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The 'Greek' Mode Domain, derivation and Table of Results. Posted
09/17/02. Some paths through some common high note-number non-
'Greek' modes passing through harmonic minor 09/18/02. Revised
with, I hope, more clarity, 09/22/02.
The 7 normal 7-note 'Greek' modes are best ordered f=lydian,
c=ionian, g=mixolydian, d=dorian, a=aeolian, e=phrygian and
b=locrian. Note that if we flat the 4th of f mode we get c mode,
then flattening the 7th gives g, then flattening the 3rd gives d,
then flattening the 6th gives a, then flattening the 2nd gives e,
then flattening the 5th gives b, then flattening the 1st takes us
back to f. Let's write them in this order with their semitone
sequences.
f lydian 2221221
c ionian 2212221
g mixolydian 2212212
d dorian 2122212
a aeolian 2122122
e phrygian 1222122
b locrian 1221222
Now let's start with the first, and take the front number off and
put it in back, and step through the series (cyclic permutation)
1 2221221
2 2212212
3 2122122
4 1221222
5 2212221
6 2122212
7 1222122
Now start with the first and take every fourth one, going around
in a circle, so
1 2221221 = f
5 2212221 = c
2 2212212 = g
6 2122212 = d
3 2122122 = a
7 1222122 = e
4 1221222 = b
And we get all the semitone sequences in 'Greek' mode order.
Lets try it for the normal 6-note tunes derived from the 'Greek'
modes.
lyd/ion 223221
ion/mix 221223
mix/dor 232212
dor/aeol 212232
aeol/phry 322122
phry/loc 122322
Again lets start with the first and cycle through, 1st note to
the rear at each step.
1 223221 = lyd/ion
2 232212 = mix/dor
3 322122 = aeol/phry
4 221223 = ion/mix
5 212232 = dor/aeol
6 122321 = phry/loc
And, the 5-notes ones, pi1- 22323, pi2- 23223, pi3- 23232, pi4-
32232, and pi5- 32322.
And so on down to 2-note scales
? 57
?? 75
Assuming that this works as well for 8-11 note scales as well as
for 7- 2 note scales, we can get the by strting at the 2 2-note
ones and get the complete domain of 'Greek' based modes from 1 to
(the) 12-note tune. We start at the 2 2-note scales and put the
sum scale between them, and add the mode# in front as the
identifer.
16|57 C____F______
80|525 C____F_G____
64|75 C______G____
Now with cyclic permutation of 525 we get the two missing
permutations of the 3-note scales as 255, and 552
66|255 C_D____G____, and 528|552 C____F____Bb_
A scale of n+1 notes always has mode number 2^(n-2) higher than
n-note scale which is the same except for one note. n in 2^(n-2)
is the position of the extra note in the 12 tone scale. This
makes our new 3-note scales only fit one way with our 2-note
ones:
528|552 C____F____Bb_
16|57 C____F______
80|525 C____F_G____
64|75 C______G____
66|255 C_D____G____
And thus we can proceed to 12 note scales, which we have done
with the results as follows:
.............................................
The 'Greek' Mode Domain. Table of all modes derivable from basic
7-note 'Greek' modes by adding or subtracting one note at a time
from adjacent modes. 2, 3, and 4 note scales won't fit on one
page, and are shown separately below the main table, with the
normal pentatonic repeated to show the connection.
Description of Table entries:
xx is a non-descriptions of unobserved mode (I'm too lazy to
figure out descriptions for modes that no one has been able to
make music from).
Top row- description of mode and mode#. * - mode is known, n -
mode not known (to me). Bottom row = semitone sequence, single
digits without spacing.
Model table entry: X shorthand description of mode (if it
has one), with f-lydian, c-ionian, g-mixolydian, d-dorian, a-
aeolian, e-phrygian, and b-locrian. n mode #, m # of tunes in
mode (of 6599 total). j - semitone sequence of mode (sum of
digits = 12 for 12 semitone scale).
X n|m
j
Note, at an entry, that the mode# to the right or left (and up or
down by one) always differs from the one you're at by 2^n (to the
nth power. This is very helpful in getting the beginning and
ending semitone sequences in the proper order. When you get 7 or
8 or 9 'ones' in the semitone sequences, they all start looking
about the same. (See file CODEMTHD.TXT for mode#.)
The "Greek' Mode Domain Table.
# of notes in scale:
5 6 7 8 9 10 11 12 (only 1 for
all 11's +1)
xx 2031|0
11112111111
xx 1519|0
1111211121
xx 1515|0 xx 1535|0
112211121 11111111121
fv1 1387|3 cv145 1531|1
11221221 1121111121
f 1386|28 fv14 1403|8 xx 2043|0
2221221 112111221 11211111111
f/c 1354|365 f+c 1402|182 cv147 1915|2
223221 22111221 1121112111
pi1 330|101 c 1370|2037 f+c+g 1914|35 xx 1919|0
22323 2212221 221112111 11111112111
c/g 346|129 c+g 1882|527 f+c+g+d 1918|3
221223 22122111 2111112111
pi2 338|13 g 858|328 c+g+d 1886|11 f+c+g+d+a 2046|4
23223 2212212 211122111 21111111111
g/d 850|102 g+d 862|46 c+g+d+a 2014|8 f+c+g+a+e 2047|1
232212 21112212 2111211111 111111111111
pi3 594|35 d 854|303 g+d+a 990|5 c+g+d+a+e 2015|0
23232 2122212 211121112 11111211111
d/a 598|288 d+a 982|83 g+d+a+e 991|1
212232 21221112 1111121112
pi4 596|35 a 726|403 d+a+e 983|1 g+d+a+e+b 1023|0
32232 2122122 111221112 11111111112
a/e 724|77 a+e 727|3 d+a+e+b 1015|1
322122 11122122 1112111112
pi5 660|0 e 725|26 a+e+b 759|1 xx 2039|0
32322 1222122 111211122 11121111111
e/b 661|3 e+b 757|1 xx 1783|0
122322 12211122 1112111211
b 693|0 xx 1781|0 xx 1791|0
1221222 122111211 11111111211
xx 1717|0 xx 1789|0
12212211 1211111211
xx 1725|0 xx 2045|0
121112211 12111111111
xx 1981|0
1211121111
xx 1983|0
11111121111
Here's the beginning of the table above:
1, 2, 3, and 4 note 'Greek' modes, with 5 show to connect with
the table above.
mode #|semitone sequence
1 2 3 4 5 = Scale notes
330
322|2523
66|255 338
64|75 82|2325
0 80|525 594
16|57 592|5232
528|552 596
532|3252
660
Column sums: 5-184, 6-704,7-3125,8-845,9-61,10-17,11-4.15-1 =
4941+1-4note (#82) = 4942 'Greek' domain tunes. Some stats:
'Greek' tunes, 2905 + (ionian) 2037 = 4942 in 66 modes. Total
tunes = 6599. Non 'Greek', are 1657 in 124 modes. 39 non-'Greek'
7-note modes account for 229 tunes, so 85 non-'Greek', non-7-note
modes account for 1428 tunes. Most are rare, but some are very
common.
I'm still trying to find out if there's any simple way to
organize the non-'Greek' domain modes. Here's a couple of long
chains where we start with 'Greek' based and end that way, but
travel through common but mostly non-'Greek' intermediates.
Going from 7-note to 12 note scale - highest probability routes.
Modes with asterisks are non-'Greek' Domain modes. hm is
harmonic minor and mm is melodic minor (both non-'Greek').
notes:
7 8 9 10 11 12
726|403 \ 2004|6
aeol d/a2v67
\ \
854|303 1750|168 2006|35 2038|11 2046|4 2047|1
dor -> av7 -> mm -> \ a/dv467 -> av3467 -> av13467
1238|21 / / 2014|8
hrm min / av36
/
1622|63 -> 1878|67
d/a6v7 dv7 \
5 6 7
338|13 342|14 1366|6
pi2 d-7 asc mel min
There are only 2 possibilities to get to harmonic minor, #1238,
by adding just one note at a time to a lower order mode, and we
have to go back to a 4-note scale to do it. Note that all modes
in the chain below are observed (barely at the start), so this
isn't just theoretical.
scheme- n|m, n = mode #, m = # of tunes in my COMBCOD3.TXT file
:i, is the semitone sequence.
Route to harmonic minor, #1238, and beyond
scale notes:
4 5 6 7
82|1 harmonic minor
A_B__D_E____
\ A_BC_D_E____ -> A_BC_D_EF___ -> A_BC_D_EF__G#
A__C_D_E____ / 86|12 214|18 1238|21
G-84|1
6 7
606|3:
A_BCdD_E__G
/
G598|288 / 1622|63
A_BC_D_E__G_ -> A_BC_D_E__Ga ->1750 in 1st chain above
other 8s from 1238
1246|1
1270|1
1494|1
other 8s from 1622
1630|2
other 9s from #1750
1751:1
1758:7
1782:3
other 10s from #2006
2007|5
2014|8
This shows the route, via familiar territory, to a good fraction
of observed 10 note tunes. [I've seen 4 11-note tunes, all in the
same mode, and an ABC of the only 12-note one I've seen is B324
among the broadside ballad tune here. These two are 'Greek'
domain modes, but the majority in the two above sequences are
not.]
The rest aren't really organized yet, but I've found 2 series
that I've labeled Ha and Hb. These have the property such that
for an n-note mode there is an n-1 'Greek' based mode without one
of its notes below it and and n+1 'Greek' based mode with one
more note above it. These have the same notes for 10 note scales,
and for 11 note scale they become the same as the 'Greek' based
modes.
Ha series:
2 3 4 5 6 7 8 9 10 - notes
2027|1122111111
1391|111121221
1514|22211121 1407|1111111221
1355|1123221 1530|221111121
362|222123 1371|11212221 2042|2211111111
46|25221 378|2211123 1883|112122111
74|2235 1362|221232 890|22111212 1887|1111122111
320|723 90|22125 1874|2322111 894|211111212
2|2,10 336|5223 602|221232 1878|21222111 1022|2111111112
18|237 848|52212 606|2111232 2006|212211111
(512)|10,2 530|2352 852|322212 734|21112122 2007|1112211111
(516)|372 534|21252 980|3221112 735|111112122
(644)|3522 662|212322 981|12221112 767|1111111122
(645)|12522 663|1112322 1013|122111112
(677)|123222 695|11121222 2037|1221111111
1701)|1232211 1719|111212211
(1709)|12122211 1727|1111112211
(1965)|121221111
(1967)|1111221111
Hb series
2 3 4 5 6 7 8 9 10 - notes
2027|1122111111
1899|112212111
1898|22212111 1407|1111111221
1866|2232111 1406|211111221
842|223212 1374|21112221 2042|2211111111
834|25212 350|2111223 2010|221211111
578|2532 342|212223 98622121112 1887|1111122111
576|732 86|21225 978|2321112 863|111112212
512|10.2 84|3225 722|232122 855|11122212 1022|2111111112
20|327 720|52122 599|1112322 1014|212111112
656|5322 597|122232 758|21211122 2007|1112211111
(206)|237 533|12252 756|3211122 1751|111221211
(266)|2253 693|321222 1749|12221211 767|1111111122
(1290)|22521 1685|1223211 765|121111122
(1322)|22232 701|12111222 2037|1221111111
(1395)|1123221 1973|122121111
(1451)|11222121 1727|1111112211
(1438)|111122121
(1967)|1111221111