An advanced idea.
We assume knowledge/understanding of the following (abstractions):
Tones are abstractions of frequencies. We only use a fixed limited number of tones. The distances between these tones are also fixed and called intervals.
The frequency and distribution of these tones is determined by the tuning system being used.
the tuning system we are using is equal-temperment tuning and contains the assumption that the semitone is defined as:
semitone = 2^(1/12): 1
All semitones in a n octave are equal to each other. (the following idea of modular scales without requiring a reference to a fixed pitch stems from this).
We define the following intervals for within one octave, and will reduce any larger intervals to one octave:
Distance in 1/2 notes name 0 Perfect unison/augmented 7th/diminished 2nd 1 minor 2nd/augmented unison 2 major 2nd/diminished 3rd 3 minor 3rd/augmented 2nd 4 major 3rd/diminished 4th 5 Perfect 4th/augmented 3rd 6 augmented 4th/diminished 5th 7 Perfect 5th/diminished 6th 8 minor 6th/augmented 5th 9 major 6th/diminished 7th 10 minor 7th/augmented 6th 11 major seventh/diminished 8ve 12 Perfect 8ve/augmented 7th
Because these terms can be cumbersome, I shall use abbreviations in either letter/number combinations or number/accidental combinations. Below is the same chart as above but abbreviated.
0 PU/aug7/dim2 ; 1/#7/bb2 1 m2/augU ; b2/#1 2 M2/dim3 ; 2/bb3 3 m3/aug2 ; b3/#2 4 M3/dim4 ; 3/b4 5 P4/aug3 ; 4/#3 6 aug4/dim5 ; #4/b5 7 P5/dim6 ; 5/bb6 8 m6/aug5 ; b6/#5 9 M6/dim7 ; 6/bb7 10 m7/aug6 ; b7/#6 11 M7/dim8 ; 7/b8 12 P8/aug7 ; 8/#7
We will use a mixture of notations #4/b5 etc. for intervallic spellings of chords and scales, m3/a2 for step patterns.
We recall that scales have step patterns that can be used to identify the scales. We can also use step patterns for chords (not commonly done though).
It is this idea of step patterns and their link to scales, chords, intervals, and modes that we will now look into in more depth. Particularly all possible scales, modes, chords, and intevals within on eoctave (in equal-tempered tuning).
We recall the greek/church modes and how we could create them by changing the step pattern one interval at a time.
Ionian = W-W-1/2-W-W-W-1/2 Dorian = W-1/2-W-W-W-1/2-W Phrygian = 1/2-W-W-W-1/2-W-W Lydian = W-W-W-1/2-W-W-1/2 Mixolydian = W-W-1/2-W-W-1/2-W Aeolian = W-1/2-W-W-1/2-W-W Locrian = 1/2-W-W-1/2-W-W-W
We now extend this idea to all discrete permutations without reference to key.
We start with the octave which has one mode, and fully covers the octave.
We next look at the intervals, which together with their inversions cover an octave. Each pair has two modes, which we call inversions.
M7-1/2 gives the modes M7-1/2 (M7 interval) 1/2-M7 (m2 interval) m7-W gives the modes m7-W (m7 interval) W-m7 (M2 interval) M6-m3 gives the modes M6-m3 (M6 interval) m3-M6 (m3 interval) m6-M3 gives the modes m6-M3 (m6 interval) M3-m6 (M3 interval) P5-P4 gives the modes P5-P4 (P5 interval) P4-P5 (P4 interval) dim5-dim5 gives the mode dim5-dim5 (dim5 interval, aug4 interval, tritone)
and that's it.
Combinations of 3 notes can be derived from taking modes of the following step patterns.
m7-1/2-1/2 (3 modes) M6-W-1/2 (6 modes) m6-W-W (3 modes) m6-m3-1/2 (6 modes) P5-M3-1/2 (6 modes) P5-m3-W (6 modes) d5-P4-1/2 (6 modes) d5-M3-W (6 modes) d5-m3-m3 (3 modes) P4-P4-W (3 modes) P4-M3-m3 (6 modes) M3-M3-M3 (1 mode)
Combinations of 4 notes can be derived from taking modes of the following step patterns.
M6-1/2-1/2-1/2 m6-W-1/2-1/2 P5-m3-1/2-1/2 P5-W-W-1/2 d5-M3-1/2-1/2 d5-m3-W-1/2 d5-W-W-W P4-P4-1/2-1/2 P4-M3-W-1/2 P4-m3-m3-1/2 P4-m3-W-W M3-M3-m3-1/2 M3-m3-m3-W M3-M3-W-W
Combinations of 5 notes can be derived from taking modes of the following step patterns.
m6-1/2-1/2-1/2-1/2 P5-W-1/2-1/2-1/2 d5-m3-1/2-1/2-1/2 d5-W-W-1/2-1/2 P4-M3-1/2-1/2-1/2 P4-m3-W-1/2-1/2 P4-W-W-W-1/2 M3-M3-W-1/2-1/2 M3-m3-m3-1/2-1/2 M3-m3-W-W-1/2 M3-W-W-W-W m3-m3-m3-W-1/2 m3-m3-W-W-W
Combinations of 6 notes can be derived from taking modes of the following step patterns.
P5-1/2-1/2-1/2-1/2-1/2 d5-W-1/2-1/2-1/2-1/2 P4-m3-1/2-1/2-1/2-1/2 P4-W-W-1/2-1/2 M3-M3-1/2-1/2-1/2-1/2 M3-m3-W-1/2-1/2-1/2 M3-W-W-W-1/2-1/2 m3-m3-m3-1/2-1/2-1/2 m3-m3-W-W-1/2-1/2 m3-W-W-W-W-1/2 W-W-W-W-W-W
Combinations of 7 notes can be derived from taking modes of the following step patterns.
d5-1/2-1/2-1/2-1/2-1/2-1/2 P4-W-1/2-1/2-1/2-1/2-1/2 M3-m3-1/2-1/2-1/2-1/2-1/2 M3-W-W-1/2-1/2-1/2-1/2 m3-m3-W-1/2-1/2-1/2-1/2 m3-W-W-W-1/2-1/2-1/2 W-W-W-W-W-1/2-1/2
Combinations of 8 notes can be derived from taking modes of the following step patterns.
P4-1/2-1/2-1/2-1/2-1/2-1/2-1/2 M3-W-1/2-1/2-1/2-1/2-1/2-1/2 m3-m3-1/2-1/2-1/2-1/2-1/2-1/2 m3-W-W-1/2-1/2-1/2-1/2-1/2 W-W-W-W-1/2-1/2-1/2-1/2
Combinations of 9 notes can be derived from taking modes of the following step patterns.
M3-1/2-1/2-1/2-1/2-1/2-1/2-1/2-1/2 m3-W-1/2-1/2-1/2-1/2-1/2-1/2-1/2 W-W-W-1/2-1/2-1/2-1/2-1/2-1/2
Combinations of 10 notes can be derived from taking modes of the following step patterns.
Combinations of 11 notes can be derived from taking modes of the following step patterns.
Combinations of 12 notes can be derived from taking modes of the following step patterns, as with the octave all modes are the same.
Of what use is this?
Maybe we'd like to find all pentatonic scales made up of 2 minor 3rds, and three whole-steps.
The item listed previously is
This can be as above or as
All other combinations are modes of these two step patterns. They are as follows:
m3-m3-W-W-W m3-W-W-W-m3 W-W-W-m3-m3 W-W-m3-m3-W W-m3-m3-W-W m3-W-m3-W-W W-m3-W-W-m3 m3-W-W-m3-W W-W-m3-W-m3 W-m3-W-m3-W
We can figure out the intervals or notes in a key.
m3-m3-W-W-W = 1,b3,b5,b6,b7 m3-W-W-W-m3 = 1,b3,4,5,6 W-W-W-m3-m3 = 1,2,3,b5,6 W-W-m3-m3-W = 1,2,3,5,b7 W-m3-m3-W-W = 1,2,4,b6,b7 m3-W-m3-W-W = 1,b3,4,b6,b7 W-m3-W-W-m3 = 1,2,4,5,6 m3-W-W-m3-W = 1,b3,4,5,b7 (Pentatonic minor) W-W-m3-W-m3 = 1,2,3,5,6 (Pentatonic major) W-m3-W-m3-W = 1,2,4,5,b7 (Egyptian)
One could create one's own chord/scale book with a much closer approximation to "all the chords" than the books currently published (i've already done this).
One could use this as a point to do research from. Consider the limitations imposed and go from there.
It is a lesson on what can be done when one small idea is taken to an extreme.
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Last updated January 1, 2004
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