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Saturday, August 18, 2018

117. Small 7-notes diatonic scale and big 10-notes diatonic (or chromatic-diatonic) scale. 5-notes, 6-notes , 8-notes and 9-notes scales with a maximal number of major or minor chord-triads.

(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!