Another almost diatonic scale

I've been playing around with general classes of scales lately. The idea is
you take two generating intervals, and make the scale up from them. You can
get different scales by changing the sizes of the intervals. One
interesting scale type is what I call 3+4 scales: 3 large and 4 small
intervals making up an octave. I'll call the large interval t and the small
one s, but be aware that these are both neutral seconds rather than a tone
and semitone.

The simplest examples of scales fitting this pattern are 7-equal, 10-equal
(t=2 steps, s=1 step) and 17-equal (t=3, s=2). Also, 31-equal fits: t=5,
s=4. I'll use 31-equal as the guide to a harmonic mapping. 24- and
38-equal are other members of this family.

The scale can be fitted to a normal keyboard, or staff notation, as follows:

C +t D +s E +s F +t G +s A +t B +s C

As far as I'm concerned, 7 note names can be used for 7 intervals regardless
of how they fit 5-limit harmony or a diatonic scale. Here, a fifth usually
is a fifth: 2t+2s. The two "wolf" fourths, A-D and F-B, happen to be close
to 11/8.

As the fifth can be divided into two equal parts, this means the resulting
neutral thirds (t+s) can be called approximations to 11/9. The remaining
third, D-F or 2s, is a minor third. The major third is 2t, although none of
these are in that 7-note scale.

These harmonies also work in 38-equal. In 17-equal, the fifth and neutral
third still work, but the minor third is more like a 7/6, and the wolf
fourth isn't much of an 11/8.

Now, this is all well and good. It offers a way of notating 17- and
38-equal and all that. However, the reason I'm sending this to the Tuning
List is that the base 7-note scale also fulfils most of the criteria for
Paul Erlich's generalized-diatonic scales. As they're such a simple class
of scales, the chances are someone has noticed this before, but I didn't
notice so I'll explain from scratch.

We're allowed some rule bending, and the main rule I'm going to bend is the
one about the definition of limit. So, I'll define the consonant limit as
including neutral thirds, perfect fifths, octave permutations thereof and
nothing else. So, that minor third and super-fourth are both dissonant --
indeed, they are the characteristic dissonances. This makes more sense in
17-equal than 31-equal. In 17-equal, 6/5 and 11/8 are so poorly tuned you
can, if you really want to, call them dissonances. 9/8 and 12/11 are more
of a problem, though.

Let's go through Paul's rules

>(0) Octave equivalence:

>There is a basic scale which repeats itself exactly at the octave,
>extending infinitely both upwards and downwards in pitch.

No problem

>(1) Scale structure:

>EITHER a or b must be satisfied.

>(Version a - distributional evenness): The basic scale has two step
>sizes, and given these step sizes, the notes are arranged in as close =
>as
>possible an approximation of an equal tuning with only as many notes =
>per
>octave as the basic scale.

This is satisfied in both 31- and 17-equal

>(Version b - tetrachordality): The basic scale has a structure
>emphasizing similarity at the Q. In particular, there is a
>"tetrachordal" structure, that is, within any octave span, the pattern
>of steps within one approximate 4:3 are replicated in another
>approximate 4:3, with the remaining "leftover" interval spanned using
>patterns of step sizes (often just one step) found in the "tetrachord."

The tetrachord is s t s. Start on B and you have s t s s t s t.

>(2) Chord structure:=20

>There exists a pattern of intervals (defined by number of scale steps,
>not specific as to exact size) which produces a complete, consonant
>chord (containing all non-equivalent consonant interval) on most scale
>degrees.

Our only consonant intervals are neutral thirds and fifths. So, consonant
triads are found in the same places on the scale as they would be in a
normal diatonic scale. "Most" here means 5 out of 7. The two dissonant
triads are B-D-F and D-F-A.

>(3) Chord relationships:

>The majority of the consonant chords have a root that lies a Q away =
>from
>the root of another consonant chord.

F-C-G and A-E are strings of fifths. So all consonant chords fulfil this
criterion.

>(4) Key coherence:

>A chord progression of no more than three consonant chords is required
>to cover the entire scale.

No problem. Chords on C, F and G cover the scale.

>(5) Tonicity:

>The notes of the scale are ordered, increasing in pitch, so that the
>first note is the root of a complete consonant chord, defined hereafter
>as the "tonic chord."

Choosing the tonic is tricky, as will become clear below. I'll go for C to
start with.

>(6) Homophonic stability:

>All characteristic dissonances are disjoint from the tonic chord, with
>the following possible exception: A characteristic dissonance may share
>a note with the tonic chord if, when played together, they form a
>consonant chord of the next higher limit (3 =DE 5, 5=DE 7, 7=DE 9).

Making C the tonic means the first part of this condition is fulfilled. The
concept of "next higher limit" is a bit tricky, seeing as I've excluded most
of the 11-limit. It may be that some other 11-limit chords can be
constructed to meet that proviso, in light of the following rule.

>(7) Melodic guidance:

>The rarest step sizes are only found adjacent to notes of the tonic
>chord.

This is the tricky one. The rarest _interval_ is the minor third which,
being a characteristic dissonance, is disjoint from the tonic. However, the
fact that these notes (D and F) are in the middle of the tonic chord (C, E
and G) may assist in pointing toward the tonic. That's the spirit of this
rule.

As there are 4 of one step size and 3 of the other, the sense of rarity
isn't that strong. Also, as the two step sizes are approximately the same
in 31-equal anyway, you can't hear the large- and small-ness. Not that I'm
the person to ask about this, though, as I don't hear the difference between
tones and semitones in a diatonic scale, consciously anyway. Still, one of
the rarer (larger) seconds (A-B) isn't adjacent to the tonic chord. Setting
F or G as the tonic complies with this rule, but causes problems with the
previous ones.

It may be that, as these scales are so close to 7-equal, you don't get a
proper sense of key center with them.

The weak version of homophonic stability:

>At least one characteristic dissonance either is disjoint from the =
tonic
>chord, or shares a note with the tonic chord such that, when played
>together, they form a consonant chord of the next highest limit (3=DE =
5,
>5=DE 7, 7=DE 9).

Works fine with F as the tonic, though.

As to the bonus rule

>P.S. The quality of approximations to consonant intervals must not be
>worse than the quality of 12-tET's approximations to the three 5-limit
>consonances (I'll accept any reasonable weighting scheme).

No problem with 31-equal. Even 17-equal looks like it fulfils it for the
intervals I've defined as consonant.

So, these scales are looking pretty good as generalized diatonics, provided
you have an idiosyncratic definition of consonant limit. Also, they do
sound quite good, but no amount of theory will explain that.

Graham
http://www.cix.co.uk/~gbreed/

On Mon, 18 Jan 1999 14:28:08 -0000, Graham Breed <g.breed@tpg.co.uk> wrote:

>We're allowed some rule bending, and the main rule I'm going to bend is the
>one about the definition of limit. So, I'll define the consonant limit as
>including neutral thirds, perfect fifths, octave permutations thereof and
>nothing else. So, that minor third and super-fourth are both dissonant --
>indeed, they are the characteristic dissonances. This makes more sense in
>17-equal than 31-equal. In 17-equal, 6/5 and 11/8 are so poorly tuned you
>can, if you really want to, call them dissonances. 9/8 and 12/11 are more
>of a problem, though.

I think of it as more of a slightly flat 19/16, the familiar 12-tet minor
third, than a mistuned 6/5. But even a perfectly tuned 19/16 sounds a bit
dissonant.

>>(1) Scale structure:
>
>>EITHER a or b must be satisfied.
>
>>(Version a - distributional evenness): The basic scale has two step
>>sizes, and given these step sizes, the notes are arranged in as close =
>>as
>>possible an approximation of an equal tuning with only as many notes =
>>per
>>octave as the basic scale.
>
>This is satisfied in both 31- and 17-equal

Interesting. One of my attempts to notate 17-equal, before I reluctantly
decided to notate it as a meantone scale, is similar to this except the
first two steps are reversed (i.e., 0-2-5-7-10-12-15-17). (In fact, it's
just a different mode of your scale, starting on the third note.)

Herman Miller wrote:

>
>
> I think of it as more of a slightly flat 19/16, the familiar 12-tet minor
> third, than a mistuned 6/5. But even a perfectly tuned 19/16 sounds a bit
> dissonant.

The 19/16 sounds better to my ear. I have never been convinced that the 6/5 is
the minor third, in fact I never use it this context since it doesn't say what
i want it to say. The 6-7-9 triad I really like!
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Herman Miller wrote:

>I think of it as more of a slightly flat 19/16, the familiar 12-tet minor
>third, than a mistuned 6/5. But even a perfectly tuned 19/16 sounds a bit
>dissonant.

19/16 lies outside the 11-limit. The problem with these scales, in the
light of Paul Erlich's rules, is that there are 11-limit intervals outside
the 11-limit "complete" chords I'm basing the harmony around. IMHO, real
complete 11-limit chords are too complex for this sort of thing.

>Interesting. One of my attempts to notate 17-equal, before I reluctantly
>decided to notate it as a meantone scale, is similar to this except the
>first two steps are reversed (i.e., 0-2-5-7-10-12-15-17). (In fact, it's
>just a different mode of your scale, starting on the third note.)

Yes, there's a fairly obvious way of tuning all notes except the D. I went
for the sharp alternative because I'm used to C-D being a major tone with JI
or whatever. The difference is that, with my way, you can choose the black
notes so that you can modulate down by two fifths. With your way, you can
modulate up by two fifths with a judicious choice of black notes. For
notation, this doesn't matter because you're not restricted to a 12 note
octave. As for this or the meantone-type notation, it all depends on
whether you think the white notes, or unaltered staff positions, should be a
5+2 scale, or any 7-note scale. I have no problem with the latter, at the
moment at least.

Oh, I did make a mistake in that post. I described the intervals in the
scale as both neutral seconds, whereas the larger one is quite large enough
to be called a major second.

FWIW, I'd write 20-equal as a 6+1 scale: 6 large and 1 small interval to
the octave. The small one I'd put on B-C, although it could be E-F. I
think that means most of the thirds are those flat major thirds. I haven't
worked it out in detail, but I offer it as a suggestion. The large
intervals will be 3 steps, the small interval 2 steps. Sharp and flat
symbols can then be used for 1 step boosts, 1 equalling 3-2.

Now, there was another message in the last digest entitled "The 5w + 2h
heptads of F sets of f". This is obviously linked to another message back
on the 11th. Could someone explain, using simple words and with some
reference to music, what all this is about? It looks like I might be
interested if I understood it. The set of scales in my 3+4 family are
roughly that described as "w-h=2". Is this like what I'd call septimally
double-positive? That is, you start with a 5+2 scale, and these scales are
the ones where the larger interval is 2 steps bigger than the smaller one.
If so, what's this stuff about exteriors and interiors and perimeters?

Graham
http://www.cix.co.uk/~gbreed/

Graham Breed wrote,

>I've been playing around with general classes of scales lately. The
idea is
>you take two generating intervals, and make the scale up from them.
You can
>get different scales by changing the sizes of the intervals. One
>interesting scale type is what I call 3+4 scales: 3 large and 4 small
>intervals making up an octave. I'll call the large interval t and the
small
>one s, but be aware that these are both neutral seconds rather than a
tone
>and semitone.

>The simplest examples of scales fitting this pattern are 7-equal,
10-equal
>(t=2 steps, s=1 step) and 17-equal (t=3, s=2). Also, 31-equal fits:
t=5,
>s=4. I'll use 31-equal as the guide to a harmonic mapping. 24- and
>38-equal are other members of this family.

>The scale can be fitted to a normal keyboard, or staff notation, as
follows:

>C +t D +s E +s F +t G +s A +t B +s C

The characteristic scale of Arabic music could be described as t s s t t
s s according to the same definitions of s and t. This is one of the
reasons I'm leaning toward 31-tET for my next guitar. Your scale has
also been identified in Arabic music but is rare at best.

>We're allowed some rule bending, and the main rule I'm going to bend is
the
>one about the definition of limit. So, I'll define the consonant limit
as
>including neutral thirds, perfect fifths, octave permutations thereof
and
>nothing else. So, that minor third and super-fourth are both dissonant
--
>indeed, they are the characteristic dissonances. This makes more sense
in
>17-equal than 31-equal. In 17-equal, 6/5 and 11/8 are so poorly tuned
you
>can, if you really want to, call them dissonances. 9/8 and 12/11 are
more
>of a problem, though.

The characteristic dissonance should really stand out as much more
dissonant that its same-generic-size variants. I don't think this works
even in 17-tET. However, with the appropriate inharmonic timbre . . .

>>(Version b - tetrachordality): The basic scale has a structure
>>emphasizing similarity at the Q. In particular, there is a
>>"tetrachordal" structure, that is, within any octave span, the pattern
>>of steps within one approximate 4:3 are replicated in another
>>approximate 4:3, with the remaining "leftover" interval spanned using
>>patterns of step sizes (often just one step) found in the
"tetrachord."

>The tetrachord is s t s. Start on B and you have s t s s t s t.

This is supposed to work in every octave span. Only three appear to work
in your scale:

s t s s t s t
s t s t s t s
t s t s s t s

The Arabic scale is tetrachordal in five octave spans:

t s s t t s s
s s t t s s t
t t s s t s s
t s s t s s t or t s s t s s t
s s t s s t t

That makes is melodically more coherent in my opinion (and judging from
actual practice).

Graham Breed wrote:

>The set of scales in my 3+4 family are roughly that described as "w-h=2".

The set (or sequence) of w-h=2 in 3 + 4 @ +t +s +s +t +s +t +s would be:

t=2 s=0
0, 2, 2, 2, 4, 4, 6, 6*

t=3 s=1
0 3 4 5 8 9 12 13
0 277 369 462 738 831 1108 1200

t=4 s=2
0 4 6 8 12 14 18 20
0 240 360 480 720 840 1080 1200

t=5 s=3
0 5 8 11 16 19 24 27
0 222 356 489 711 844 1067 1200

t=6 s=4
0 6 10 14 20 24 30 34
0 212 353 494 706 847 1059 1200

t=7 s=5
0 7 12 17 24 29 36 41
0 205 351 498 702 849 1054 1200

etc.

>Is this like what I'd call septimally double-positive? That is, you start
with a 5+2 scale, and these scales are the ones where the larger interval is
2 steps bigger than the smaller one.

No. (Unless it also happens to be.)

>what's this stuff about exteriors and interiors and perimeters?

You could use n"F" sets of n"f" to explicitly define an exterior, interior,
and perimeter sets of equidistant divisions of the octave for the 3 + 4 @ +t
+s +s +t +s +t +s heptad where n"F" = 7, n"f" = 3, and columns one through
seven depict w - h = 1 through w - h = 7:

3 6 2 5 1 4 7
10 13 9 12 8 11 14
17 20 16 19 15 18 21

Exterior set @ 2, 5, 1, 4, 7, 8, 11, and 14:

Es Es Es Es Es
Es Es Es

Perimeter set @ 3, 6, 9, 12, 15, 18, and 21:

Ps Ps
Ps Ps
Ps Ps
Ps

Interior set @ 10, 13, 17, 20, 16, and 19:

Is Is
Is Is Is Is

>The simplest examples of scales fitting this pattern are 7-equal, 10-equal
(t=2 steps, s=1 step) and 17-equal (t=3, s=2). Also, 31-equal fits: t=5,
s=4.

Ostensibly, an interior set would delimit the �simplest� n-tET examples of
the particular heptads characteristic "f" and "F" properties�**

t=2 s=1
0 2 3 4 6 7 9 10
0 240 360 480 720 840 1080 1200

t=3 s=1
0 3 4 5 8 9 12 13
0 277 369 462 738 831 1108 1200

t=3 s=2
0 3 5 7 10 12 15 17
0 212 353 494 706 847 1059 1200

t=4 s=2
0 4 6 8 12 14 18 20
0 240 360 480 720 840 1080 1200

t=4 s=1
0 4 5 6 10 11 15 16
0 300 375 450 750 825 1125 1200

t=5 s=1
0 5 6 7 12 13 18 19
0 316 379 442 758 821 1137 1200

Mapped to nS sets of nt, every n-tET>ntnS/nSnt (i.e., 21tET) yields a heptad
were h or (using your definition) s>0 of the perimeter set, and s>-n of the
exterior set:

w-h=1
------------------------------------------------
Es
Is
Is
24
31
38
45
52
59
66
73
80
87
94
101
108
115
122
129
136
143
150
157
164
171
178
185
192
199
206
213
220
227
234
241
248
255
262
269
276
283
290
297
304
311
318
325
332
339
346
353
360
367
374
381
388
395
402
409
416
423
430
437
444
451
458
465
472
479
486
493
500
507
514
521
528
535
542
549
556
563
570
577
584
591
598
605
612
619
626
633
640
647
654
661
668
675
682
689
696
703
710
717
724
731
738
745
752
759
766
773
780
787
794
801
808
815
822
829
836
843
850
857
864
871
878
885
892
899
906
913
920
927
934
941
948
955
962
969
976
983
990
997
1004
1011
1018
1025
1032
1039
1046
1053
1060
1067
1074
1081
1088
1095
1102
1109
1116
1123
1130
1137
1144
1151
1158
1165
1172
1179
1186
1193
1200

w-h=2
------------------------------------------------
Ps
Is
Is
27
34
41
48
55
62
69
76
83
90
97
104
111
118
125
132
139
146
153
160
167
174
181
188
195
202
209
216
223
230
237
244
251
258
265
272
279
286
293
300
307
314
321
328
335
342
349
356
363
370
377
384
391
398
405
412
419
426
433
440
447
454
461
468
475
482
489
496
503
510
517
524
531
538
545
552
559
566
573
580
587
594
601
608
615
622
629
636
643
650
657
664
671
678
685
692
699
706
713
720
727
734
741
748
755
762
769
776
783
790
797
804
811
818
825
832
839
846
853
860
867
874
881
888
895
902
909
916
923
930
937
944
951
958
965
972
979
986
993
1000
1007
1014
1021
1028
1035
1042
1049
1056
1063
1070
1077
1084
1091
1098
1105
1112
1119
1126
1133
1140
1147
1154
1161
1168
1175
1182
1189
1196

w-h=3
------------------------------------------------
Es
Ps
Is
23
30
37
44
51
58
65
72
79
86
93
100
107
114
121
128
135
142
149
156
163
170
177
184
191
198
205
212
219
226
233
240
247
254
261
268
275
282
289
296
303
310
317
324
331
338
345
352
359
366
373
380
387
394
401
408
415
422
429
436
443
450
457
464
471
478
485
492
499
506
513
520
527
534
541
548
555
562
569
576
583
590
597
604
611
618
625
632
639
646
653
660
667
674
681
688
695
702
709
716
723
730
737
744
751
758
765
772
779
786
793
800
807
814
821
828
835
842
849
856
863
870
877
884
891
898
905
912
919
926
933
940
947
954
961
968
975
982
989
996
1003
1010
1017
1024
1031
1038
1045
1052
1059
1066
1073
1080
1087
1094
1101
1108
1115
1122
1129
1136
1143
1150
1157
1164
1171
1178
1185
1192
1199

w-h=4
------------------------------------------------
Es
Ps
Is
26
33
40
47
54
61
68
75
82
89
96
103
110
117
124
131
138
145
152
159
166
173
180
187
194
201
208
215
222
229
236
243
250
257
264
271
278
285
292
299
306
313
320
327
334
341
348
355
362
369
376
383
390
397
404
411
418
425
432
439
446
453
460
467
474
481
488
495
502
509
516
523
530
537
544
551
558
565
572
579
586
593
600
607
614
621
628
635
642
649
656
663
670
677
684
691
698
705
712
719
726
733
740
747
754
761
768
775
782
789
796
803
810
817
824
831
838
845
852
859
866
873
880
887
894
901
908
915
922
929
936
943
950
957
964
971
978
985
992
999
1006
1013
1020
1027
1034
1041
1048
1055
1062
1069
1076
1083
1090
1097
1104
1111
1118
1125
1132
1139
1146
1153
1160
1167
1174
1181
1188
1195

w-h=5
------------------------------------------------
Es
Es
Ps
22
29
36
43
50
57
64
71
78
85
92
99
106
113
120
127
134
141
148
155
162
169
176
183
190
197
204
211
218
225
232
239
246
253
260
267
274
281
288
295
302
309
316
323
330
337
344
351
358
365
372
379
386
393
400
407
414
421
428
435
442
449
456
463
470
477
484
491
498
505
512
519
526
533
540
547
554
561
568
575
582
589
596
603
610
617
624
631
638
645
652
659
666
673
680
687
694
701
708
715
722
729
736
743
750
757
764
771
778
785
792
799
806
813
820
827
834
841
848
855
862
869
876
883
890
897
904
911
918
925
932
939
946
953
960
967
974
981
988
995
1002
1009
1016
1023
1030
1037
1044
1051
1058
1065
1072
1079
1086
1093
1100
1107
1114
1121
1128
1135
1142
1149
1156
1163
1170
1177
1184
1191
1198

w-h=6
------------------------------------------------
Es
Es
Ps
25
32
39
46
53
60
67
74
81
88
95
102
109
116
123
130
137
144
151
158
165
172
179
186
193
200
207
214
221
228
235
242
249
256
263
270
277
284
291
298
305
312
319
326
333
340
347
354
361
368
375
382
389
396
403
410
417
424
431
438
445
452
459
466
473
480
487
494
501
508
515
522
529
536
543
550
557
564
571
578
585
592
599
606
613
620
627
634
641
648
655
662
669
676
683
690
697
704
711
718
725
732
739
746
753
760
767
774
781
788
795
802
809
816
823
830
837
844
851
858
865
872
879
886
893
900
907
914
921
928
935
942
949
956
963
970
977
984
991
998
1005
1012
1019
1026
1033
1040
1047
1054
1061
1068
1075
1082
1089
1096
1103
1110
1117
1124
1131
1138
1145
1152
1159
1166
1173
1180
1187
1194

w-h=7
------------------------------------------------
Es
Es
Ps
28
35
42
49
56
63
70
77
84
91
98
105
112
119
126
133
140
147
154
161
168
175
182
189
196
203
210
217
224
231
238
245
252
259
266
273
280
287
294
301
308
315
322
329
336
343
350
357
364
371
378
385
392
399
406
413
420
427
434
441
448
455
462
469
476
483
490
497
504
511
518
525
532
539
546
553
560
567
574
581
588
595
602
609
616
623
630
637
644
651
658
665
672
679
686
693
700
707
714
721
728
735
742
749
756
763
770
777
784
791
798
805
812
819
826
833
840
847
854
861
868
875
882
889
896
903
910
917
924
931
938
945
952
959
966
973
980
987
994
1001
1008
1015
1022
1029
1036
1043
1050
1057
1064
1071
1078
1085
1092
1099
1106
1113
1120
1127
1134
1141
1148
1155
1162
1169
1176
1183
1190
1197

Respectfully,
Dan

*I include Interior sets in these kinds of examples solely for illustrative
purposes.

**If thought of in terms of addressing 3rd and 6th size (as opposed to
fourth and fifth size), you could recast "f" and "F" as something on the
order of nt and nS(where "n" would be neutral, "t" would be third, and "S"
would be sixth�).

FWIW:
Perimeter set got jogged by the email process... In the event that this
should happen once again, the bracketed tET's also show the positioning...

[3] [6] 2 5 1 4 7
10 13 [9] [12] 8 11 14
17 20 16 19 [15] [18] [21]

Perimeter set @ 3, 6, 9, 12, 15, 18, and 21:

Ps Ps
Ps Ps
Ps Ps Ps

Dan

>The characteristic scale of Arabic music could be described as t s s t t
>s s according to the same definitions of s and t. This is one of the
>reasons I'm leaning toward 31-tET for my next guitar. Your scale has
>also been identified in Arabic music but is rare at best.

Cool, nice to see we're thinking along the same lines. And what's one note
between friends? Any more info on the Arabic stuff would be welcomed.

>The characteristic dissonance should really stand out as much more
>dissonant that its same-generic-size variants. I don't think this works
>even in 17-tET. However, with the appropriate inharmonic timbre . . .

I had a play with this on my TX81Z. It did exaggereate the difference, but
I really need more than 4 tones. I won't be pursuing this any further,
because it is quite perverse to work that hard to reduce the consonance.
I'd rather embrace the "characteristic consonances" and use 31- or 38-equal
for actual music.

<ASIDE>

To respond to David Keenan from another thread. I'm convinced about
11-limit harmony now, or at least some 11-limit intervals. Intervals around
11/8 do have something their immediate neighbours don't. There's also
something about neutral thirds, but I don't know if this has anything to do
with the 11/9 approximation. Whether or not this is due to the "11-ness"
I've abandoned as a meaningless question. I hope my ear hasn't been trained
in any way so as to make it hear different things to my intended listeners.
Familiarity must count for something, though. Oh, and this works for a
variety of timbres.

When you get to the 19-limit these questions become even more meaningless.
ANY interval will be close to some 19-limit interval. When people start to
talk 19-limit, I find it best to smile sympathetically and change the
subject.

</ASIDE>

>>The tetrachord is s t s. Start on B and you have s t s s t s t.

>This is supposed to work in every octave span. Only three appear to work
>in your scale:

Oh, sorry. But this is an either/or condition, so the maximal evenness
(that's the approximation to 7-equal thing, isn't it?) is enough. It's
still only my bizzare definition of completeness that stops this from being
a generalized diatonic scale.

>The Arabic scale is tetrachordal in five octave spans:

>t s s t t s s
>s s t t s s t
>t t s s t s s
>t s s t s s t or t s s t s s t
>s s t s s t t

>That makes is melodically more coherent in my opinion (and judging from
>actual practice).

Ah, but it doesn't pass your strict melodic criterea whereas my scale does.
I accept yours gives stronger melodies: I tried it last night. I think I'll
keep my original scale as the base and throw in Bb as an accidental.

>>The characteristic scale of Arabic music could be described as t s s t
t
>>s s according to the same definitions of s and t. This is one of the
>>reasons I'm leaning toward 31-tET for my next guitar. Your scale has
>>also been identified in Arabic music but is rare at best.

>Cool, nice to see we're thinking along the same lines. And what's one
note
>between friends? Any more info on the Arabic stuff would be welcomed.

I would point you to Manuel's scale list at the Mills ftp site, but its
gone! Can it be found somewhere else?

>>>The tetrachord is s t s. Start on B and you have s t s s t s t.

>>This is supposed to work in every octave span. Only three appear to
work
>>in your scale:

>Oh, sorry. But this is an either/or condition, so the maximal evenness
>(that's the approximation to 7-equal thing, isn't it?) is enough. It's
>still only my bizzare definition of completeness that stops this from
being
>a generalized diatonic scale.

Well, I'm not a big fan of maximal evenness, as my paper makes clear. If
I could rewrite the contest, I'd eliminate that, but maybe allow for a
structure that spans a 5:4 and occurs 3 times in every octave span (as
in the ME 22-out-of-41).

>>The Arabic scale is tetrachordal in five octave spans:

>>t s s t t s s
>>s s t t s s t
>>t t s s t s s
>>t s s t s s t or t s s t s s t
>>s s t s s t t

>>That makes is melodically more coherent in my opinion (and judging
from
>>actual practice).

>Ah, but it doesn't pass your strict melodic criterea whereas my scale
does.

Because of the maximal evenness? Again, I'd like to eliminate that
provision.

>I accept yours gives stronger melodies: I tried it last night. I think
I'll
>keep my original scale as the base and throw in Bb as an accidental.

Nice! The typical "Arabic" preset on synths lowers E and B by a
quarter-tone (relative to 12-tET).

> I would point you to Manuel's scale list at the Mills ftp site, but its
> gone! Can it be found somewhere else?

Yes, it's not the most recent collection though.
This is the file list:
ftp://byrd.math.uga.edu/pub/music/dx7/scales/scales.doc
And the archive of scales:
ftp://byrd.math.uga.edu/pub/music/dx7/scales/scales.zip

Manuel Op de Coul coul@ezh.nl