Yahoo Tuning Groups Ultimate Backup tuning 2001: A MOS Odyssey (Part 2)

2001: A MOS Odyssey (Part 2)

After giving more careful study to Erv Wilson's letter to John
Chalmers about MOS scales:

http://www.anaphoria.com/mos.PDF

and spending some time at the Dan Stearns Institute of Mosological
Studies as well as Dave Keenan's site, I give you part 2 of our 2001:
A MOS Odyssey.

In part one we looked at how the Prime Series Ratios provides us with
various widths of fifth generators, which reveal vectors to many
lower primes; and actually targeting the thirds of many lower primes
as the special goal of the scales. Here we will stay within the same
harmonic sensibilities, and for economy of discussion, plus allowing
for more illustrations; we will focus again on 12 tone MOS. Another
benefit is that there may be many aspiring tunists reading this
article who may only have the capability to tune to a 12
pitch "octave tuning", and may not find the discussion of MOS scales
with more than 12 pitches of interest, where they may not be able to
tune a synth to play them.

In this installment of 2001: A MOS Odyssey we take an even closer and
more thorough look at this fascinating world, except from the angle
of the Classic Fourth Generator MOS. As Stearns and Wilson show, for
a Fourth Generator to create a Moment of Symmetry, it must fit within
the range of 2/5 to 3/7 of an octave, which in simple terms is around
480 to 514 cents. I conjecture that this range of 2/5 to 3/7 of an
octave, and it's compliment 3/5 to 4/7 for our Fifth Generators,
really defines the literal borders of what we may call a "fourth"
or "fifth" in tunings intended for timbres with a near linear
harmonic series. In tuning terms we know this represents quite a
large expanse.

The exploration of prime vectors (the manner in which high primes
audibly "connect" to lower ones) is greatly facilitated by the use of
the Prime Series Ratios. The scale used here provides us with 257
Prime Series Ratios within the 2/5 to 3/7 of an octave. That is 257
different MOS possibilities in a range of about 34 cents. This allows
us to draw many fine distinctions and exacting connections between
the high and low primes. As one might transpose, there are many other
many other possibilities to target with this same logic, but here we
seek to find which widths of Fourth Generators will audibly touch the
thirds of select lower number primes. This is really quite an
exciting prismatic effect, where as the size of the generator shrinks
or grows, we see refraction of myriad prime vectors, which join
together the simple and the complex number strata; giving us
intervals which are audibly the same as the approximated target
intervals.

Now that we know what the borders are, let us examine what may serve
us as a kind of abstract middle point - or starting point, in which a
12 tone MOS pattern is generated with an ratio approximating an equal
temperament fourth generator of around 500 cents. Here we'll use
311/233 @ 499.902. If we repeatedly stack the generator we get the
values below when reduced within the octave:

499.902, 999.803, 299.704, 799.606, 99.507, 599.409, 1099.310,
399.212, 899.113, 199.015, 698.917

Note that the 3rd chain of the generator gives us the minor 3rd, and
the 8th chain gives us the major 3rd. This is merely to illustrate
the initial intervallic roles at this starting point, but as we will
see below; proportional to the generator size, the role of these
beginning points will change radically. Even though this just
represents an abstract starting point, you could still give the above
scale as a gift to your least favored in-laws, this way nothing goes
to waste! The symmetry of this scale is interesting though:

0.000
99.508
99.508
100.689
99.508
100.689
99.508
99.508
100.689
99.508
100.689
99.508
100.689

5L,7s

The next question in the readers mind must be: "Well, what would the
scales at the extreme borders look and sound like?" I thought you'd
ask this, and naturally it was my first thought too. To learn which
ratios in the "8,479 Tone, Prime Series Ratio, Non-Octave Scale"(1.),
would be the break points either side of 2/5 to 3/7 of an octave,
became the immediate goal. The border ratios found being: 991/751 @
480.087 cents, and 467/347 @ 514.184 cents. Below we see the extreme
poles of the Fourth Generator MOS:

Generator
991/751 @ 480.087 cents

Cents Steps
0 0
0.433 0.433
0.866 0.433
240.260 239.394
240.693 0.433
480.087 239.394
480.519 0.433
480.952 0.433
720.346 239.394
720.779 0.433
960.173 239.394
960.606 0.433
961.039 0.433

Generator
467/347 @ 514.184 cents

Cents Steps
0 0
169.817 169.817
339.633 169.817
341.890 2.257
511.707 169.817
513.963 2.257
683.780 169.817
853.597 169.817
855.853 2.257
1025.670 169.817
1027.927 2.257
1197.743 169.817
1200.000 2.257

Now these are indeed some very bizarre scales. Contrast these with
the above starting point ET MOS, and you'll see how the scale degrees
morph into other functional intervals relative to the generator size.

It is interesting to take a look at some of the Scala data for these
two scales, which reveals some very unique properties:

Generator
991/751 @ 480.087 cents
Scale has Myhill's property
generators: 5 of 480.0866 cents and 7 of 719.9134 cents
Number of recognizable fifths : 18, average 719.721 cents

Generator
467/347 @ 514.184 cents
Scale has Myhill's property
generators: 5 of 514.1843 cents and 7 of 685.8157 cents
Scale is distributional even
Scale is a Constant Structure
Number of recognizable fifths : 15, average 686.005 cents

It is remarkable how these two scales actually have more recognizable
fifths than the above ET example, which has 11! This is due to the
scale, being right at the point of "passing through the looking
glass", as Erv Wilson says eloquently in his letter, and creating
unisons. Just one more ratio in either direction would cause the
values to fall outside the boundaries; no longer being in MOS
territory. This little paradoxical finding will sure put some "Just"
in your "Intonation", and force you to consider an extreme form of it.

Now that we know what lies at the extreme borders of our Odyssey, let
us now explore and harvest the vast treasures of the middle world,
but first let's prepare ourselves for the treasure hunt, by looking
at a few of the target flavors we seek along our journey.

Taken from Margo Schulter's table

http://www.egroups.com/message/tuning/17799

of Neo-Gothic flavors, we will seek:

--------------------------------------------------------------------
flavor ratio zone rough size description/example
--------------------------------------------------------------------
49-flavor 49:40 ~351 cents neutral 3rd -- 17-tET
17-flavor 21:17 ~366 cents submajor 3rd -- 46-tET
5-flavor 5:4 ~386 cents Pythagorean schisma 3rd
3-flavor 81:64 ~408 cents Pythagorean major 3rd
11-flavor 14:11 ~418 cents regular 3rd -- 29-tET or 46-tET
23-flavor 23:18 ~424 cents regular 3rd -- 17-tET
7-flavor 9:7 ~435 cents regular 3rd -- 22-tET, 3-7 JI
13-flavor 13:10 ~454 cents "diesis 3rd" -- 29-tET

(Note: Here's a lady after my own heart with this 23 prime ratio!)

And combining these with a few additional favorites for good measure,
we have our table of target thirds:

Ratio Cents
22/19 254
7/6 267
27/23 278
20/17 281
19/16 298
25/21 302
6/5 316
23/19 331
17/14 336
28/23 341
11/9 347
49/40 351
27/22 355
16/13 359
21/17 366
26/21 370
5/4 386
24/19 404
81/64 408
19/15 409
14/11 418
23/18 424
32/25 427
9/7 435
22/17 446
13/10 454

It is an important point to make here again, that what we are really
seeking with our generating intervals, are the multiplied widths of
an approximate 4/3, which will give us our target thirds. Of course
as above, by multiplying our generator by 3, we get our minor 3rds,
and multiplied by 8, we get our major 3rds, but it is important to
note that this is also dependent upon being within a small band
around our abstract starting point of the ET MOS.

As before, we will begin with the narrow fourth generators, and
progress through the ranks. I have adopted Stearns modal rotation
scheme, because it best shows a kind of exploded view of intervallic
interconnections. Notice how that relative to the generator size, the
3rd and 8th chains will affect the columns for the minor and major
thirds.

Our first scale:

Scale 01: 5L,7s
Generator
557/421 484.629

0 23 46 254 277 485 508 531 739 762 969 992 1200
0 23 231 254 461 485 508 715 739 946 969 1177 1200
0 208 231 438 461 485 692 715 923 946 1154 1177 1200
0 23 231 254 277 485 508 715 739 946 969 992 1200
0 208 231 254 461 485 692 715 923 946 969 1177 1200
0 23 46 254 277 485 508 715 739 762 969 992 1200
0 23 231 254 461 485 692 715 739 946 969 1177 1200
0 208 231 438 461 669 692 715 923 946 1154 1177 1200
0 23 231 254 461 485 508 715 739 946 969 992 1200
0 208 231 438 461 485 692 715 923 946 969 1177 1200
0 23 231 254 277 485 508 715 739 762 969 992 1200
0 208 231 254 461 485 692 715 739 946 969 1177 1200

557 = 22/19 254 (9), 27/23 278 (4), 9/7 438 (3)

To show the connections of the prime vectors, I will use a notation
for the above scale and all to follow: 557 = 22/19 254 (9), 27/23 278
(4). "557" being the prime limit of the generator, then the targeted
ratios (with cents) which connect to this limit, and in parentheses
the number of occurrences of the interval (as suggested by Paul
Erlich). It is worth noting that this scale has 3 occurrences of a
near 9/7 @ 438.343 cents, and has a whopping 17 recognizable fifths
at an average of 707.204 cents! We can easily see (and hear) that the
557 Prime generator "audibly" connects to primes 3, 7, 11, 19, and
23; giving us intervals which are all but indistinguishable from the
thirds of these lower primes. Perhaps it is a little
more "conventional" to look for these connections from low to high
primes, but here we shatter this convention, and reverse the order of
our view, by looking downward from the Prime Series number strata. To
put it oxymoronically; it's the same difference. Please note that
here if the thirds fall within about 3 cents of the target ratios, it
is considered in the notation, which can include some paradoxical
entries (see below).

This is a very interesting "comma MOS" too, with the 23.143 being
close to the Pythagorean comma. Some may find difficulty in making
music with commas in a scale such as this, while others (such as
myself, David Beardsley and Margo Schulter) find great melodic
and "adaptive" power in such small steps. Of course one must agree
with Stearns, in that "the answer is more trivial the further you are
away from ideal", and while this kind of MOS scale may be interesting
and novel from a theoretical point of view, it will have less musical
utility than MOS which has more even steps, but can serve as a kind
of special effects scale where one may require such a flavor in a
given composition.

Scale 02: 5L,7s
Generator
691/521 488.883

0 44 89 267 311 489 533 578 756 800 978 1022 1200
0 44 222 267 444 489 533 711 756 933 978 1156 1200
0 178 222 400 444 489 667 711 889 933 1111 1156 1200
0 44 222 267 311 489 533 711 756 933 978 1022 1200
0 178 222 267 444 489 667 711 889 933 978 1156 1200
0 44 89 267 311 489 533 711 756 800 978 1022 1200
0 44 222 267 444 489 667 711 756 933 978 1156 1200
0 178 222 400 444 622 667 711 889 933 1111 1156 1200
0 44 222 267 444 489 533 711 756 933 978 1022 1200
0 178 222 400 444 489 667 711 889 933 978 1156 1200
0 44 222 267 311 489 533 711 756 800 978 1022 1200
0 178 222 267 444 489 667 711 756 933 978 1156 1200

691 = 7/6 267 (8), 22/17 446 (8)

Scale 03: 5L,7s
Generator
1171/881 492.633

0 63 126 278 341 493 556 619 771 834 985 1048 1200
0 63 215 278 429 493 556 707 771 922 985 1137 1200
0 152 215 366 429 493 644 707 859 922 1074 1137 1200
0 63 215 278 341 493 556 707 771 922 985 1048 1200
0 152 215 278 429 493 644 707 859 922 985 1137 1200
0 63 126 278 341 493 556 707 771 834 985 1048 1200
0 63 215 278 429 493 644 707 771 922 985 1137 1200
0 152 215 366 429 581 644 707 859 922 1074 1137 1200
0 63 215 278 429 493 556 707 771 922 985 1048 1200
0 152 215 366 429 493 644 707 859 922 985 1137 1200
0 63 215 278 341 493 556 707 771 834 985 1048 1200
0 152 215 278 429 493 644 707 771 922 985 1137 1200

1171 = 27/23 278 (9), 21/17 366 (3), 28/23 341 (4), 32/25 427 (8)

(Our first taste of Neo-Gothic flavors!)

Scale 04: 5L,7s
Generator
613/461 493.344

0 67 133 280 347 493 560 627 773 840 987 1053 1200
0 67 213 280 427 493 560 707 773 920 987 1133 1200
0 147 213 360 427 493 640 707 853 920 1067 1133 1200
0 67 213 280 347 493 560 707 773 920 987 1053 1200
0 147 213 280 427 493 640 707 853 920 987 1133 1200
0 67 133 280 347 493 560 707 773 840 987 1053 1200
0 67 213 280 427 493 640 707 773 920 987 1133 1200
0 147 213 360 427 573 640 707 853 920 1067 1133 1200
0 67 213 280 427 493 560 707 773 920 987 1053 1200
0 147 213 360 427 493 640 707 853 920 987 1133 1200
0 67 213 280 347 493 560 707 773 840 987 1053 1200
0 147 213 280 427 493 640 707 773 920 987 1133 1200

613 = 20/17 281 (9), 16/13 359 (3), 11/9 347 (4), 32/25 427 (8)

Scale 05: 5L,7s
Generator
677/509 493.788

0 69 138 281 350 494 563 632 775 844 988 1057 1200
0 69 212 281 425 494 563 706 775 919 988 1131 1200
0 143 212 356 425 494 637 706 850 919 1062 1131 1200
0 69 212 281 350 494 563 706 775 919 988 1057 1200
0 143 212 281 425 494 637 706 850 919 988 1131 1200
0 69 138 281 350 494 563 706 775 844 988 1057 1200
0 69 212 281 425 494 637 706 775 919 988 1131 1200
0 143 212 356 425 568 637 706 850 919 1062 1131 1200
0 69 212 281 425 494 563 706 775 919 988 1057 1200
0 143 212 356 425 494 637 706 850 919 988 1131 1200
0 69 212 281 350 494 563 706 775 844 988 1057 1200
0 143 212 281 425 494 637 706 775 919 988 1131 1200

677 = 20/17 281 (9), 27/22 355 (3), 49/40 351 (4), 23/18 424 (8)

Scale 06: 5L,7s
Generator
281/211 495.993

0 80 160 288 368 496 576 656 784 864 992 1072 1200
0 80 208 288 416 496 576 704 784 912 992 1120 1200
0 128 208 336 416 496 624 704 832 912 1040 1120 1200
0 80 208 288 368 496 576 704 784 912 992 1072 1200
0 128 208 288 416 496 624 704 832 912 992 1120 1200
0 80 160 288 368 496 576 704 784 864 992 1072 1200
0 80 208 288 416 496 624 704 784 912 992 1120 1200
0 128 208 336 416 544 624 704 832 912 1040 1120 1200
0 80 208 288 416 496 576 704 784 912 992 1072 1200
0 128 208 336 416 496 624 704 832 912 992 1120 1200
0 80 208 288 368 496 576 704 784 864 992 1072 1200
0 128 208 288 416 496 624 704 784 912 992 1120 1200

281 = 17/14 336 (3), 21/17 366 (4), 26/21 370 (4), 14/11 418 (8)

(We're cooking up some Neo-Gothic flavors in the MOS kitchen now!)

Scale 07: 5L,7s
Generator
1151/863 498.546

0 93 185 296 388 499 591 684 794 887 997 1090 1200
0 93 203 296 406 499 591 701 794 904 997 1107 1200
0 110 203 313 406 499 609 701 812 904 1015 1107 1200
0 93 203 296 388 499 591 701 794 904 997 1090 1200
0 110 203 296 406 499 609 701 812 904 997 1107 1200
0 93 185 296 388 499 591 701 794 887 997 1090 1200
0 93 203 296 406 499 609 701 794 904 997 1107 1200
0 110 203 313 406 516 609 701 812 904 1015 1107 1200
0 93 203 296 406 499 591 701 794 904 997 1090 1200
0 110 203 313 406 499 609 701 812 904 997 1107 1200
0 93 203 296 388 499 591 701 794 887 997 1090 1200
0 110 203 296 406 499 609 701 794 904 997 1107 1200

1151 = 19/16 298 (9), 6/5 316 (3), 5/4 386 (4), 24/19 404 (8), 81/64
408 (8)

Scale 08: 5L,7s
Generator
463/347 499.292

0 96 193 298 394 499 596 692 797 894 999 1095 1200
0 96 201 298 403 499 596 701 797 902 999 1104 1200
0 105 201 306 403 499 604 701 806 902 1007 1104 1200
0 96 201 298 394 499 596 701 797 902 999 1095 1200
0 105 201 298 403 499 604 701 806 902 999 1104 1200
0 96 193 298 394 499 596 701 797 894 999 1095 1200
0 96 201 298 403 499 604 701 797 902 999 1104 1200
0 105 201 306 403 508 604 701 806 902 1007 1104 1200
0 96 201 298 403 499 596 701 797 902 999 1095 1200
0 105 201 306 403 499 604 701 806 902 999 1104 1200
0 96 201 298 394 499 596 701 797 894 999 1095 1200
0 105 201 298 403 499 604 701 797 902 999 1104 1200

463 = 19/16 298 (9), 24/19 404 (8)

Scale 09: 7L,5s
Generator
223/167 500.635

0 103 206 302 405 501 604 707 803 906 1001 1104 1200
0 103 199 302 397 501 604 699 803 898 1001 1097 1200
0 96 199 294 397 501 596 699 795 898 994 1097 1200
0 103 199 302 405 501 604 699 803 898 1001 1104 1200
0 96 199 302 397 501 596 699 795 898 1001 1097 1200
0 103 206 302 405 501 604 699 803 906 1001 1104 1200
0 103 199 302 397 501 596 699 803 898 1001 1097 1200
0 96 199 294 397 493 596 699 795 898 994 1097 1200
0 103 199 302 397 501 604 699 803 898 1001 1104 1200
0 96 199 294 397 501 596 699 795 898 1001 1097 1200
0 103 199 302 405 501 604 699 803 906 1001 1104 1200
0 96 199 302 397 501 596 699 803 898 1001 1097 1200

223 = 25/21 302 (9), 24/19 404 (4)

Notice that here we have passed the 5/12 barrier, and now our MOS
scales have 7L,5s steps. Here too, the fifths begin to narrow as the
fourths grows wider.

Scale 10: 7L,5s
Generator
947/709 501.095

0 105 211 303 409 501 607 712 804 910 1002 1108 1200
0 105 198 303 396 501 607 699 804 897 1002 1095 1200
0 92 198 290 396 501 593 699 791 897 989 1095 1200
0 105 198 303 409 501 607 699 804 897 1002 1108 1200
0 92 198 303 396 501 593 699 791 897 1002 1095 1200
0 105 211 303 409 501 607 699 804 910 1002 1108 1200
0 105 198 303 396 501 593 699 804 897 1002 1095 1200
0 92 198 290 396 488 593 699 791 897 989 1095 1200
0 105 198 303 396 501 607 699 804 897 1002 1108 1200
0 92 198 290 396 501 593 699 791 897 1002 1095 1200
0 105 198 303 409 501 607 699 804 910 1002 1108 1200
0 92 198 303 396 501 593 699 804 897 1002 1095 1200

947 = 25/21 302 (9), 19/15 409 (4)

Scale 11: 7L,5s
Generator
79/59 505.365

0 127 254 316 443 505 632 759 821 948 1011 1138 1200
0 127 189 316 379 505 632 695 821 884 1011 1073 1200
0 62 189 252 379 505 568 695 757 884 946 1073 1200
0 127 189 316 443 505 632 695 821 884 1011 1138 1200
0 62 189 316 379 505 568 695 757 884 1011 1073 1200
0 127 254 316 443 505 632 695 821 948 1011 1138 1200
0 127 189 316 379 505 568 695 821 884 1011 1073 1200
0 62 189 252 379 441 568 695 757 884 946 1073 1200
0 127 189 316 379 505 632 695 821 884 1011 1138 1200
0 62 189 252 379 505 568 695 757 884 1011 1073 1200
0 127 189 316 443 505 632 695 821 948 1011 1138 1200
0 62 189 316 379 505 568 695 821 884 1011 1073 1200

79 = 22/19 254 (5), 6/5 316 (9), 22/17 446 (4)

Scale 12: 7L,5s
Generator
859/641 506.801

0 134 268 320 454 507 641 775 827 961 1014 1148 1200
0 134 186 320 373 507 641 693 827 880 1014 1066 1200
0 52 186 239 373 507 559 693 746 880 932 1066 1200
0 134 186 320 454 507 641 693 827 880 1014 1148 1200
0 52 186 320 373 507 559 693 746 880 1014 1066 1200
0 134 268 320 454 507 641 693 827 961 1014 1148 1200
0 134 186 320 373 507 559 693 827 880 1014 1066 1200
0 52 186 239 373 425 559 693 746 880 932 1066 1200
0 134 186 320 373 507 641 693 827 880 1014 1148 1200
0 52 186 239 373 507 559 693 746 880 1014 1066 1200
0 134 186 320 454 507 641 693 827 961 1014 1148 1200
0 52 186 320 373 507 559 693 827 880 1014 1066 1200

859 = 26/21 370 (8), 13/10 454 (4), 23/18 424 (1)

Scale 13: 7L,5s
Generator
619/461 510.207

0 151 302 331 482 510 661 812 841 992 1020 1171 1200
0 151 180 331 359 510 661 690 841 869 1020 1049 1200
0 29 180 208 359 510 539 690 718 869 898 1049 1200
0 151 180 331 482 510 661 690 841 869 1020 1171 1200
0 29 180 331 359 510 539 690 718 869 1020 1049 1200
0 151 302 331 482 510 661 690 841 992 1020 1171 1200
0 151 180 331 359 510 539 690 841 869 1020 1049 1200
0 29 180 208 359 388 539 690 718 869 898 1049 1200
0 151 180 331 359 510 661 690 841 869 1020 1171 1200
0 29 180 208 359 510 539 690 718 869 1020 1049 1200
0 151 180 331 482 510 661 690 841 992 1020 1171 1200
0 29 180 331 359 510 539 690 841 869 1020 1049 1200

619 = 25/21 302 (2), 23/19 331 (9), 16/13 359 (8)

Scale 14: 7L,5s
Generator
953/709 512.029

0 160 320 336 496 512 672 832 848 1008 1024 1184 1200
0 160 176 336 352 512 672 688 848 864 1024 1040 1200
0 16 176 192 352 512 528 688 704 864 880 1040 1200
0 160 176 336 496 512 672 688 848 864 1024 1184 1200
0 16 176 336 352 512 528 688 704 864 1024 1040 1200
0 160 320 336 496 512 672 688 848 1008 1024 1184 1200
0 160 176 336 352 512 528 688 848 864 1024 1040 1200
0 16 176 192 352 368 528 688 704 864 880 1040 1200
0 160 176 336 352 512 672 688 848 864 1024 1184 1200
0 16 176 192 352 512 528 688 704 864 1024 1040 1200
0 160 176 336 496 512 672 688 848 1008 1024 1184 1200
0 16 176 336 352 512 528 688 848 864 1024 1040 1200

953 = 17/14 336 (9), 49/40 351 (8)

So we have seen a nearly 360 degree rotation of the MOS prism for a
range of rational prime 4th generators; revealing flavors of thirds
which could satisfy many RI/JI appetites.

Thanks,

Jacky Ligon

Notes:

1. This is the scale I was referring to when I spoke of tuning Joe
Monzo's work. It is a subset of the "108,000 Tone, Prime Series
Ratio, Non-Octave Scale". : )

2. Special thanks to Dan Stearns (for his patient tutelage), Margo
Schulter, Kraig Grady, Dave Keenan, Paul Erlich, Graham Breed and
Robert Walker for all your valuable posts and web pages, from which I
have learned greatly.

3. Steal This Scale: Feel free to explore any of these scales, and
please let me know what you think. So far I've only improvised on
them, and there are many interesting ones to play. Let there be
Moments of Symmetrical bliss throughout the land!

2001: A MOS Odyssey (Part 2)

🔗ligonj@northstate.net

🔗PERLICH@ACADIAN-ASSET.COM

🔗ligonj@northstate.net

🔗D.Stearns <STEARNS@CAPECOD.NET>