(This post has not been written completely yet)
We have mentioned in previous posts in this book, that the diatonic scale has the optimal mathematical property of being the only 7-notes scale (within the chromatic Bach 12-notes scale) with he maximum number of major and minor chord-triads.
An interesting next question is what are the corresponding such scales with the same maximal property but with a larger number of notes? E.g. 8,9, or 10 notes?
We will answer it for the case of 10-notes scale. It is known that the 7-notes diatonic scale contains 3 major triad-chords , and 3 minor triads-chords plus a diminished triad chord. E.g. the C major mode diatonic scale has the 3 , C, F, G major and the 3, D, E, A minor chords. It is natural to think of a scale that you can also have these 3 minor as major chords too , or the 3 majors as minor chords too. In other words all the C, F, G, as minor too or all the D, E, A as major too , but not both! This is a 10-notes scale C, D, D#, E, F, G, G#, A, Bb, B, with interval pattern in semitones , 2-1-1-1-2-1-1-1-1-1 or a cyclic permutation (mode) of it C, C# ,D, E, F, F# ,G, G#, A, B again with interval pattern ,
1-1-2-1-1-1-1-1-2-1 which is a cyclic permutation of the previous thus according to our definition of a scale in this book, the same scale in different modes. Notice that the pattern of sharps is the known f-d-g but with the original non-sharp notes too!
The 11-notes maximal harmonic scale is obviously the 1-1-1-1-1-1-1-1-1-1-1-2 as there is no other 11-notes scale!
There are instrument players e.g. Quena flute players, that have customized their hole system to a 8-front + 1 thump hole system that corresponds to this 10-notes big diatonic (or chromatic-diatonic scale) E.g. Here
https://www.youtube.com/watch?v=6olStHr43aE
Obviously Xiao flutes or Shakuhachi flutes or Irish whistles , or recorders etc, can be made with such 8+1 holes 10-notes big diatonic scale.
If we take the power chords 1, 4, 5 of a diatonic scale , we get by their notes a
4-notes scale. E,g, in C major it would be C -G -C , F-C-F , G-D-G , thus C-D-F-G-C and in semitones
2-3-2-5.
On the other hand if we take a minor chord e.g. C-Eb-G then by adding a major 7nth Bb we get a
Cm7 chord C-Eb-G-B-C which is by itself a
4-notes maximal harmonic scale with interval structure
3-4-3-2 , which already has two chords in it the C minor and the B major
In the next we classify, 5-notes, 6-notes, 8-notes and 9-notes scales with maximum number of major or minor chord-triads.
5-notes maximal harmonic pentatonic (Greek-Richter pentatonic scale)
intervals patterns 4-3-2-2-1 e.g. c4-e4-g4-a4-b4-c5 or I ,III , V, vi, vii, I
(It is essentially A chord with 7nth and with 6th)
Notice that this scale suggests the chord progression by taking chords of the full 7-notes diatonic scale based on the above notes of the maximal harmonic 5-notes scale which has the highest statistical frequency in popular songs: 1(7) 4 5 6m E.g. in C major scale it would be C7, F , G Am.
Analternative version is the inverse 2-2-3-4-1
This inverse version when tuned at an harmonica asin the table below, will give at the blow row a major chord and at the draw row a major an interval of fith apart
E,g, Fmajor-Cmajor
6-notes maximal harmonic (Celtic minor)
intervals patterns 2-2-3-2-2-1 e.g. c4-d4-e4-g4-a4-b4-c5 or I ,II , III, V, vi, vii, I
(It is essentially a chord with 7nth, 6th and 2nd. A mode of it is known also the (raised) Celtic minor scale of the hang-drums )
Notice that this scale also suggests a chord progression by taking chords of the full 7-notes diatonic scale based on the above notes of the maximal harmonic 5-notes scale which has highest statistical frequency in popular songs: 1(7) , 2m, 4 ,5, 6m E.g. in C major scale it would be C7,Dm, F , G Am. This progression includes both the Blues progression 1 2m 5 , and the most popular 1,4,5,6m.
One very close it is os the Cretan minor scale with interval structure 4-1-2-2-2-1 (e.g. c4-e4-g4-a4-b4-c5) with 4 chords 2 major two minor C , G major and Am Em minor.
8-notes maximal harmonic
intervals patterns 2-2-1-1-1-2-2-1 e.g. c4-d4-e4-f4-f#4-g4-a4-b4-c5
(mode of it known also as the Bebop dominant scale)
It seems that this maximal 8-notes harmonic scale (also known as bebop dominant) has the property of having higher density of minor or major chords per number of notes compared to the 7-notes diatonic . Already the maximal harmonic scales have maximum number of chord among other scales with the same number of notes. It seems also that the maximum density of major/minor chords per number of notes has the 12-notes chromatic scale.
9-notes maximal harmonic
intervals patterns 1-1-2-1-1-1-2-2-1 e.g. c4-c4#-d4-e4-f4-f#4-g4-a4-b4-c5
I have made flutes , from overtone flutes with maximal harmonic such 5-notes, 6-notes, 8-tone , 9-tone, 10-tone , 11-tone scales!
