-G E-A G-A
4 1 3 3 1 2
14 / 6
2.3333
C Ninth No Seventh
C-E-G-D
C-E C-G C-D E-G E-D G-D
4 1 2 3 2 1
14 / 6
2.3333
C Major
C-E-G
C-E C-G E-G
4 1 3
8 / 3
2.6667
C minor
C-Eb-G
C-Eb C-G Eb-G
3 1 4
8 / 3
2.6667
C Ninth
C-E-G-Bb-D
C-E C-G C-Bb C-D E-G E-Bb E-D G-Bb G-D Bb-D
4 1 2 2 3 6 2 3 1 4
28 / 10
2.8000
C Major Seventh
C-E-G-B
C-E C-G C-B E-G E-B G-B
4 1 5 3 1 4
18 / 6
3.0000
C Thirteenth
C-E-G-A-Bb
C-E C-G C-A C-Bb E-G E-A E-Bb G-A G-Bb A-Bb
4 1 2 3 3 6 1 3 2 5
30 / 10
3.0000
C Diminished Triad
C-Eb-Gb
C-Eb C-Gb Eb-Gb
3 6 3
12 / 3
4.0000
C minor Major Seventh
C-Eb-G-B
C-Eb C-G C-B Eb-G Eb-B G-B
3 1 5 4 8 4
25 / 6
4.1667
C Diminished Seventh
C-Eb-Gb-Bbb
C-Eb C-Gb C-Bbb Eb-Gb Eb-Bbb Gb-Bbb
3 6 9 3 6 3
30 / 6
5.0000
C Augmented Triad
C-E-G#
C-E C-G# E-G#
4 8 4
16 / 3
5.3333
A similar principle may be applied to all chords and scales to indicate levels of dissonance and consonance.
