2001: A MOS Odyssey

2001 A MOS Odyssey

Inspired by the many and various "fifth generator" posts of late, I
would like to interject some of the treasures found among the Prime
Series, with regard to various widths of prime ratio fifths, which
generate scales capable of optimizing chosen harmonic intervals. In
these scales, thirds of 3-17 Limit are optimized:

Scales 1-10, constructed from Prime Series Ratios chains of fifths
(seen in order of generator widths):

#1

Generator Approximated Ratios
709/479 678.912
0
110.208
157.824
268.032 7/6 266.871
315.648 6/5 315.641
473.472
631.296
678.912
789.12
836.736
994.56
1152.384
1200

#2

Generator Approximated Ratios
877/587 695.060
0
65.417
190.119
255.536
380.238
445.656 22/17 446.363
570.358
695.06
760.477
885.179
950.596
1075.298
1200

#3

Generator Approximated Ratios
1087/727 696.390
0
74.727
192.779
267.506 7/6 266.871
385.558 5/4 386.314
460.286
578.338
696.39
771.117
889.169
963.896
1081.948
1200

#4

Generator Approximated Ratios
809/541 696.613
0
76.293
193.227
269.52 7/6 266.871
386.453 5/4 386.314
462.747
579.68
696.613
772.907
889.84
966.133
1083.067
1200

#5

Generator Approximated Ratios
271/181 698.764
0
91.346
197.528
288.874 13/11 289.210
395.055
486.402
592.583
698.764
790.11
896.291
987.638
1093.819
1200

#6

Generator Approximated Ratios
461/307 703.834
0
126.836
207.667
334.503 17/14 336.130
415.335 14/11 417.508
542.171
623.002
703.834
830.67
911.501
1038.337
1119.169
1200

#7 (for Margo Schulter)

Generator Approximated Ratios
359/239 704.368
0
130.575
208.736
339.311
417.471 14/11 417.508
548.047
626.207
704.368
834.943
913.104
1043.679
1121.839
1200

#8

Generator Approximated Ratios
269/179 705.176
0
136.231
210.352
346.583 11/9 347.408
420.704 14/11 417.508
556.935
631.055
705.176
841.407
915.528
1051.759
1125.88
1200

#9

Generator Approximated Ratios
191/127 706.493
0
145.451
212.986
358.437 16/13 359.472
425.972
571.423
638.958
706.493
851.944
919.479
1064.93
1132.465
1200

#10

Generator Approximated Ratios
631/419 708.828
0
161.794
217.655
379.449
435.311 9/7 435.084
597.105
652.966
708.828
870.622
926.483
1088.277
1144.139
1200

Feel free to try these out if you enjoy MOS. They are quite lovely
sounding actually.

Thanks,

Jacky Ligon

Hi Jacky,

The first one wouldn't be MOS as it's a three-stepsize 12-note set of
5 small 3 medium and 4 large stepsizes.

For a twelve note set to be MOS with a fifth generator the fifth must
be greater than 1:2^(4/7), less than 1:2^(3/5) and not equal to
1:2^(7/12).

The ones that fall between 4/7ths and 7/12ths of an octave will have 5
small and 7 large stepsizes. The ones that fall between 7/12ths and
3/5ths of an octave will have 7 small and 5 large steps.

--Dan Stearns

Hi Jacky,

Here is a little piece using the MIDI acoustic guitar and Cor Anglais patches in your
hypermos large prime scale 10.

http://members.nbci.com/tune_smithy/2001_a_MOS_Odyssy_scale10.mid

thanks

Robert

--- In tuning@egroups.com, "Robert Walker" <robert_walker@r...> wrote:
> Hi Jacky,
>
> Here is a little piece using the MIDI acoustic guitar and Cor
Anglais patches in your
> hypermos large prime scale 10.
>
> http://members.nbci.com/tune_smithy/2001_a_MOS_Odyssy_scale10.mid
>
> thanks
>
> Robert

Thanks so much for this Robert! It played wonderfully on my system,
and is a very lovely piece. What a powerful mood!

I've been told that you are a good melodist, and now I understand why.
Very cool!

Jacky Ligon

P.S. I'm playing this in CakeWalk 9, and I'm not sure if it's playing
at your intended tempo. What was the intended bpm, so I can be sure?

Hi Jacky,

> Thanks so much for this Robert! It played wonderfully on my system,
> and is a very lovely piece. What a powerful mood!

> I've been told that you are a good melodist, and now I understand why.
> Very cool!

Thanks.

It was very easy to write in your lovely scale. Just came like that,
straight away.

> P.S. I'm playing this in CakeWalk 9, and I'm not sure if it's playing
> at your intended tempo. What was the intended bpm, so I can be sure?

crotchet = 80 (smallest division heard = quaver).

Robert

Hi Jacky!

Though I'm sure you're having fun with your single-chain-of-fifths MOSs with
various rational fifths, inspired perhaps by

http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm,

I would urge you not to ignore the possibilities of double-chain-of-fifths
tunings and scales as in:

http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

You could approximate the half-octave with 99/70, or 1393/985, or a
convergent of intermediate complexity . . .

Also, in the scales you posted, it might be valuable to state how many times
each near-just interval occurs -- for example, in a 12-tone scale with
631/419 fifths, 9:7 occurs 8 times.

Hi Jacky,

I've done a gif of the score for

http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.mid

as
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_low_res.gif
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p1.gif
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p2.gif

Also, here is the nwc (NoteWorthy Composer) file:
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.nwc

The nwc file is meant to be played through score retuning software,
such as FTS, hooked to play button of NWC, e.g. via Hubi's loopback cable.

The lyric line for the Cor Anglais shows the scale to use.

Idea is, C is 1/1, C# is 161.794 cents, and so on.

I did it in the "diatonic mode" in your scale, with a few accidentals, which
happens to be a nice mode to play in, but isn't anything like a major scale:

I.e. degrees 0 2 4 5 7 9 11 12:

1/1 217.655 cents 435.311 cents 597.105 cents 708.828 cents 926.483 cents 1144.14 cents
2/1

9/7 = 435.084 cents

= neutral third, augmented fourth, quarter tone leading note.

I've labelled the clefs as "Acoustic guitar patch" and "Cor anglais patch",
as I don't expect them to be playable on the actual instruments particularly.

The Cor anglais patch goes well below the range of Cor Anglais (F - c'').

Robert

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Hi Jacky!
>
> Though I'm sure you're having fun with your single-chain-of-fifths
MOSs with
> various rational fifths, inspired perhaps by
>
> http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm,

Paul,

I did go and read this when you mentioned it recently, which did help
the focus of my paper. It is actually lots of fun indeed!!! : )

>
> I would urge you not to ignore the possibilities of double-chain-of-
fifths
> tunings and scales as in:
>
> http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm
>
> You could approximate the half-octave with 99/70, or 1393/985, or a
> convergent of intermediate complexity . . .

Thanks for this! Got it coming up now.

>
> Also, in the scales you posted, it might be valuable to state how
many times
> each near-just interval occurs -- for example, in a 12-tone scale
with
> 631/419 fifths, 9:7 occurs 8 times.

This is a great idea Paul! I've got the better part of "part 2" ready
to post, and I'll definitely take this into serious consideration.
Unison Vectors anyone?

Jacky Ligon

P.S. What are 12 tone scales with less than the 5s7L and 7L5s called,
which have only two step sizes and are generated by an interval
around 9/7? Ran into these whilst exploring. I know they aren't MOS,
but are interesting all the same. Any terms will help. I found some
of your old MOS posts today, but haven't absorbed yet.

Jacky wrote,

>P.S. What are 12 tone scales with less than the 5s7L and 7L5s called,

Less than the 5s7L and 7l5s? What does that mean?

>which have only two step sizes and are generated by an interval
>around 9/7? Ran into these whilst exploring. I know they aren't MOS,

Any scale with a single generator and only two step sizes is an MOS. Perhaps
you had something else in mind?

--- In tuning@egroups.com, "Robert Walker" <robert_walker@r...> wrote:
> Hi Jacky,
>
> I've done a gif of the score for
>
>
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.m
id
>
> as
>
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_l
ow_res.gif
>
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p
1.gif
>
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10_p
2.gif
>
> Also, here is the nwc (NoteWorthy Composer) file:
>
http://homepage.ntlworld.com/robertwalker/2001_a_MOS_Odyssey_scale10.n
wc

Robert!

This is fantastic! Thanks so much! The quality of the gifs is great.
You must have saved them at higher than 72ppi.

>
> I did it in the "diatonic mode" in your scale, with a few
accidentals, which
> happens to be a nice mode to play in, but isn't anything like a
major scale:
>
> I.e. degrees 0 2 4 5 7 9 11 12:
>
> 1/1 217.655 cents 435.311 cents 597.105 cents 708.828 cents 926.483
cents 1144.14 cents
> 2/1
>
> 9/7 = 435.084 cents
>
> = neutral third, augmented fourth, quarter tone leading note.
>

It's beautiful what you did with it. Seems that you found "the
pattern" quickly. There's an innate pattern in every scale, that is
only revealed by playing the scale. You tapped the essense.

I wonder how diffcult it might be to create a MOS animation, which
would be fed with a data table of hundreds fourth generators ranging
between 2/5 and 3/7 octave, whilst showing graphically, how as the
size of the generator grows or shrinks, how this affects the size of
scale degrees? Perhaps even treating the 2/5 to 3/7 as a 360 degree
rotation. It would be an interesting visualization, as one could see
how the roles of the scale degrees changes relative to the generator
size. I can see this in my mind!

Erv Wilson says "it passes through the looking glass". So cool!

Thanks,

Jacky Ligon

Paul H. Erlich wrote,

<<Any scale with a single generator and only two step sizes is an MOS.
Perhaps you had something else in mind?>>

With all the exploratory posts involving three-term series and indexes
I thought I'd repost this bit from the other list as when I say
"indexing", two-term series are really what it's all about, as they
are very well defined.

When I started using indexing it was because I wanted to answer the
following question: given any arbitrary scale constructed of "a"
amount of small steps and "b" amount of large steps, what range can an
interval safely occupy so as to result in scale of two stepsizes and
Myhill's property... ?

Jacky seems to be fond of twelve note sets (or at least tends to give
his examples as such), so I'll use those as an example.

These would be the possible twelve note two-stepsize indexes.

[1,11]
[2,10]
[3,9]
[4,8]
[5,7]
[6,6]
[7,5]
[8,4]
[9,3]
[10,2]
[11,1]

Here's the procedure:

1) Reduce "a" and "b" by their GCD.

2) Scale the periodicity accordingly. So for example I'll assume "P" =
1:2 (as is usually the case, but "P" can be anything you'd like in
this process, its complete fluidity is a built in feature), and I'll
use the first reducible index from the above twelve note examples;
[2,10]. This reduces to [1,5]. Now if the index is reducible than the
periodicity is as well, so "P" must be logarithmically scaled by the
GCD, which in this case was 2. So P = 1:2 becomes P = 1:2^(1/2).

3) Convert the index, i.e., [a,b], into adjacent fractions. (Adjacent
fractions are two fractions that differ by 1 when cross-multiplied.)

4) These adjacent fractions can then be used to seed a Stern-Brocot
Tree:

<http://206.4.57.253/editorial/knot/SB_tree.html>

Now x, y, and x+y (as depicted at the above link) define the borders
of what range an interval can safely occupy so as to result in scale
of two stepsizes and Myhill's property.

Lets look at the first twelve note index in the above example; [1,11].
Following the procedure I just outlined would give the following x, y,
and x+y:

0/1 1/11
1/12

Now any interval > 1:2^(0/1) and < 1:2^(1/12) results in a [1,11]
scale. And any interval > 1:2^(1/12) and < 1:2^(1/11) results in the
opposite scale index of [11,1].

Of course the closer you are to any of the borders the more trivial
and less meaningful the 'answer' is. In the Apical weighting scheme I
use, I set the 'default' equal scale as x+2y (as depicted at the above
Stern-Brocot Tree link).

This weighting is the mean of the Golden and Silver constant, and it
also sets up a useful condition when moving to theoretical n-term
scales. Because x+2y is essentially a tempered version of thefollowing
scaling condition:

Let stepsizes be small to large alphabetized variables so that
two-stepsize = [a,b], three-stepsize = [a,b,c] and so forth and so on
where the alphabetized variables are any whole numbers.

Then assign each variable an uppercase fixed size:

A = (LOG(2)-LOG(1))*(1200/LOG(2))
B = (LOG(3)-LOG(1))*(1200/LOG(2))
C = (LOG(4)-LOG(1))*(1200/LOG(2))

etc.

Let this be a fixed interval template. This allows for a convenient
and aurally agreeable distribution of stepsizes amongst any [a,b,...]
index of:

a/b = log(3)/log(2)
c/b = log (4)/log(3)
d/c = log(5)/log(4)

etc.

Next you scale a given [a,b,...] index by a given periodicity (P).
This will give us a percentage (X) to scale the fixed A,B,... interval
template with.

So for a two-stepsize index you'd have:

(a*A)+(b*B)/P = X

For a three-stepsize index:

(a*A)+(b*B)+(c*C)/P = X

And so forth and so on...

Now A/X, B/X,... will give a given [a,b,...] index corresponding
stepsizes scaled to P.

So as I said before, x+2y in a two-term index is essentially a
tempered version of this. It would seem to follow suit that this would
be the case in other n-term scales as well... though no known parallel
condition to adjacent fractions exists that I'm aware of here, so
putting all the pieces together has proved very difficult. But for the
two-term indexes everything works exceptionally well, and may provide
a different inroad to the world of MOS scales.

--Dan Stearns

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Jacky wrote,
>
> >P.S. What are 12 tone scales with less than the 5s7L and 7L5s
called,
>
> Less than the 5s7L and 7l5s? What does that mean?

Oops! Meant to say 5s7L and 7s5L.

>
> >which have only two step sizes and are generated by an interval
> >around 9/7? Ran into these whilst exploring. I know they aren't
MOS,

This is in error. Disregard.

>
> Any scale with a single generator and only two step sizes is an
MOS. Perhaps
> you had something else in mind?

Thanks for the clarification. I can't seem to find the scale right
now. Was playing with this during having a raging fever this weekend.
Onward!

Jacky

I wrote,

<<it also sets up a useful condition when moving to theoretical n-term
scales. Because x+2y is essentially a tempered version of the
following scaling condition:>>

That's incorrect -- 2x+3y would be the tempered version of that
scaling condition. And though useful for n-term extrapolations, this
is not related to the mean of the Golden and Silver constant in the
manner I was pointing at, so please scratch that end part.

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
>
> With all the exploratory posts involving three-term series and
indexes
> I thought I'd repost this bit from the other list as when I say
> "indexing", two-term series are really what it's all about, as they
> are very well defined.
>

I want to thank Dan for his valuable post here. I think once I have
totally grasped a working knowledge of the indexes, the code of MOS
will be fully revealed.

For a friend who is working through these MOS posts, I'd like to also
include one of my cross posts from the New JI list:

MOS scales (Moments of Symmetry), can be generally "generated" by
stacking chains of intervals, then reducing them to fit within the
2/1. In this case it was with chains of exotic high prime ratio
fifths.

Many times when theorists seek to use fifths such as some of the ones
in my post, and intend to optimize the thirds in this manner, they
will sometimes choose more mathematical means, such as "weighting"
intervals with constants (an irrational approach). Please note that I
actively use both.

The extremely interesting thing for me, is that we can also
use "rational" means to achieve exactly the same thing. The Prime
Series Ratios provides us many such valuable things, and stands as a
mostly unexplored area of JI/RI.

Here's how #6 is generated:
#6

Generator Approximated Ratios
461/307 703.834
0
126.836
207.667
334.503 17/14 336.130
415.335 14/11 417.508
542.171
623.002
703.834
830.67
911.501
1038.337
1119.169
1200

If we take the ratio 461/307 @ 703.834 cents, and we
repeatedly "chain" it 11 times, we get the below cents values:

0
703.834
1407.667
2111.501
2815.335
3519.169
4223.002
4926.836
5630.67
6334.503
7038.337
7742.171

Then when we reduce this within the 2/1, we have:

0
703.834
207.667
911.501
415.335
1119.169
623.002
126.836
830.67
334.503
1038.337
542.171

Next we sort and add the 2/1:

0
126.836
207.667
334.503
415.335
542.171
623.002
703.834
830.67
911.501
1038.337
1119.169
1200

This is one of my favorites of this set, for the fact that it gives
you good 11 and 17 thirds:

334.503 17/14 336.130
415.335 14/11 417.508

A wonderful property!

Thanks,

Jacky Ligon

Jacky Ligon wrote,

<<Many times when theorists seek to use fifths such as some of the
ones in my post, and intend to optimize the thirds in this manner,
they will sometimes choose more mathematical means, such as
"weighting" intervals with constants (an irrational approach).>>

I used the weighted fifth tuning of Kornerup's as a model for
generalizing two-term indexes.

When a two-term index is converted into an adjacent fraction and the
adjacent fractions are taken as a Fibonacci series, you can see the
generators (the numerators) working their way towards a Phi-weighting.
So there's a built-in logic, or internal consistency in this... and
using this framework it's easy to see that the 3rd term of a Fibonacci
series is the "m" and the 4th term is the "n" in an m-out-of-n set,
and that the 1st and 2nd terms are the "small" and "large" steps
respectively.

Here's the single generator, Phi-weighted, twelve note two-term sets:

[1,11]

0/1, 1/11, 1/12, 2/23, ...

0 103 207 310 413 516 620 723 826 930 1033 1136 1200
0 103 207 310 413 516 620 723 826 930 1033 1097 1200
0 103 207 310 413 516 620 723 826 930 993 1097 1200
0 103 207 310 413 516 620 723 826 890 993 1097 1200
0 103 207 310 413 516 620 723 787 890 993 1097 1200
0 103 207 310 413 516 620 684 787 890 993 1097 1200
0 103 207 310 413 516 580 684 787 890 993 1097 1200
0 103 207 310 413 477 580 684 787 890 993 1097 1200
0 103 207 310 374 477 580 684 787 890 993 1097 1200
0 103 207 270 374 477 580 684 787 890 993 1097 1200
0 103 167 270 374 477 580 684 787 890 993 1097 1200
0 64 167 270 374 477 580 684 787 890 993 1097 1200

[2,8]

0/1, 1/5, 1/6, 2/11, ...

P = 1:2^(1/2)

0 107 214 320 427 534 600 707 814 920 1027 1134 1200
0 107 214 320 427 493 600 707 814 920 1027 1093 1200
0 107 214 320 386 493 600 707 814 920 986 1093 1200
0 107 214 280 386 493 600 707 814 880 986 1093 1200
0 107 173 280 386 493 600 707 773 880 986 1093 1200
0 66 173 280 386 493 600 666 773 880 986 1093 1200

[3,9]

0/1, 1/3, 1/4, 2/7, ...

P = 1:2^(1/3)

0 111 221 332 400 511 621 732 800 911 1021 1132 1200
0 111 221 289 400 511 621 689 800 911 1021 1089 1200
0 111 179 289 400 511 579 689 800 911 979 1089 1200
0 68 179 289 400 468 579 689 800 868 979 1089 1200

[4,8]

0/1, 1/2, 1/3, 2/5, ...

P = 1:2^(1/4)

0 115 229 300 415 529 600 715 829 900 1015 1129 1200
0 115 185 300 415 485 600 715 785 900 1015 1085 1200
0 71 185 300 371 485 600 671 785 900 971 1085 1200

[5,7]

3/5, 4/7, 7/12, 11/19, ...

0 74 192 266 385 458 577 696 770 889 962 1081 1200
0 119 192 311 385 504 623 696 815 889 1008 1126 1200
0 74 192 266 385 504 577 696 770 889 1008 1081 1200
0 119 192 311 430 504 623 696 815 934 1008 1126 1200
0 74 192 311 385 504 577 696 815 889 1008 1081 1200
0 119 238 311 430 504 623 742 815 934 1008 1126 1200
0 119 192 311 385 504 623 696 815 889 1008 1081 1200
0 74 192 266 385 504 577 696 770 889 962 1081 1200
0 119 192 311 430 504 623 696 815 889 1008 1126 1200
0 74 192 311 385 504 577 696 770 889 1008 1081 1200
0 119 238 311 430 504 623 696 815 934 1008 1126 1200
0 119 192 311 385 504 577 696 815 889 1008 1081 1200

[6,6]

P = 1:2^(1/6)

0/1, 1/1, 1/2, 2/3, ...

0 124 200 324 400 524 600 724 800 924 1000 1124 1200
0 76 200 276 400 476 600 676 800 876 1000 1076 1200

[7,5]

4/7, 3/5, 7/12, 10/17, ...

0 129 208 337 416 545 625 704 833 912 1041 1120 1200
0 80 208 288 416 496 575 704 784 912 992 1071 1200
0 129 208 337 416 496 625 704 833 912 992 1120 1200
0 80 208 288 367 496 575 704 784 863 992 1071 1200
0 129 208 288 416 496 625 704 784 912 992 1120 1200
0 80 159 288 367 496 575 655 784 863 992 1071 1200
0 80 208 288 416 496 575 704 784 912 992 1120 1200
0 129 208 337 416 496 625 704 833 912 1041 1120 1200
0 80 208 288 367 496 575 704 784 912 992 1071 1200
0 129 208 288 416 496 625 704 833 912 992 1120 1200
0 80 159 288 367 496 575 704 784 863 992 1071 1200
0 80 208 288 416 496 625 704 784 912 992 1120 1200

[8,4]

1/2, 0/1, 1/3, 1/4, ...

P = 1:2^(1/4)

0 83 166 300 383 466 600 683 766 900 983 1066 1200
0 83 217 300 383 517 600 683 817 900 983 1117 1200
0 134 217 300 434 517 600 734 817 900 1034 1117 1200

[9,3]

1/3, 0/1, 1/4, 1/5, ...

P = 1:2^(1/3)

0 87 173 260 400 487 573 660 800 887 973 1060 1200
0 87 173 313 400 487 573 713 800 887 973 1113 1200
0 87 227 313 400 487 627 713 800 887 1027 1113 1200
0 140 227 313 400 540 627 713 800 940 1027 1113 1200

[10,2]

1/5, 0/1, 1/6, 1/7, ...

P = 1:2^(1/2)

0 91 181 272 363 453 600 691 781 872 963 1053 1200
0 91 181 272 363 509 600 691 781 872 963 1109 1200
0 91 181 272 419 509 600 691 781 872 1019 1109 1200
0 91 181 328 419 509 600 691 781 928 1019 1109 1200
0 91 237 328 419 509 600 691 837 928 1019 1109 1200
0 147 237 328 419 509 600 747 837 928 1019 1109 1200

[11,1]

1/11, 0/1, 1/12, 1/13, ...

0 95 190 285 380 476 571 666 761 856 951 1046 1200
0 95 190 285 380 476 571 666 761 856 951 1105 1200
0 95 190 285 380 476 571 666 761 856 1010 1105 1200
0 95 190 285 380 476 571 666 761 915 1010 1105 1200
0 95 190 285 380 476 571 666 820 915 1010 1105 1200
0 95 190 285 380 476 571 724 820 915 1010 1105 1200
0 95 190 285 380 476 629 724 820 915 1010 1105 1200
0 95 190 285 380 534 629 724 820 915 1010 1105 1200
0 95 190 285 439 534 629 724 820 915 1010 1105 1200
0 95 190 344 439 534 629 724 820 915 1010 1105 1200
0 95 249 344 439 534 629 724 820 915 1010 1105 1200
0 154 249 344 439 534 629 724 820 915 1010 1105 1200

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

Dan,

It's interesting how the weighted fifth also creates some interesting
harmonic alignments!

Thanks for this second post. I was studying your last one this
evening, and this will help me to grasp it better. So each line/scale
is generated by the next weighted adjacent fraction?

Thanks,

Jacky Ligon

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
>
> [2,8]
>
> 0/1, 1/5, 1/6, 2/11, ...
>
> P = 1:2^(1/2)
>
> 0 107 214 320 427 534 600 707 814 920 1027 1134 1200
> 0 107 214 320 427 493 600 707 814 920 1027 1093 1200
> 0 107 214 320 386 493 600 707 814 920 986 1093 1200
> 0 107 214 280 386 493 600 707 814 880 986 1093 1200
> 0 107 173 280 386 493 600 707 773 880 986 1093 1200
> 0 66 173 280 386 493 600 666 773 880 986 1093 1200
>

Dan,

Should not this [2,8] index reduce by its GDC and result in [1,4]?

Forgive me if I've missed something here.

Jacky Ligon

Jacky Ligon wrote,

<<Should not this [2,8] index reduce by its GDC and result in [1,4]?>>

As they're all supposed to be twelve note indexes, that should've read
[2,10] and not [2,8]. Just a typo, the rest of the example is correct.
Thanks for catching it.

--Dan Stearns

--- In tuning@egroups.com, ligonj@n... wrote:
> --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> >
> > [2,8]
> >
> > 0/1, 1/5, 1/6, 2/11, ...
> >
> > P = 1:2^(1/2)
> >
> > 0 107 214 320 427 534 600 707 814 920 1027 1134 1200
> > 0 107 214 320 427 493 600 707 814 920 1027 1093 1200
> > 0 107 214 320 386 493 600 707 814 920 986 1093 1200
> > 0 107 214 280 386 493 600 707 814 880 986 1093 1200
> > 0 107 173 280 386 493 600 707 773 880 986 1093 1200
> > 0 66 173 280 386 493 600 666 773 880 986 1093 1200
> >

Dan,

I just happened to notice that the last scale here contains what was
know in ancient times as "The Evil Fifth". This evil generator of
2086812/1419857 @ 666.666 will yield a 9s1L 9 Tone MOS. I've been too
afraid to compose with this though, since "Evil" is not my bag.
Perhaps I'll pull it out around Halloween.

666.666 The Evil 5th
0: 1/1 0.000 unison, perfect prime
1: 133.332 cents 133.332
2: 266.664 cents 266.664
3: 399.996 cents 399.996
4: 533.328 cents 533.328
5: 2086812/1419857 666.666
6: 799.998 cents 799.998
7: 933.330 cents 933.330
8: 1066.662 cents 1066.662
9: 1199.994 cents 1199.994

0: 133.332
1: 133.332
2: 133.332
3: 133.332
4: 133.332
5: 133.338
6: 133.332
7: 133.332
8: 133.332
9: 133.332

Number of notes : 9
Smallest interval : 133.332 cents
Average interval (divided octave) : 133.333 cents
Average / Smallest interval : 1.000005
Largest interval of one step : 133.338 cents
Largest / Average interval : 1.000040
Largest / Smallest interval : 1.000045
Least squares average interval : 133.3327 cents
Median interval of one step : 133.332 cents
Most common interval of one step : 133.332 cents, amount: 8
Interval standard deviation : 0.0019 cents
Interval skew : 0.0000 cents
Scale is strictly proper
Scale has Myhill's property
generators: 1 of 133.3320 cents and 8 of 1066.6620 cents
Scale is maximal even
Scale is a mode of a 9-tone equal temperament
Scale contains two identical pentachords
Scale is a Constant Structure
Number of different intervals : 16 = 2.00000 / class
Smallest interval difference : 0.006 cents
Most common intervals : 133.332 cents & inv., amount: 8
Number of recognisable fifths : 0
Scale is a chain of 4 triads 0.0 533.328 1066.662 cents
Most common triad is 0.0 133.332 266.664 cents, amount: 9
Rothenberg stability : 1.000000 = 1
Lumma stability : 0.999960
Rothenberg efficiency : 0.740741 redundancy :
0.259259
Prime limit : 17 (not all pitches rational)
Limited transpositions :
1 2 3 4 5 6 7 8
Inversional symmetry on degrees :
0 1 2 3 4 5 6 7 8
Inversional symmetry on intervals :
0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9

}: )

Jacky Ligon

Dan,

What if "P" is irrational?

How would one deal with an irrational non-octave "P"?

Perhaps a slightly stretched of compressed octave as "P"?

I'm uncertain how to apply your formula to this possibility.

Thanks,

Jacky

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
>
> When a two-term index is converted into an adjacent fraction and the
> adjacent fractions are taken as a Fibonacci series, you can see the
> generators (the numerators) working their way towards a Phi-
weighting.
> So there's a built-in logic, or internal consistency in this... and
> using this framework it's easy to see that the 3rd term of a
Fibonacci
> series is the "m" and the 4th term is the "n" in an m-out-of-n set,
> and that the 1st and 2nd terms are the "small" and "large" steps
> respectively.
>
> Here's the single generator, Phi-weighted, twelve note two-term
sets:
>
> [2,10]
>
> 0/1, 1/5, 1/6, 2/11, ...
>
>
> [5,7]
>
> 3/5, 4/7, 7/12, 11/19, ...
>
> 0 74 192 266 385 458 577 696 770 889 962 1081 1200
> 0 119 192 311 385 504 623 696 815 889 1008 1126 1200
> 0 74 192 266 385 504 577 696 770 889 1008 1081 1200
> 0 119 192 311 430 504 623 696 815 934 1008 1126 1200
> 0 74 192 311 385 504 577 696 815 889 1008 1081 1200
> 0 119 238 311 430 504 623 742 815 934 1008 1126 1200
> 0 119 192 311 385 504 623 696 815 889 1008 1081 1200
> 0 74 192 266 385 504 577 696 770 889 962 1081 1200
> 0 119 192 311 430 504 623 696 815 889 1008 1126 1200
> 0 74 192 311 385 504 577 696 770 889 1008 1081 1200
> 0 119 238 311 430 504 623 696 815 934 1008 1126 1200
> 0 119 192 311 385 504 577 696 815 889 1008 1081 1200
>

Dan,

Now I am truly lost in space! I thought the [5,7] index should be
2/5, 3/7, 5/12.

Right-Wrong?

I can follow your logic well up to the point of converting the
indexes for [5,7] and [7,5]. Could you please explain in slow motion
again how to convert these indexes into adjacent fractions? Even
though I think I know the correct answers to be:

[5,7] 2/5, 3/7, 5/12
[7,5] 4/7, 3/5, 7/12

I still think I'm getting confused by something here.

Thanks for your continued assistance,

Jacky Ligon

I can feel that Dan.

Jacky Ligon wrote,

<<I thought the [5,7] index should be 2/5, 3/7,>>

As they're both adjacent fractions it doesn't really matter. One would
just be the compliment of the other.

A generator (or adjacent fraction) here always has a analogous '5th'
and '4th', as in a circle (or spiral/whatever) of fifths. I think this
(circle of fifths/fourths) is a handy mindset to adopt when looking at
this so long as one remembers that this is just a mental reference
point, and that the generators are whatever they are... so in other
words, they may or may not be fifths and fourths but the whole
generalization is constructed in a like manner.

--Dan Stearns

Jacky Ligon wrote,

<<What if "P" is irrational? How would one deal with an irrational
non-octave "P"? Perhaps a slightly stretched of compressed octave as
"P"?>>

Yes, the value of "P" is completely fluid in the algorithm so that you
can make it anything you'd like.

So

X = P/((a+W*b))*(A+W*B)

where

"P" = a given periodicity

"W" = a given weight

"A"/"a", "B"/"b" = the two adjacent fractions of a given [a,b] index

and "X" = the resulting weighted generator

Now say you wanted an arbitrary scale of 3 small steps and 5 large
steps where your periodicity is 1225� and your large step is exactly
two of your small steps...

1225/((3+2*5))*(1+2*2) = X

So your generator, "X", would be ~471.154�. And the resulting rotation
of scales would be:

0 188 377 471 660 848 942 1131 1225
0 188 283 471 660 754 942 1037 1225
0 94 283 471 565 754 848 1037 1225
0 188 377 471 660 754 942 1131 1225
0 188 283 471 565 754 942 1037 1225
0 94 283 377 565 754 848 1037 1225
0 188 283 471 660 754 942 1131 1225
0 94 283 471 565 754 942 1037 1225

Let me know if that helps.

thanks,

--Dan Stearns

Jacky Ligon wrote,

<<This evil generator of 2086812/1419857 @ 666.666 will yield a 9s1L
9 Tone MOS.>>

Hi Jacky,

If you used the 9th multiple of 2086812/1419857 for your octave you'd
have an [8,1] scale, and if you used the 2/1 you'd have a [7,2] scale.
But in either case you have nine equal no matter how you look at it,
and the difference in stepsizes is completely trivial in any ears on
fashion.

<<I've been too afraid to compose with this though, since "Evil" is
not my bag. Perhaps I'll pull it out around Halloween.>>

Now, now... 9-tET is a "nice" tuning: See no evil, hear no evil!

A while back Enrique Ubieta and "bimodalism" came up. I did a lot of
microtonal experimentation with his "bimodal triad" (a 1-b3-3-5 four
note chord that combines the minor and major triad). And one of the
things I determined was that 9 equal, or preferably some optimal
tweaking thereof, was most likely the theoretical best tuning to
accomplish Ubieta's bimodal theory; which would harmonize all scale
degrees to the bimodal triad.

And speaking of Enrique, he sent me three of his bimodal compositions
just the other day, two guitar pieces and his "Sui Generis Suite" for
harpsichord. "Bimodal" (one of the guitar pieces) seems to me like it
really should be a repertory type piece, analogous to say Brouwer's
guitar pieces... it's a very fine piece indeed.

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> Jacky Ligon wrote,
>
> <<This evil generator of 2086812/1419857 @ 666.666 will yield a
9s1L
> 9 Tone MOS.>>
>
> Hi Jacky,
>
> If you used the 9th multiple of 2086812/1419857 for your octave
you'd
> have an [8,1] scale, and if you used the 2/1 you'd have a [7,2]
scale.
> But in either case you have nine equal no matter how you look at it,
> and the difference in stepsizes is completely trivial in any ears on
> fashion.

This is so interesting! I didn't even recognize that it was 9ET. I
did notice the approximated 7/6 was a member of the "Evil Scale".

>
>
> <<I've been too afraid to compose with this though, since "Evil" is
> not my bag. Perhaps I'll pull it out around Halloween.>>
>
> Now, now... 9-tET is a "nice" tuning: See no evil, hear no evil!

He, he! Very funny!

>
> A while back Enrique Ubieta and "bimodalism" came up. I did a lot of
> microtonal experimentation with his "bimodal triad" (a 1-b3-3-5 four
> note chord that combines the minor and major triad). And one of the
> things I determined was that 9 equal, or preferably some optimal
> tweaking thereof, was most likely the theoretical best tuning to
> accomplish Ubieta's bimodal theory; which would harmonize all scale
> degrees to the bimodal triad.

Would it be ok to show an example of one of his scales? BTW, I've
always been hugely interested in bi-tonal chords, which were an
important part of my past 12 Tone keyboard technique.

A funny story: I went to audition for a "Prog-Rock" band back in the
90s. My main interest in them was they played in other meters than 4.
After the practice, the band leader (a fabulously talented guitarist)
and I had a little jam session when everyone else was gone. He ask
for me to begin an improvisation based on my ideas (we'd focused on
his entirely until this point), so I improvised with one hand in a
certain tonality, then the other transposed down by a tritone, and a
different third. He went "Wait a minute! What was that?!!!" He got me
to play the pattern slowly - and then he totally nailed the mode!
Then it was on! It was great to have someone relate to this - and in
Prog Rock this should be no problem. It was great fun, and I think
our private jam was the best thing of the evening. They liked what I
played, but I didn't accept the gig because of stylistic differences,
and they were too far away to be practical.

Of course Ornette had taken all this to levels inconceived of. He
*IS* the multi-tonal man. Why did Jamaladeen Tacuma just pop into my
mind?

>
> And speaking of Enrique, he sent me three of his bimodal
compositions
> just the other day, two guitar pieces and his "Sui Generis Suite"
for
> harpsichord. "Bimodal" (one of the guitar pieces) seems to me like
it
> really should be a repertory type piece, analogous to say Brouwer's
> guitar pieces... it's a very fine piece indeed.

Well (no, not a Zappa "Well"), I got to tell you that I love Brouwer,
and this is a great recommendation. Is this a commercial release? Any
chance to hear this?

Well maybe Evil ain't so bad after all!

: )

Jacky Ligon

<<Would it be ok to show an example of one of his scales>>

Here's some links:

<http://www.ubieta.com/bio.htm>
<http://www.ubieta.com/bimodalism/BimodalHarmony.htm>

It is my impression that Ubieta himself operates completely within the
usual twelve-tone equal tempered way of going about things... and when
he brings up the harmonic series (etc.) and any sort of teleology he
appears to only mean it in a purely conjectural manner; "the truth
from the facts inclines us to think so" is what he once wrote me...

The scales that I last posted about this were 14 and 23-tET temperings
of an 18:21:23 identity where if you were to lattice out the 18:21:23
scale in these temperaments you'd have a situation somewhat analogous
to the diatonic scale where the comma -- a 7889/7776 in the following
14-tET example -- is absorbed by the temperament and an additional
consonant chord results. In this case an additional "bimodal chord".

These peculiar diatonic-like extrapolations came about by taking the
first two terms of the bimodal chord -- i.e., the minor and major
thirds -- as an identity.

I then came up with a little template to generate 9-tone scales where
each scale degree is connected to a centralized tonic:

b-a
|
|
|
|
|
ai | b
\ | /
\ | /
\ | /
\ | /
\|/
(a+b)i----------t---------a+b
/|\
/ | \
/ | \
/ | \
/ | \
bi | a
|
|
|
|
|
(b-a)i

where

"a" and "b" simply represent a minor and major third respectively
(which are generally culled from an identity in the form of t:a:b)

"i" indicates an inversion, or 'inverted'

and "t" is any given pitch or tonic

So with that in mind, here's the scale that results when t:a:b =
18:21:23

23/21
/|\
/ | \
/ | \
/ | \
/ | \
12/7----+---23/18
/|\ | /|\
/ | \ | / | \
/ | \ | / | \
/ | \ | / | \
/ | \|/ | \
216/161--+----1/1----+--161/108
\ | /|\ | /
\ | / | \ | /
\ | / | \ | /
\ | / | \ | /
\|/ | \|/
36/23---+----7/6
\ | /
\ | /
\ | /
\ | /
\|/
42/23

And here's the 14-tET example with the additional consonant chord:

8-----------2
/|\ /|\
/ | \ / | \
/ | \ / | \
/ | \ / | \
/ | \ / | \
3-----+----11-----+-----5
\ | /|\ | /|\
\ | / | \ | / | \
\ | / | \ | / | \
\ | / | \ | / | \
\|/ | \|/ | \
6-----+-----0-----+-----8
\ | /|\ | /|\
\ | / | \ | / | \
\ | / | \ | / | \
\ | / | \ | / | \
\|/ | \|/ | \
9-----+-----3-----+----11
\ | / \ | /
\ | / \ | /
\ | / \ | /
\ | / \ | /
\|/ \|/
12-----------6

In 14-tET this is a proper scale. A proper scale has instances of
shared intervals amongst interval classes, and in this case the shared
(or "ambiguous") intervals are the augmented and diminished 3rds and
4ths, the 5ths and
6ths, and the 7ths and 8ths:

0 171 257 429 514 686 771 943 1029 1200
0 86 257 343 514 600 771 857 1029 1200
0 171 257 429 514 686 771 943 1114 1200
0 86 257 343 514 600 771 943 1029 1200
0 171 257 429 514 686 857 943 1114 1200
0 86 257 343 514 686 771 943 1029 1200
0 171 257 429 600 686 857 943 1114 1200
0 86 257 429 514 686 771 943 1029 1200
0 171 343 429 600 686 857 943 1114 1200

23-tET, like 14, is consistent (that is if you even allow that the
full bimodal chord can be rightfully seen in such a light here), and
thereby allows all instances of the identity's cross-set or interval
matrix to always be the best rounded (LOG(N)-LOG(D))*(T/LOG(2))
representations (were "N" and "D" are the numerator and denominator of
the relevant consonant ratios, and "T" is the temperament).

However, unlike 14-tET, 23-tET is strictly proper, and this allows for
uniquely articulated representations of all interval classes. 23-tET
recognizes the diaschisma like 285768/279841 while still hiding the
syntonic comma like 7889/7776:

I. 0 157 261 417 522 678 783 939 1043 1200
II. 0 104 261 365 522 626 783 887 1043 1200
III. 0 157 261 417 522 678 783 939 1096 1200
IV. 0 104 261 365 522 626 783 939 1043 1200
V. 0 157 261 417 522 678 835 939 1096 1200
VI. 0 104 261 365 522 678 783 939 1043 1200
VII. 0 157 261 417 574 678 835 939 1096 1200
VIII. 0 104 261 417 522 678 783 939 1043 1200
IX. 0 157 313 417 574 678 835 939 1096 1200
X. 0 157 261 417 522 678 783 939 1043 1200

Anyway, these were some of the types of ideas that I was looking at
and experimenting with. But as I said in the previous post I
eventually decided that a simple 9 equal, or preferably some optimally
tweaked variations thereof, would seem to be the scale that is most
theoretically in step with Ubieta's bimodalism.

--Dan Stearns

--- In tuning@y..., ligonj@n... wrote:

/tuning/topicId_17744.html#17939

> Well maybe Evil ain't so bad after all!

Who put out that incredible blues album called "Live Evil"?
Was it Howlin' Wolf?

-monz

MONZ@JUNO.COM wrote:
>
>
> --- In tuning@y..., ligonj@n... wrote:
>
> /tuning/topicId_17744.html#17939
>
> > Well maybe Evil ain't so bad after all!
>
> Who put out that incredible blues album called "Live Evil"?
> Was it Howlin' Wolf?

Miles Davis.

Although there were a couple of live
Howlin Wolf albums, I don't think there were any called Live Evil.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm
* http://mp3.com/davidbeardsley