In ancient Greece according to Aristoxenos, there were 3 musical gendres , the diatonic , the chromatic and enharmonic which were also tetrachords (total length an interval of 4th) and pentachords (total legth an interval of 5th and consisting from an initial or final tone and a tetrachord as before) and by combing a tetrachord with a pentachord they got various 7-notes scales inside an octave . In other words the 7-notes scale was of the form Tetrachord1+2+tetrachord2 , in which case it was called two "divorced" tercahords or Tetrachord1+tetrachord2+2 in which case it was called two adjoint tetrachords. 1) the Diatonic was balanced feelings (that eventually after more than 1 thousand years resulted to the 7-notes diatonic scale, and a simplified rounding of their tetrcahords to a single one was the 2-2-1 and most probably all permutations of it ) the 2) Chromatic was sad feelings (that eventually after more than 1 thousand years resulted to the 7-notes harmonic minor and double harmonic minor scales, blues scale etc , And a simplified rounding of their tetrachords to a single one was the 3-1-1 nd most probably all permutations of it ) and 3) the enharmonic was happy feelings (used major 3rd intervals and a simplified rounding to a single one was the 4-(1/2)-(1/2)) and most probably all permutations of it ) and has many similarities with the Japanese 5-notes scales like the Akebono with interval structure 4 - 2 - 1 - 4 - 1 e.g. at the mode 2-1-4-1-4 e.g. A B C E F A. (See e.g. the video https://www.youtube.com/watch?v=vv7CO-nVFj8 )
In Ancient Greece, Byzantine and middle east, the 7-notes scales were created by combining 4-notes (tetrachords) that have total length an interval of 4th and 5-notes (pentachords) that have total length and interval of 7th. In this way all such 7-notes scales contain the power-5 chord 1-5-1' (e.g. C3-G3-C4).
The representation of such tetrachords and pentachords can be with the sequence of intervals in semitones. Thus to find all of them is in mathematical terminology to find all partitions of 5 in ti 3 numbers (having sum 5) and all partitions of 7 in to 4 numbers (having sum 7).
The solutions is the next
ALL 6 TETRACHORDS
DIATONIC GENDRE (FAMILY)
(containing only 2 and 1)
2-2-1 syntono major , Ionian/mixolydian (known also as Rast/Tsargiak/Mahour)
1-2-2 syntono minor , Phrygian/Locrian (known also as Qurdi)
2-1-2 minor, Dorian/Aeolian (known also as Bousselik/Nichavent/Ousak)
CHROMATIC GENDRE (FAMILY)
(containing 3)
1-3-1 2nd chromatic (known also as Hijaz)
1-1-3 parachromatic tonal
3-1-1 1st chromatic
ENHARMONIC GENDRE (FAMILY)
1/2-1/2-4
1/2-4-1/2
4-1/2-1/2
ENHARMONIC GENDRE (FAMILY)
1/2-1/2-4
1/2-4-1/2
4-1/2-1/2
All pentachords contain a single power-5 chord that power-5 chord which is defined by the end-notes of the pentachord.
THE DIATONIC FAMILY
(containing only 2, and 1)
2-2-2-1 Major Lydian Contains a major chord (known also as Natzi/Zaouil)
2-2-1-2 Major Ionian/Mixolydian Contains a major chord.(known also as Rast/Mahour)
2-1-2-2 Minor Dorian Contains a minor chord (known also as Bousselik/Nichavent)
1-2-2-2 Minor Frygian/Aeolian Contains a minor chord (known also as Qourdi)
THE CHROMATIC FAMILY
(containing 3 )
1-1-3-2
1-1-2-3
1-3-2-1 Contains a major chord and a diminished chord
1-2-3-1 Contains a minor chord and a diminished chord
1-2-1-3 Contains a minor chord and a diminished chord
1-3-1-2 Contains a major chord and a diminished chord (known also as Hijaz)
3-2-1-1 Contains a diminished chord
2-3-1-1
3-1-2-1 Contains a major chord and a diminished chord
2-1-3-1 Contains a minor chord and a diminished chord (known also as Nevesser/Nikriz)
3-1-1-2 Contains a major chord (known also as Saz-kar)
2-1-1-3 Contains a minor chord (known also as Samba)
THE ENHARMONIAN GENDRE (FAMILY)
(containing 4 )
1-1-1-4 Contains a minor chord
1-1-4-1
1-4-1-1
4-1-1-1 Contains a major chord
By combining a treachord and a pentachord we get a 7-notes scale. We remind the reader that in this book by scale we mean a sequence of notes giving an octave and all cyclic permutations of them are considered the same scale but also 7 different modes of it.
As a rough estimation of an upper bound on the number of the scales of this genealogy we calculate that if x is the number of different such scales then x<=6*20*2*2=480, because we will combine 6 tetrachords with 18 pentachords thus 6*20=120 such combinations, but also combining first 18 pentachords and then 6 tetrachords will give twice as many, thus 6*2*20=240 . And by taking also the inverse order on all notes of them at most 6*20*2*2=480 scales. Of course some of them will turn out to be simply modes (cyclic permutations ) of some previous or next, thus 480 is an upper bound not an exact number.
In this way we can get the next scale
1) The diatonic
2) The harmonic minor
3) The inverse of the harmonic minor
4) The Neopolitan minor scale
5) The inverse of the Neopolitan minor scale
6) The double harmonic minor
7) The melodic minor
8) The double melodic minor or Arabic scale or Neopolitan major
9) Parachromatic Byzantine scale (1-1-3-2-1-1-3)
10) Inverse parachromatic Byzantine
11) Debussy's 7-tonic (2-1-1-1-1-3-3) .
etc
Modes of the above 1-8 scales include the Byzantines minor and Byzantine double minor , Hungarian minor, Gypsy major and minor, Rast scale, Nikriz scale, Hitzaz scale, Hitzaskar scale , Nihavent scale, Neveser scale, 7-notes Sambach scale, Ousak scale , Qurdi scale , kartzigiar scale etc
Some japanese folk scales
Also arabic scales (projected in the Bach equally temperament 12-notes chromatic scale ) under the names Shouzinak , Housam, Shegiach , Moustaar , Persian or todi theta scale, the Enigmatic scale,
Debussy's 7-tonic (2-1-1-1-1-3-3) etc
etc
See also post 21.
And many more beautiful scales that are not known with any name!
In ancient Greece according to Aristoxenus the tetrachords where 6 and of 3 generations.
They divided the semitone in to 6 equal segments, or the octave in to 72 segments or the pure 4th in to 30 segments.
Then the tetrachords were the next (the integers are Aristoxenus segments)
ENHARMONIC GENDRE
3-3-24 or semitone/2 -- semitone/2 --2 tones
CHROMATIC GENDRE
Soft 4-4-22 or 1/3 tone 1/3 tone 22 segments
4.5 -4.5 -21
tonal 6-6-18 or 1 semitone 1 semitone 3 semitones
DIATONIC GENDRE
uniform 6-9-15
syntono 6-12-12
Now two tetrachords make a scale of an octave in two ways
The disjunctive way = tetrachord+1tone+tetrachord=5+2+5 semitones=12 semitones
Or the adjoint way = tetrachord+tetrachord+1 tone=5+5+2 semitones=12 semitones
E.g. syntono+1tone+syntono=diatonic scale lydian mode
or syntono+syntono+1tone=diatonic scale Locrian mode
or tonal+tonal+1tone=6-6-18-6-6-18-12 which is a mode ofthe by now known as inverse persian todi theta scale
The same with the tonal+1tone+tonal.
etc
See also post 227
For reasons of completeness we enumerate here the trichords which are of length an interval of 3rd.
All of them are of the DIATONIC FAMILY
1-2
2-1
2-2
OF SCPECIAL INTEREST IS THE 7-NOTE SCALE 2-2-1-2-1-2-2 E.G. C-D-E-F-G-G#-Bb-C WHICH IS DERIVED FROM THE FIRST 13 OVERTONES HARMONIC SERIES IN A NATURAL TRUMPET. AND IT IS CONSTRUCTED FROM THE 4-CHORD 2-2-1 AND THE 5-CHORD 2-1-2-2. IT IS THE MELODIC MINOR OR ACOUSTIC OVERTONES SCALE IF THE 7NTH HARMONIC E.G. IN C-D-E-F-G-G#-B-C if we interprete the 7nth overtone as B rather than Bb. BUT It is the inverse 7-notes scale of the first 13 overtones e.g. in a string or natural trumpet C-D-E-F-G-G#-B-C if we perceive THE 7NTH HARMONIC AS B
ALL 7-NOTES SCALES OF THE ENHARMONIC GENDRE BY COMBINING A DIATONIC TETRACHORD WITH AN ENHARMONIC PENTACHORD
2-2-1-1-1-1-4
2-2-1-4-1-1-1
2-2-1-1-4-1-1
2-2-1-1-1-4-1
1-2-2-1-1-1-4
1-2-2-4-1-1-1
1-2-2-1-4-1-1
1-2-2-1-1-4-1
2-1-2-1-1-1-4
2-1-2-4-1-1-1
2-1-2-1-4-1-1
2-1-2-1-1-4-1
ALL 6-NOTES (ENHARMONIC GENDER) SCALES THAT CONTAIN THE POWER-5 CHORD AND AT LEAST ONE MAJOR 3RD INTERVAL
Many of the irish melodies belong to such scales!
Because the power chord is contained, then there must exist an interval of 4th and an interval of 5th. If the interval of 4th contains a major 3rd then the only solutions are
4-1, 1-4 . Otherwise the interval of major 3rd must be contained inside the interval of 5th, and thus we have the 5-chords of the enharmonic gender
1-1-1-4 Contains a minor chord
1-1-4-1
1-4-1-1
4-1-1-1 Contains a major chord
In total we enumerate at first all such scales with 2 major 3rd intervals
1) 4-1-1-1-1-4
2) 1-4-1-1-1-4
3) 1-1-4-1-1-4
If there is only one major 3rd interval (4 semitones)
Then we combine the 1-4 , 4-1 with any 5-chord in the diatonic or chromatic gender
4) 1-4-2-2-2-1
5) 1-4-2-2-1-2
6) 1-4-2-1-2-2
7) 1-4-1-2-2-2
8) 1-4-1-1-3-2
9) 1-4-1-3-2-1
10) 1-4-1-2-3-1
11) 1-4-1-2-1-3
12) 1-4-1-3-1-2
13) 1-4-2-3-1-1
14) 1-4-3-1-2-1
15) 1-4-2-1-3-1
16) 1-4-3-1-1-2
17) 1-4-2-1-1-3
18) 4-1-2-2-1-2
19) 4-1-2-1-2-2
20) 4-1-1-2-2-2
21) 4-1-1-1-3-2
22) 4-1-1-2-3-1
23) 4-1-1-2-1-3
24) 4-1-3-1-1-2
25) 4-1-2-1-1-3
Among these 25 scales we filter the combinationof diatonic 4-chords and only one major third
thus haveing the 7 diatonic-enharmonic hexatonic scales
I have seydel harmonicas with these dtaitonic-enharmonic hexatonic scales
4) 1-4-2-2-2-1 (HEX ARABIC OR HEX NEAPOLITAN MAJOR 1M =inverse of 20) )
5) 1-4-2-2-1-2 (HEX MELODIC MINOR 1M-2d inverse of 19))
6) 1-4-2-1-2-2 (HEX DIATONICENHARMONIC 1m-2d)
7) 1-4-1-2-2-2 (HEX AKEBONO-CRETAN 1M-7m)
18) 4-1-2-2-1-2 (HEX ENHARMONIC 1M-7d)
19) 4-1-2-1-2-2 (HEX HMIN ENHARM 1m-7d inverse of 5) )
20) 4-1-1-2-2-2 (HEX ARABIC OR HEX NEAPOLITAN MAJOR 1m)
Among these 25 scales we filter those that contain the 4-3, or 3-4 which is triad chord as with more harmonic sound
Major chord 4-3:
14) 1-4-3-1-2-1
16) 1-4-3-1-1-2
By cyclic permutation we may re-write them as
14) 4-3-1-2-1-1
16) 4-3-1-1-2-1
16) 4-3-1-1-2-1
Minor chord 3-4:
23) 4-1-1-2-1-3
25) 4-1-2-1-1-3
By cyclic permutation we may re-write them as
23) 3-4-1-1-2-1
25) 3-4-1-2-1-1
ALL 5-NOTES SCALES THAT CONTAIN THE POWER-5 CHORD AND AT LEAST ONE MAJOR 3RD (4 SEMITONES)
This already mean that we know tow notes of the scale . The other 3
1) All in the lower interval of 5th . Thus a 5-chord as above and then an interval of 4th or 5 semitones.
1) 1-1-1-4 -5
2) 1-1-4-1-5
3) 1-4-1-1-5
4) 4-1-1-1 -5
2) All in the upper interval of 4th . Thus a 4-chord as above as as upper 3 notes and as lower notes the 3-4, 4-3 E.g.
5) 4-3-2-2-1 Known in this blog as the inverse maximal harmonic 5-tonic scale (see post 117 and 204 )
6) 4-3-1-2-2
7) 4-3-2-1-2
8) 4-3-1-3-1
9) 4-3-1-1-3
10) 4-3-3-1-1
and
11) 3-4-2-2-1
6) 3-4-1-2-2
7) 3-4-2-1-2
8) 3-4-1-3-1
9) 3-4-1-1-3
10) 3-4-3-1-1
3) 2 notes in the lower interval of 5th
3-2-2, 2-2-3, 4-2-1, 4-1-2 , 2-1-4, 1-2-4, 3-3-1, 1-3-3, 3-1-3, 1-1-5, 5-1-1, 1-5-1
and one in the upper interval of 4th
2-3, 3-2, 1-4 , 4-1 Thus
11) 3-2-2-4-1
12) 3-2-2-1-4
13) 2-2-3-1-4
14) 2-2-3-4-1 Known also as Greek pentatonic 2-2-3-4-1 (known also as Raga Chitthakarshini)
15) 4-2-1-2-3
16) 4-2-1-3-2
17) 4-2-1-1-4
18) 4-2-1-4-1 known also as the Japanese Hirajoshi 5-notes scale and In 5-notes scale and Iwato 5-notes scale and as Akebono 5-notes scale
19) 4-1-2-3-2
20) 4-1-2-1-4
21) 4-1-2-4-1
22) 2-1-4-2-3 known also as the Japanese Insen 5-notes scale
23) 1-2-4-2-3
24) 1-2-4-3-2
25) 1-2-4-4-1
26) 3-3-1-1-4
27) 3-3-1-4-1
28) 1-3-3-4-1
29) 3-1-3-1-4
30) 3-1-3-4-1
31) 1-1-5-1-4
32) 1-1-5-4-1
33) 5-1-1-1-4
34) 5-1-1-4-1
35) 1-5-1-4-1
36) 1-5-1-1-4
4) 1 note in the lower interval of 5th and 2 notes in the upper interval of 4th
thus 1-6, 6-1 and 1-1-1-2, 1-1-2-1, 1-2-1-1, 2-1-1-1, but none contains 4 , thus no new 5-notes scales.
2-1-4-1-4
