Re: Towards a hyper MOS

I've been mulling it over, wondering if one needs two generators.
After all, the 5-limit j.i. scale is made using two generators
5/4 and 3/2

(Hope this shows the returns okay - sending it as html)

See diagram of 5-limit j.i. scale by Monz:

http://www.egroups.com/message/tuning/12963

diatonic = 1st 2 rows.

So a natural thought is to continue the two line lattice round the octave.

Start with 1/1 5/4 3/2 2/1

= 1/1 5/4 6/5 4/3

= M S L

Expand as steps
1/1 5/4 6/5 5/4 6/5,...

Stop whenever there are three interval sizes:

4 notes =

5/4 6/5 5/4 16/15

L - M - L S

Substitution:

M -> L

S -> M

L -> L S

1: 1 16/15 111.731 cents minor diatonic semitone
1: 1 6/5 315.641 cents minor third
1: 2 5/4 386.314 cents major third
2: 2 4/3 498.045 cents perfect fourth
2: 2 3/2 701.955 cents perfect fifth
3: 2 8/5 813.686 cents minor sixth
3: 1 5/3 884.359 cents major sixth, BP sixth
3: 1 15/8 1088.269 cents classic major seventh
So that one is okay.

7 notes works, as we know.

This way of doing it makes

1/1 9/8 5/4 45/32 3/2 27/16 15/8 2/1
=
9/8 10/9 9/8 16/15 -- 9/8 10/9 16/15
(-- = point where usual j.i. diatonic scale starts)

L M - L S - L M - S
Substitution from previous scale
L -> L M
M -> L S
S ->S (unchanged)

So after 17 repetitions of 1/1 5/4 3/2 and reducing into octave one gets the scale:
1/1 135/128 2187/2048 9/8 1215/1024 5/4 81/64 45/32 729/512 3/2 405/256 27/16 3645/2048 15/8 243/128 2/1
steps
135/128 81/80 256/243 135/128 256/243 81/80 135/128 256/243 81/80 256/243 135/128 81/80 256/243 135/128 256/243 81/80
256/243

L S M - L M - S L M - S M - L S M - L M - S M

Substitution from previous scale
L -> L S M, S L M, or L M S
M -> L M
S -> S M

Checking in SCALA, this scale does indeed have three interval sizes for each degree!

Interval class, Number of incidences, Size:
1: 5 81/80 21.506 cents syntonic comma, Didymus comma
1: 7 256/243 90.225 cents Pythagorean limma
1: 5 135/128 92.179 cents major limma, large chroma
2: 7 16/15 111.731 cents minor diatonic semitone
2: 3 2187/2048 113.685 cents apotome
2: 7 10/9 182.404 cents minor whole tone
3: 2 4096/3645 201.956 cents
3: 13 9/8 203.910 cents major whole tone
3: 2 2560/2187 272.629 cents
4: 3 729/640 225.416 cents
4: 11 32/27 294.135 cents Pythagorean minor third
4: 3 1215/1024 296.089 cents
5: 8 6/5 315.641 cents minor third
5: 1 8192/6561 384.360 cents Pythagorean diminished fourth
5: 8 5/4 386.314 cents major third
6: 4 512/405 405.866 cents narrow diminished fourth
6: 9 81/64 407.820 cents Pythagorean major third
6: 4 320/243 476.539 cents grave fourth
7: 1 6561/5120 429.326 cents
7: 15 4/3 498.045 cents perfect fourth
7: 1 10935/8192 499.999 cents fourth + schisma, 5-limit approximation to ET fourth
8: 6 27/20 519.551 cents acute fourth
8: 5 1024/729 588.270 cents Pythagorean diminished fifth
8: 6 45/32 590.224 cents tritone
9: 6 64/45 609.776 cents 2nd tritone
9: 5 729/512 611.730 cents Pythagorean tritone
9: 6 40/27 680.449 cents grave fifth
10: 1 16384/10935 700.001 cents fifth - schisma, 5-limit approximation to ET fifth
10: 15 3/2 701.955 cents perfect fifth
10: 1 10240/6561 770.674 cents
11: 4 243/160 723.461 cents acute fifth
11: 9 128/81 792.180 cents Pythagorean minor sixth
11: 4 405/256 794.134 cents wide augmented fifth
12: 8 8/5 813.686 cents minor sixth
12: 1 6561/4096 815.640 cents Pythagorean augmented fifth
12: 8 5/3 884.359 cents major sixth, BP sixth
13: 3 2048/1215 903.911 cents
13: 11 27/16 905.865 cents Pythagorean major sixth
13: 3 1280/729 974.584 cents
14: 2 2187/1280 927.371 cents
14: 13 16/9 996.090 cents Pythagorean minor seventh
14: 2 3645/2048 998.044 cents
15: 7 9/5 1017.596 cents just minor seventh, BP seventh
15: 3 4096/2187 1086.315 cents Pythagorean diminished octave
15: 7 15/8 1088.269 cents classic major seventh
16: 5 256/135 1107.821 cents octave - major limma
16: 7 243/128 1109.775 cents Pythagorean major seventh
16: 5 160/81 1178.494 cents octave - syntonic comma

Next time round, smallest interval is
1.95372 cents

So I think we are onto something here...

Robert

Robert Walker wrote,

> So a natural thought is to continue the two line lattice round the
octave.

The problem, at least as far as a three-stepsize parallel to
MOS/Myhill's property is concerned, is that this is not the least bit
generalized, as you know it will only "work" when it works (no 5-tone,
5/4 and 3/2 generator L M S, etc.).

If there were an easy way to answer a question like 'for any given L M
S scale, say 2L 5M 3S for example, the two generators are such and
such' then a two generator three-stepsize MOS/Myhill's property
parallel would be achieved.

Also if you look at your big 17-tone scale, it's really just a
two-stepsize cardinality, 5s12L scale any practical way you look at
it.

--Dan Stearns

Dan Stearn wrote,

>The problem, at least as far as a three-stepsize parallel to
>MOS/Myhill's property is concerned, is that this is not the least bit
>generalized, as you know it will only "work" when it works (no 5-tone,
>5/4 and 3/2 generator L M S, etc.).

I'd like to know why it works though, it can't just be coincidence for a
scale that big, there must be some general principle behind it. Well almost
certainly, one does get mathematical coincidences sometimes that lead
one astray.

>If there were an easy way to answer a question like 'for any given L M
>S scale, say 2L 5M 3S for example, the two generators are such and
>such' then a two generator three-stepsize MOS/Myhill's property
>parallel would be achieved.

Yes, at present the substitutions aren't that promising, just included them
to show what they are.

>Also if you look at your big 17-tone scale, it's really just a
>two-stepsize cardinality, 5s12L scale any practical way you look at
>it.

In cents, step sizes are

92.1787 cents 90.225 cents 21.5063 cents

Basicly two step sizes I agree.

Left out one note in the scale, (step sizes were shown okay) - should be

1/1 135/128 2187/2048 9/8 1215/1024 5/4 81/64 10935/8192 45/32 729/512 3/2 405/256 27/16 3645/2048 15/8 243/128 2/1

(left out the 10935/8192)

It's only a first step, may lead somewhere, or maybe nowhere.

As a next step, tried same experiment with 1/1 7/6 3/2

Here is 1st scale with 3 sizes of notes:

1/1 49/48 7/6 49/32 7/4 2/1 (6 notes this time)
steps
49/48 8/7 21/16 8/7 8/7

It works. (I leave out the SCALA results here as anyone with a copy of the prog. can easily duplicate them).

Using
1/1 6/5 3/2
scale:
1/1 27/25 6/5 81/50 9/5 2/1
steps:
27/25 10/9 27/20 10/9 10/9 (5 notes)

Works!

Using
1/1 11/8 3/2
scale
1/1 33/32 9/8 297/256 11/8 3/2 99/64 27/16 2/1
steps:
33/32 12/11 33/32 32/27 12/11 33/32 12/11 32/27 (8 notes this time)

Works again.

I definitely smell a theorem here!

Let's try something else instead of 3/2:

1/1 5/4 7/4
scale:
1/1 35/32 5/4 49/32 7/4 2/1 (5 notes)
steps:
35/32 8/7 49/40 8/7 8/7
Works!

1/1 11/8 13/8

But they don't all work

1/1 11/8 143/64

when expanded in this way,

has 4 note scale with 3 step sizes

1/1 143/128 11/8 1573/1024 2/1

steps:
143/128 16/13 143/128 2048/1573
S M S L
for interval class 2 has just two sizes:
SM and SL

But even this one works if you go up one more note

scale:
1/1 143/128 20449/16384 11/8 1573/1024 2/1 (5 notes)
steps:
143/128 143/128 2048/1859 143/128 2048/1573
Works!

I definitely smell a theorem here! Would be a bit premature to
make an exact conjecture for it though,

Robert

Hyper MOS search results:

I've added option to FTS to find the hyper-mos scales.

It will be in next upload of beta if anyone wants a go - I'll do one later tonight / early tomorrow morning, at whatever
stage it has reached.

Some new exs.:

1/1 7/5 3/2
yields 12 note scale:
1/1 21/20 9/8 189/160 81/64 1701/1280 7/5 3/2 63/40 27/16 567/320 243/128 2/1
steps
21/20 15/14 21/20 15/14 21/20 256/243 15/14 21/20 15/14 21/20 15/14 256/243

For 1/1 5/4 3/2, the 17 note scale is largest one FTS found in search to 1000 notes..

1/1 9/8 5/4 yields lots of scales

E.g.

1/1 1125/1024 9/8 625/512 5/4 5625/4096 45/32 3125/2048 25/16 28125/16384 225/128 15625/8192 125/64 2/1

steps

1/1 1125/1024 128/125 625/576 128/125 1125/1024 128/125 625/576 128/125 1125/1024 128/125 625/576 128/125 128/125

and

1/1 140625/131072 1125/1024 9/8 78125/65536 625/512 5/4 703125/524288 5625/4096 45/32 390625/262144 3125/2048 25/16
3515625/2097152 28125/16384 225/128 1953125/1048576 15625/8192 125/64 2/1
(18 notes)

steps:
140625/131072 128/125 128/125 78125/73728 128/125 128/125 140625/131072 128/125 128/125 78125/73728 128/125 128/125
140625/131072 128/125 128/125 78125/73728 128/125 128/125 128/125

All work.

What about using some other interval to reduce into instead of the octave?

Try 3 instead of 2, and 1/1 5/4 3/2:

1/1 9/8 5/4 45/32 3/2 27/16 15/8 135/64 9/4 81/32 45/16 3/1

steps

9/8 10/9 9/8 16/15 9/8 10/9 9/8 16/15 9/8 10/9 16/15

Works!

Try:

1/1 34/25 11/7
scale:
1/1 187/175 121/98 22627/17150 34/25 11/7 2057/1225 1331/686 2/1
steps:
187/175 275/238 187/175 1372/1331 275/238 187/175 275/238 1372/1331

And that one doesn't work!
In cents and L M S notation:
115 cents 250 cents 115 cents 52.5 cents 250 cents 115 cents 250 cents 52.5 cents
M L M S L M L S

Numbers are okay for all the classes except, only two intervals of class 4 (in the LMS notation, LMMS and LLMS)

Oh well...

Go on to 14 notes and you get

1/1 248897/240100 187/175 161051/134456 121/98 30116537/23529800 22627/17150 34/25 14641/9604 11/7 2737867/1680700
2057/1225 1771561/941192 1331/686 2/1

steps: 248897/240100 1372/1331 366025/326536 1372/1331 248897/240100 1372/1331 1372/1331 366025/326536 1372/1331
248897/240100 1372/1331 366025/326536 1372/1331 1372/1331
(in cents the three sizes are 115 cents 312 cents 365 cents)

Still doesn't work - three interval sizes for all except 7 step intervals.
The 20 note scale has three interval sizes for all except 10 step intervals.

That's the last MOS for this one before 1000 notes.

So it is an exception, but a very ordered one!

It would be nice to have a conjecture something like this:

Conjecture

Let a, b be two step sizes. Let c be any other positive number.

Make the sequence a, a+b, a+b+a, a+b+a+b, ...

I.e. a, a+b, 2a+b, 2a+2b, 3a+2b,...

Define R(a,b,c,n) as the above sequence truncated at n terms, reduced modulo c, and the terms re-arranged
in order of increasing size..

Suppose that (some extra conditions)

Then R(a,b,c,n) has three sizes of interval for each interval class.

Where by an interval of interval class k one means the usual definition - an interval consisting of k steps.

But what are the extra conditions we need?

Robert

Robert Walker wrote,

>Checking in SCALA, this scale does indeed have three interval sizes for
each degree!

Three specific interval sizes for each generic interval size! Wow! Is this a
mode of any previously identified 17-tone scale?

>So I think we are onto something here...

I'd certainly say so . . . though your construction seemed kind of haphazard
. . . can you think of a more systematic way of describing the construction
of hyper-MOS scales? And what happened to the Tribonacci-like sequence we
were expecting?

Robert Walker wrote,

> I'd like to know why it works though, it can't just be coincidence
for a scale that big, there must be some general principle behind it.
Well almost certainly, one does get mathematical coincidences
sometimes that lead one astray.

Hi Robert. I notice that if you take your two generator SML idea and
let "y" and "z" (where y>z) be the generators so S = y-z, M = z, and L
= p-y; where "p" is both your theoretical 1/1 and whatever you define
as your periodicity (traditionally the octave, 2/1), you would have a
basic SML "tree" of:

z
/ \
/ \
p-----y

And if you were to scale a given SML index by percentages of "P", you
would have a generalized two generator, three-stepsize model relative
to whatever you initially call y and z that works only if the "root
runners" in your expansion process agree with the index.

So say we stick with your original 2:3 and 4:5 generators:

5/4
/ \
/ \
1/1---3/2

Now lets say we want to make an SML 5-tone scale that both
accomplishes a two generator, three-stepsize MOS and jives with your
"root runner" expansion. 2S 2M 1L would do the job, as y = 443, z =
244, and L = 314, M = 244, and S = 199 giving a scale of 0 244 443 687
886 1200.

244---687
/ \ / \
/ \ / \
0----443---886

0 244 443 687 886 1200
0 199 443 642 956 1200
0 244 443 757 1001 1200
0 199 513 757 956 1200
0 314 558 757 1001 1200

What I'm interested in is a similar, but completely generalized
approach not tied to special conditions (in other words one that works
for any given SML index). I think that that would probably have to do
away with the root runners and therefore the two generator idea... but
I'm really not sure?

--Dan Stearns

I think that a generalized three-stepsize scale can be achieved for
any SML using a "scaled by a percentage of P" method. (Though that
lacks a neat ordering rule of the type that is conveniently supplied
by either a single or double generator method.)

But I now think a completely generalized three-stepsize Myhill's
property is not possible -- does everybody but me know this already?
I'm sure there's some fairly simple mathematical condition as to why
this is so, if as I think it is.

Is MOS by some definition 'a scale constructed of two and only two
stepsizes'? Can Myhill's property be seen as a separate condition? If
so, I think generalized n-stepsize scales are achievable, if not, I
think it's time to forget about it... ?

--Dan Stearns

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

> Is MOS by some definition 'a scale constructed of two and only two
> stepsizes'?

That's equivalent to MOS, but not directly the definition.

> Can Myhill's property be seen as a separate condition?

Yes, though a logically equivalent one.

> If
> so, I think generalized n-stepsize scales are achievable, if not, I
> think it's time to forget about it... ?

Not sure what you're getting at, but keep going, this has been very interesting
. . .

Paul Erlich wrote,

<< Not sure what you're getting at, but keep going, this has been very
interesting . . . >>

I'll try and explain the 'scaled by P method' a bit, and that should
make what I'm trying to get at here a bit clearer.

If you first replace the initials of stepsizes with small to large
alphabetized variables, say two-stepsize = [a,b], three-stepsize =
[a,b,c] and so on and so forth where the alphabetized variables are
any whole number... you could the use some stepsize template, I'll use
a harmonic series type template where a = 1:2, b = 1:3, c = 1:4, etc.
So say we had a two-stepsize [2,5] scale where P is the octave as it
usually is... and with this template you would scale "a" and "b" by
the sum of 2 * (LOG(2)-LOG(1))*(1200/LOG(2)) and 5 *
(LOG(3)-LOG(1))*(1200/LOG(2)) divided by 1200... and this would give a
rounded "a" and "b" of 121 and 192 cents:

0 192 383 504 696 887 1079 1200
0 192 313 504 696 887 1008 1200
0 121 313 504 696 817 1008 1200
0 192 383 575 696 887 1079 1200
0 192 383 504 696 887 1008 1200
0 192 313 504 696 817 1008 1200
0 121 313 504 625 817 1008 1200

Using the same process with an [a,b,c] scale of [2,2,3] you'd have a =
107, b = 170, and c = 215:

0 215 385 493 707 878 1093 1200
0 170 278 493 663 878 985 1200
0 107 322 493 707 815 1030 1200
0 215 385 600 707 922 1093 1200
0 170 385 493 707 878 985 1200
0 215 322 537 707 815 1030 1200
0 107 322 493 600 815 985 1200

Now if you allow that MOS is simply 'a scale constructed of two and
only two stepsizes', it's easy to see a generalized n-stepsize rule
using this [a,b,...] scaled by P method.

A single generator chain which is reduced by P and truncated at a+b
will always result in Myhill's property within a P given the correct
generator size... I can see no such n-stepsize parallel.

--Dan Stearns

This thread is very interesting. I wish I had time to work on it too.
I eagerly await your further results.

Regards,
-- Dave Keenan

>But I now think a completely generalized three-stepsize Myhill's
>property is not possible -- does everybody but me know this already?
>I'm sure there's some fairly simple mathematical condition as to why
>this is so, if as I think it is.

Dan, what do you mean here? 3-step-size scales are possible, and I
believe I've seen some in your posts.

-Carl

Carl Lumma wrote,

<< Dan, what do you mean here? 3-step-size scales are possible, and I
believe I've seen some in your posts. >>

What I meant was a *completely* generalized three-stepsize Myhill's
property... in other words a condition which would allow *any* given
SML scale to render a three-stepsize scale with, as Robert Walker
would say, "Tryhill's" property -- it ain't gonna happen!

But there's surely an easy rule/process that will render only those
that do work... and it seems to me that this is what Robert Walker is
looking at.

The posts I've been doing on this thread have concentrated on
generalizing three-stepsize MOS where "three-stepsize MOS" simply says
'a scale constructed of three and only three stepsizes'. I've come up
with what I think is a very nice way to do this and now trying I'm
trying to dream up some sort of an appropriately generalized and
non-generator derived ordering/organization rule...

Anyone got any ideas?

--Dan Stearns

Dan wrote,

>The posts I've been doing on this thread have concentrated on
>generalizing three-stepsize MOS where "three-stepsize MOS" simply says
>'a scale constructed of three and only three stepsizes'.

That seems a poor terminology since MOS does not simply mean "a scale
constructed of two and only two stepsizes" -- the melodic minor scale being
one counterexample.

Paul H. Erlich wrote,

<< That seems a poor terminology since MOS does not simply mean "a
scale constructed of two and only two stepsizes" -- the melodic minor
scale being one counterexample. >>

Yes, your quite right. And by abandoning the generator idea, using the
term MOS is even less of a good idea. I'm not sure what to call it
really... but before I even start to worry too much about any of that
I've got to come up with some type of a non-generator global ordering
rule for three-stepsize indexes -- any ideas?

--Dan Stearns

"Yes, your quite right. And by abandoning the generator idea, using the
term MOS is even less of a good idea. I'm not sure what to call it
really... but before I even start to worry too much about any of that
I've got to come up with some type of a non-generator global ordering
rule for three-stepsize indexes -- any ideas?"

To me, an important feature of any "scale-formerly-known-as-a-MOS" :) is
the interval by which specific intervals of the same generic interval class
differ from each other, which I will call the "interval of coloration",
because a difference of it will not affect the generic interval class, but
only the specific interval or "color". As an example, the interval of
coloration of the meantone diatonic would be the chromatic semitone, because
the difference between any two diatonic intervals with the same number (such
as "thirds" or "sevenths") is a chromatic semitone. In an n-tone MOS, the
interval formed from n generators is the interval of coloration.
When this idea is expanded to two dimensions, there are three intervals
of coloration, one of which is the sum of the other two. In the case of the
just diatonic these are 81/80 + 25/24 = 135/128. This seems a potential way
to tablulate the scales.

Hi David Clampitt,

Sorry not to have posted earlier - been v. pre-occupied with finishing
release of Fractal Tune Smithy 1.09, and wanted to think over some
fragmentary results I had before replying.

I've realised from your post that I made a mistake in the proof of my
attempted theorem about scales with prime numbers of notes, as you gave a
counter example with 5 notes that is trivalent and not pairwise
well-formed.

> entailment of pairwise well-formedness. A counterexample is Vieru's
> Bacovia mode, the complement of the pairwise well-formed Hungarian or
> gypsy minor: e.g. C E F Aflat B. This has trivalence but is not
> pairwise well-formed.

steps 4 1 3 3 1

L S M M S
1: L M S
2: LS SM MS
3: LMS MMS (LMS again) LSS
changes M <-> L and M <-> S.

My mistake was to assume without a detailed proof that as you go round the
circle of the scale, the intervals can only change by single use of each
rule such as L -> M etc. In his counter example, L -> M -> L, and then M ->
S -> M, which means the rule L->S doesn't need to be used. Don't know how I
missed that!

The counter examples I found so far that are trivalent, and pairwise well
formed, and not made using alternating double generators have all been
seven note scales in the pattern
1 3 1 1 3 1

Scales in this category include:
Modes of 12-tet
0 2 1 3 1 1 3 1 Minor Gipsy, Mela Simhendramadhyama, Raga Madhava Manohari,
Maqam Suzdil, Nawa Athar, Double Harmonic Minor, Niavent: Greece

1 3 1 2 1 3 1 Major Gipsy, Persian, Double Harmonic Major, Bhairav That,
Mela Mayamalavagaula, Raga Paraj, Second Byzantine mode, Hitzaskiar:
Greece, Maqam Zengule, Hijaz Kar

is a mode of the same scale.

and the one I mentioned before:
arist_penh2.scl | Permuted Aristoxenos's Enharmonion, 3 + 24 + 3 parts
0 cents 50 cents 450 cents 500 cents 700 cents 750 cents 1150 cents 1200
cents

and it is easy to make more examples.

However, haven't been able to extend this pattern to larger scales.

I wonder if there are any other patterns for pairwise well formed scales.

As a first step towards finding out more about the structure of pairwise
well formed scales, can such a scale always be spanned using two generating
intervals in some order?

By spanning, I mean that by starting with one note, you can get to all the
others in turn by using the two generators in some sequence. So this is an
attempt to generalise from alternating generators to any sequence of two
generators.

After making two of the step sizes equivalent by rescaling them, since
resulting scale is Myhill, it has a generator, so one of the interval
classes will have one interval size with a unique instance in the scale
(the other interval size in this class being the generator).

In the original scale, this class will also have an interval size with a
unique instance, and the other two interval sizes for this class are then
candidates to span the scale.

For a scale with a prime number of notes, then successive intervals in the
interval class span the scale (as any interval class has a number of steps
co-prime with the number of steps in the scale), so result follows.

For scales with non-prime numbers of notes, then the interval class for the
Myhill scale that results on making two of the step sizes equivalent must
span the Myhill scale, (and so the number of steps in this class has to be
co-prime with the number of steps in the scale).

So again the result follows.

Perhaps not a very strong property to establish for these scales, but it is
something...

Robert

Welcome to the Alternate Tunings Mailing List.Hi Gene,

As I remember it, in fact Dan's
method here seems to be a rather general one involving
creating a trivalent scale given a choice of 3 numbers,
and some constants. At least as general as the normal
way of making a MOS in terms of the family of
scales you can make with it.

I vaguely remember that there was some connection
with n-ets maybe, using those numbers to select
particular subsets of a larger n-et, or maybe that was
some other closely related topic, we are talking about
a discussion that took place over five years ago now
and my memory of it isn't that clear. At any rate
there is a particular construction that
he is describing there which generates those
scales in a recursive way.

It took a lot of explaining and I came very close to understanding
it when we had to break off.

In that post, he uses a number
of shorthands like silver weighting etc and his a b c notation,
all of which have to be explained before one can know what they mean
- they do refer to particular very specific constructions
that need to be described and understood, and there
are many assumed details that one can't
know from reading the post, just
like a mathematical paper relying on
results and definitions in earlier papers.

Also do you have any ideas about his conjecture that
the only completely rational trivalent scales are the
ones with an odd number of notes? He may even
have proved it or got very close
to doing so, I remember discussing that with
him too, and it is rather intriguing.

In our discussions, we found that it
takes time on both sides, for the
mathematician listener to figure out
what he is saying, also for him to work out
how to say it and to answer all the questions.
I think what it is is that he has figured out some of
the mathematical techniques he needed
largely for himself, so has got used to rather
individual ways of presenting it, but whenever I have had the
opportunity to follow it through, the reasoning
has been sound, the results interesting, and
often coming at the ideas with a novel slant.
There is a lot more to what he says than
often appears at first sight when one reads it.
It is well worth the effort on both sides if
one has the time to do it.

If no-one else can describe his technique to me, I
hope to come back to the discussion
when I have more time.

BTW I did a comprehensive search of the
SCALA archive for trivalent scales and higher
valency scales ,and you will still find it somewhere in the
files area under my name probably.
I can't remember how I did it, except that it
involved some programming and I have a vague
idea I may have somehow used FTS to prepare
the list of scales, but can't remember how.

Just did a search of my desktop for "trivalent" and found a note
about it where I mention that the search turned up
only one trivalent scale with an even number of notes
which was:

arabic1.scl | From Fortuna. Try C or G major
>
> 1/1 100 cents 200 cents 300 cents 350 cents 500 cents 600 cents 700
cents 800 cents 900
> cents 1000 cents 1050 cents 2/1
>
> Or in 24-tet:
> 0 2 4 6 7 10 12 14 16 18 20 21 24
> steps:
> 2 2 2 1 3 2 2 2 2 2 1 3

It also turned up a number of modes of et scales
as trivalent, and they were all with even
number of notes, so I wonder if it goes both
ways, that all et trivalent scales have an
even number of notes and that all rational
trivalent scales have an odd number,
at least the data seems to suggest
that as a conjecture.

Thanks,

Robert

----- Original Message -----
From: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Sent: Wednesday, October 25, 2006 11:50 AM
Subject: [tuning] Digest Number 4177

Welcome to the Alternate Tunings Mailing List.
Messages In This Digest (6 Messages)
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Aaron Wolf
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
2. Re: Kirkwood gaps scale From: George D. Secor
3. Re: History again - first mention of 55-division? From: threesixesinarow
View All Topics | Create New Topic Messages
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Aaron Wolf" backfromthesilo@yahoo.com backfromthesilo
Tue Oct 24, 2006 8:32 am (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Thanks for the enthusiastic encouragement. I wasn't going to
> > bother, but fine... here it is:
> >
> > http://ozmusic.com/aaron/mp3/WMeetAgain/WMA.pdf
> >
> > I wrote in pencil and I scanned it quickly, it isn't perfect.
>
> Thanks! I'll try to set aside time this weekend to crunch on
> this.
>
> > I've been trying to figure out what notation will be easiest
> > for me to read musically, easiest to use when editing on the
> > computer, and easiest to explain to others...
>
> From my point of view, you aren't there yet. :)
>
> What did you think of my suggestion (did it make any sense
> to you)?
>

I understand a little, but I feel neither of us is really there.
I do want to start having commas more apparent.

My idea for something to be taught to actual singers is something like
this:

How about this? a notation that indicates the fundamental of each
chord in such a way that is simple and clear, and if it is a
pythagorean note, then indicate what power of 3 that it is (in other
words, like the old time barbershopper's "clock" system that says how
many fifths away on the circle of fifths we are). Then use different
note heads to simply indicate a 3, 5, 7 etc identity in relation to
the fundamental. If that could be combined with a very simple
indication of melodic comma shifts (no need to specify different
commas), that should cover everything. What do you think?

-Aaron

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Messages in this topic (10)
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 8:59 pm (PST)
> My idea for something to be taught to actual singers is something
> like this:
>
> How about this? a notation that indicates the fundamental of
> each chord in such a way that is simple and clear, and if it is
> a pythagorean note, then indicate what power of 3 that it is
> (in other words, like the old time barbershopper's "clock"
> system that says how many fifths away on the circle of fifths
> we are). Then use different note heads to simply indicate a
> 3, 5, 7 etc identity in relation to the fundamental. If that
> could be combined with a very simple indication of melodic
> comma shifts (no need to specify different commas), that should
> cover everything. What do you think?

Sounds good, except I don't like the part about the fundamental's
absolute pitch. First off, what happens if it isn't a power
of 3? Secondly, I don't think performers (or composers) need
to know this. There's no way a bass is going to be able to say,
"Oh, 3129/8080, I was a bit flat there, wasn't I?"

Performers are more likely to be interested in which note
remain unchanged or nearly unchanged between chords. They
can then tune pure to it/them. For nearly unchanged notes,
yes I agree plenty of mileage could be gotten out of only one
type of "nearly". I like drawing lines to make the common
tones obvious. Then one doesn't even have to read music (or
both clefs) to know that he's got to match the bari's previous
pitch. Dotted lines make a good nearly, with perhaps a plus
or minus sign above them to indicate direction.

That takes care of shifts. Drift I think is best shown as a
cumulative cents offset from concert pitch. Every time the
offset changes direction -- say it's been going flat and
starts going sharp -- the current offset should be printed
above the barline. Say we start out at A=440, go 20 cents
flat every other bar for 5 bars, then go 20 cents sharp every
other bar for 5 bars and end on A=440. The notation would
show "-100" above the 6th barline and "+-0" above the last
barline. The point is that this is something that should
happen naturally if one tunes pure and respects the shifts
notation (above). It's just there as a check so you can
troubleshoot whether you're going flat 'cause you're tired
or because the composer/arranger intended it.

-Carl

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Messages in this topic (10)
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 10:06 pm (PST)
[I wrote...]
> Just take this example:
>
> http://lumma.org/music/score/Retrofit_JI.pdf
> (this is 4MB, might take a while to load)
>
> It's a formative showing of the notation I suggested in my
> previous message, but I haven't gone through and made sure
> the commas don't accumulate. I really should get around to
> doing that one day.

There are 24 root changes in the piece:

9/8
3/2
7/4
9/8
6/5
9/8
16/9
8/7
28/15
60/49
10/7
3/2
4/3
9/8
27/16
3/2
64/35
7/4
9/8
21/16
4/3
9/8
8/5
9/8

This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
or about 94 cents. I don't know if that's up or down, but
I don't think it'll be a problem in a 3-minute piece with
24 root changes.

-Carl

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Messages in this topic (10)
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 11:40 pm (PST)
> This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
> or about 94 cents. I don't know if that's up or down, but
> I don't think it'll be a problem in a 3-minute piece with
> 24 root changes.

One *can* write a neoclassical piece in 12-tET, fancifully
assign it an 11-limit adaptive JI tuning, and come out OK.

-Carl

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Messages in this topic (10)
2. Re: Kirkwood gaps scale
Posted by: "George D. Secor" gdsecor@yahoo.com gdsecor
Tue Oct 24, 2006 2:13 pm (PST)
--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Robert walker" <robertwalker@>
wrote:
>
> > I had a go at retuning it to 17-et - a similarly sized semitone
> though not otherwise very close, anyway, this is the mode I used,
> closest to your scale:
> > 0 3 4 6 7 10 13 14 17
> > and here is the result:
>
> Here are retunings to 31, 34 (the sharp 7 version), 46, 68, and 99,
> as well as the JI original. I'm fond of the extra shimmer 99 adds
on,
> though it generally sounds a lot like the JI version. Anyway, people
> can compare to 31 and see if sharp fifths are really melodically
> better as George claims.
>
> http://bahamas.eshockhost.com/~xenharmo/midi/examples/kirkwood/

Hi Gene & Robert,

I got a chance to listen to the Kirkwood gaps (original & all of the
retunings) last night (several times) to decide if I had any
preferences. I observed that 34 is the only tuning that has both a
good 5/4 and 5/3 and also has the Archytas comma tempered out in the
upward jump of the fifth between 7/4 and 4/3 (at 0:27-0:28). But I
can't say that it's clearly my favorite for that reason, because I
also liked 46 and 68 better than the others.

In the paper, I didn't mean to give the impression that wide fifths
are necessarily better melodically under all circumstances -- I was
writing specifically about a diatonic scale, where wide fifths result
in significantly narrower-than-12ET diatonic semitones. Several
times in the paper (pp. 59, 74, 75), I made statements to the effect
that chromatic semitones in 31-ET could be used to enhance the
melodic effect of that (narrow-fifth) tuning, so the size of the
fifths is, under more general circumstances, beside the point.

If your tuning has dozens of tones/octave, you then have
opportunities to alter the mood on the fly by making subtle
substitutions in your scale subset, so it's not wise to draw
conclusions too quickly about one of these tunings vs. another.

--George

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Messages in this topic (9)
3. Re: History again - first mention of 55-division?
Posted by: "threesixesinarow" CACCOLA@NET1PLUS.COM threesixesinarow
Tue Oct 24, 2006 4:33 pm (PST)
--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> ...
> What Sauveur said in 1701 would be good to know too, though I'm
> waiting for that article on order.

http://gallica.bnf.fr/ark:/12148/bpt6k3503q/f6.item
"Système générale des Intervalles des Sons" 1701, page 299
http://gallica.bnf.fr/ark:/12148/bpt6k3489j/f420.table
"Table générale des Systemes temperés de Musique." 1707, article page
203, commentary page 117
http://gallica.bnf.fr/ark:/12148/bpt6k35149
"Table generale des Systemes temperés de Musique " 1711, article page
307, commentary page 80
http://gallica.bnf.fr/ark:/12148/bpt6k3516x/f449.table
"Rapport des Sons des Cordes d'Instruments de Musique aux Fléches des
Cordes; Et nouvelle détermination des Sons fixes." 1713, page 324

all in _Histoire de l'Académie royale des sciences avec les mémoires
de mathématique et de physique tirés des registres de cette Académie_

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Messages in this topic (3) Recent Activity
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Welcome to the Alternate Tunings Mailing List.Hi Gene,

As I remember it, in fact Dan's
method here seems to be a rather general one involving
creating a trivalent scale given a choice of 3 numbers,
and some constants. At least as general as the normal
way of making a MOS in terms of the family of
scales you can make with it.

I vaguely remember that there was some connection
with n-ets maybe, using those numbers to select
particular subsets of a larger n-et, or maybe that was
some other closely related topic, we are talking about
a discussion that took place over five years ago now
and my memory of it isn't that clear. At any rate
there is a particular construction that
he is describing there which generates those
scales in a recursive way.

It took a lot of explaining and I came very close to understanding
it when we had to break off.

In that post, he uses a number
of shorthands like silver weighting etc and his a b c notation,
all of which have to be explained before one can know what they mean
- they do refer to particular very specific constructions
that need to be described and understood, and there
are many assumed details that one can't
know from reading the post, just
like a mathematical paper relying on
results and definitions in earlier papers.

Also do you have any ideas about his conjecture that
the only completely rational trivalent scales are the
ones with an odd number of notes? He may even
have proved it or got very close
to doing so, I remember discussing that with
him too, and it is rather intriguing.

In our discussions, we found that it
takes time on both sides, for the
mathematician listener to figure out
what he is saying, also for him to work out
how to say it and to answer all the questions.
I think what it is is that he has figured out some of
the mathematical techniques he needed
largely for himself, so has got used to rather
individual ways of presenting it, but whenever I have had the
opportunity to follow it through, the reasoning
has been sound, the results interesting, and
often coming at the ideas with a novel slant.
It is well worth the effort on both sides if
one has the time to do it.

If no-one else can describe his technique to me, I
hope to come back to the discussion
when I have more time.

BTW I did a comprehensive search of the
SCALA archive for trivalent scales and higher
valency scales ,and you will still find it somewhere in the
files area under my name probably.
I can't remember how I did it, except that it
involved some programming and I have a vague
idea I may have somehow used FTS to prepare
the list of scales, but can't remember how.

Just did a search of my desktop for "trivalent" and found a note
about it where I mention that the search turned up
only one trivalent scale with an even number of notes
which was:

arabic1.scl | From Fortuna. Try C or G major
>
> 1/1 100 cents 200 cents 300 cents 350 cents 500 cents 600 cents 700
cents 800 cents 900
> cents 1000 cents 1050 cents 2/1
>
> Or in 24-tet:
> 0 2 4 6 7 10 12 14 16 18 20 21 24
> steps:
> 2 2 2 1 3 2 2 2 2 2 1 3

It also turned up a number of modes of et scales
as trivalent, and they were all with even
number of notes, so I wonder if it goes both
ways, that all et trivalent scales have an
even number of notes and that all rational
trivalent scales have an odd number,
at least the data seems to suggest
that as a conjecture.

Thanks,

Robert

----- Original Message -----
From: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Sent: Wednesday, October 25, 2006 11:50 AM
Subject: [tuning] Digest Number 4177

Welcome to the Alternate Tunings Mailing List.
Messages In This Digest (6 Messages)
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Aaron Wolf
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
2. Re: Kirkwood gaps scale From: George D. Secor
3. Re: History again - first mention of 55-division? From: threesixesinarow
View All Topics | Create New Topic Messages
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Aaron Wolf" backfromthesilo@yahoo.com backfromthesilo
Tue Oct 24, 2006 8:32 am (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Thanks for the enthusiastic encouragement. I wasn't going to
> > bother, but fine... here it is:
> >
> > http://ozmusic.com/aaron/mp3/WMeetAgain/WMA.pdf
> >
> > I wrote in pencil and I scanned it quickly, it isn't perfect.
>
> Thanks! I'll try to set aside time this weekend to crunch on
> this.
>
> > I've been trying to figure out what notation will be easiest
> > for me to read musically, easiest to use when editing on the
> > computer, and easiest to explain to others...
>
> From my point of view, you aren't there yet. :)
>
> What did you think of my suggestion (did it make any sense
> to you)?
>

I understand a little, but I feel neither of us is really there.
I do want to start having commas more apparent.

My idea for something to be taught to actual singers is something like
this:

How about this? a notation that indicates the fundamental of each
chord in such a way that is simple and clear, and if it is a
pythagorean note, then indicate what power of 3 that it is (in other
words, like the old time barbershopper's "clock" system that says how
many fifths away on the circle of fifths we are). Then use different
note heads to simply indicate a 3, 5, 7 etc identity in relation to
the fundamental. If that could be combined with a very simple
indication of melodic comma shifts (no need to specify different
commas), that should cover everything. What do you think?

-Aaron

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Messages in this topic (10)
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 8:59 pm (PST)
> My idea for something to be taught to actual singers is something
> like this:
>
> How about this? a notation that indicates the fundamental of
> each chord in such a way that is simple and clear, and if it is
> a pythagorean note, then indicate what power of 3 that it is
> (in other words, like the old time barbershopper's "clock"
> system that says how many fifths away on the circle of fifths
> we are). Then use different note heads to simply indicate a
> 3, 5, 7 etc identity in relation to the fundamental. If that
> could be combined with a very simple indication of melodic
> comma shifts (no need to specify different commas), that should
> cover everything. What do you think?

Sounds good, except I don't like the part about the fundamental's
absolute pitch. First off, what happens if it isn't a power
of 3? Secondly, I don't think performers (or composers) need
to know this. There's no way a bass is going to be able to say,
"Oh, 3129/8080, I was a bit flat there, wasn't I?"

Performers are more likely to be interested in which note
remain unchanged or nearly unchanged between chords. They
can then tune pure to it/them. For nearly unchanged notes,
yes I agree plenty of mileage could be gotten out of only one
type of "nearly". I like drawing lines to make the common
tones obvious. Then one doesn't even have to read music (or
both clefs) to know that he's got to match the bari's previous
pitch. Dotted lines make a good nearly, with perhaps a plus
or minus sign above them to indicate direction.

That takes care of shifts. Drift I think is best shown as a
cumulative cents offset from concert pitch. Every time the
offset changes direction -- say it's been going flat and
starts going sharp -- the current offset should be printed
above the barline. Say we start out at A=440, go 20 cents
flat every other bar for 5 bars, then go 20 cents sharp every
other bar for 5 bars and end on A=440. The notation would
show "-100" above the 6th barline and "+-0" above the last
barline. The point is that this is something that should
happen naturally if one tunes pure and respects the shifts
notation (above). It's just there as a check so you can
troubleshoot whether you're going flat 'cause you're tired
or because the composer/arranger intended it.

-Carl

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Messages in this topic (10)
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 10:06 pm (PST)
[I wrote...]
> Just take this example:
>
> http://lumma.org/music/score/Retrofit_JI.pdf
> (this is 4MB, might take a while to load)
>
> It's a formative showing of the notation I suggested in my
> previous message, but I haven't gone through and made sure
> the commas don't accumulate. I really should get around to
> doing that one day.

There are 24 root changes in the piece:

9/8
3/2
7/4
9/8
6/5
9/8
16/9
8/7
28/15
60/49
10/7
3/2
4/3
9/8
27/16
3/2
64/35
7/4
9/8
21/16
4/3
9/8
8/5
9/8

This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
or about 94 cents. I don't know if that's up or down, but
I don't think it'll be a problem in a 3-minute piece with
24 root changes.

-Carl

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Messages in this topic (10)
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 11:40 pm (PST)
> This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
> or about 94 cents. I don't know if that's up or down, but
> I don't think it'll be a problem in a 3-minute piece with
> 24 root changes.

One *can* write a neoclassical piece in 12-tET, fancifully
assign it an 11-limit adaptive JI tuning, and come out OK.

-Carl

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Messages in this topic (10)
2. Re: Kirkwood gaps scale
Posted by: "George D. Secor" gdsecor@yahoo.com gdsecor
Tue Oct 24, 2006 2:13 pm (PST)
--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Robert walker" <robertwalker@>
wrote:
>
> > I had a go at retuning it to 17-et - a similarly sized semitone
> though not otherwise very close, anyway, this is the mode I used,
> closest to your scale:
> > 0 3 4 6 7 10 13 14 17
> > and here is the result:
>
> Here are retunings to 31, 34 (the sharp 7 version), 46, 68, and 99,
> as well as the JI original. I'm fond of the extra shimmer 99 adds
on,
> though it generally sounds a lot like the JI version. Anyway, people
> can compare to 31 and see if sharp fifths are really melodically
> better as George claims.
>
> http://bahamas.eshockhost.com/~xenharmo/midi/examples/kirkwood/

Hi Gene & Robert,

I got a chance to listen to the Kirkwood gaps (original & all of the
retunings) last night (several times) to decide if I had any
preferences. I observed that 34 is the only tuning that has both a
good 5/4 and 5/3 and also has the Archytas comma tempered out in the
upward jump of the fifth between 7/4 and 4/3 (at 0:27-0:28). But I
can't say that it's clearly my favorite for that reason, because I
also liked 46 and 68 better than the others.

In the paper, I didn't mean to give the impression that wide fifths
are necessarily better melodically under all circumstances -- I was
writing specifically about a diatonic scale, where wide fifths result
in significantly narrower-than-12ET diatonic semitones. Several
times in the paper (pp. 59, 74, 75), I made statements to the effect
that chromatic semitones in 31-ET could be used to enhance the
melodic effect of that (narrow-fifth) tuning, so the size of the
fifths is, under more general circumstances, beside the point.

If your tuning has dozens of tones/octave, you then have
opportunities to alter the mood on the fly by making subtle
substitutions in your scale subset, so it's not wise to draw
conclusions too quickly about one of these tunings vs. another.

--George

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Messages in this topic (9)
3. Re: History again - first mention of 55-division?
Posted by: "threesixesinarow" CACCOLA@NET1PLUS.COM threesixesinarow
Tue Oct 24, 2006 4:33 pm (PST)
--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> ...
> What Sauveur said in 1701 would be good to know too, though I'm
> waiting for that article on order.

http://gallica.bnf.fr/ark:/12148/bpt6k3503q/f6.item
"Système générale des Intervalles des Sons" 1701, page 299
http://gallica.bnf.fr/ark:/12148/bpt6k3489j/f420.table
"Table générale des Systemes temperés de Musique." 1707, article page
203, commentary page 117
http://gallica.bnf.fr/ark:/12148/bpt6k35149
"Table generale des Systemes temperés de Musique " 1711, article page
307, commentary page 80
http://gallica.bnf.fr/ark:/12148/bpt6k3516x/f449.table
"Rapport des Sons des Cordes d'Instruments de Musique aux Fléches des
Cordes; Et nouvelle détermination des Sons fixes." 1713, page 324

all in _Histoire de l'Académie royale des sciences avec les mémoires
de mathématique et de physique tirés des registres de cette Académie_

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Welcome to the Alternate Tunings Mailing List.Hi Gene,

As I remember it, in fact Dan's
method here seems to be a rather general one involving
creating a trivalent scale given a choice of 3 numbers,
and some constants. At least as general as the normal
way of making a MOS in terms of the family of
scales you can make with it.

I vaguely remember that there was some connection
with n-ets maybe, using those numbers to select
particular subsets of a larger n-et, or maybe that was
some other closely related topic, we are talking about
a discussion that took place over five years ago now
and my memory of it isn't that clear. At any rate
there is a particular construction that
he is describing there which generates those
scales in a recursive way.

It took a lot of explaining and I came very close to understanding
it when we had to break off.

In that post, he uses a number
of shorthands like silver weighting etc and his a b c notation,
all of which have to be explained before one can know what they mean
- they do refer to particular very specific constructions
that need to be described and understood, and there
are many assumed details that one can't
know from reading the post, just
like a mathematical paper relying on
results and definitions in earlier papers.

Also do you have any ideas about his conjecture that
the only completely rational trivalent scales are the
ones with an odd number of notes? He may even
have proved it or got very close
to doing so, I remember discussing that with
him too, and it is rather intriguing.

In our discussions, we found that it
takes time on both sides, for the
mathematician listener to figure out
what he is saying, also for him to work out
how to say it and to answer all the questions.
I think what it is is that he has figured out some of
the mathematical techniques he needed
largely for himself, so has got used to rather
individual ways of presenting it, but whenever I have had the
opportunity to follow it through, the reasoning
has been sound, the results interesting, and
often coming at the ideas with a novel slant.
There is a lot more to what he says than
often appears at first sight when one reads it.
It is well worth the effort on both sides if
one has the time to do it.

If no-one else can describe his technique to me, I
hope to come back to the discussion
when I have more time.

BTW I did a comprehensive search of the
SCALA archive for trivalent scales and higher
valency scales ,and you will still find it somewhere in the
files area under my name probably.
I can't remember how I did it, except that it
involved some programming and I have a vague
idea I may have somehow used FTS to prepare
the list of scales, but can't remember how.

Just did a search of my desktop for "trivalent" and found a note
about it where I mention that the search turned up
only one trivalent scale with an even number of notes
which was:

arabic1.scl | From Fortuna. Try C or G major
>
> 1/1 100 cents 200 cents 300 cents 350 cents 500 cents 600 cents 700
cents 800 cents 900
> cents 1000 cents 1050 cents 2/1
>
> Or in 24-tet:
> 0 2 4 6 7 10 12 14 16 18 20 21 24
> steps:
> 2 2 2 1 3 2 2 2 2 2 1 3

It also turned up a number of modes of et scales
as trivalent, and they were all with even
number of notes, so I wonder if it goes both
ways, that all et trivalent scales have an
even number of notes and that all rational
trivalent scales have an odd number,
at least the data seems to suggest
that as a conjecture.

Thanks,

Robert

----- Original Message -----
From: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Sent: Wednesday, October 25, 2006 11:50 AM
Subject: [tuning] Digest Number 4177

Welcome to the Alternate Tunings Mailing List.
Messages In This Digest (6 Messages)
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Aaron Wolf
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
2. Re: Kirkwood gaps scale From: George D. Secor
3. Re: History again - first mention of 55-division? From: threesixesinarow
View All Topics | Create New Topic Messages
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Aaron Wolf" backfromthesilo@yahoo.com backfromthesilo
Tue Oct 24, 2006 8:32 am (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Thanks for the enthusiastic encouragement. I wasn't going to
> > bother, but fine... here it is:
> >
> > http://ozmusic.com/aaron/mp3/WMeetAgain/WMA.pdf
> >
> > I wrote in pencil and I scanned it quickly, it isn't perfect.
>
> Thanks! I'll try to set aside time this weekend to crunch on
> this.
>
> > I've been trying to figure out what notation will be easiest
> > for me to read musically, easiest to use when editing on the
> > computer, and easiest to explain to others...
>
> From my point of view, you aren't there yet. :)
>
> What did you think of my suggestion (did it make any sense
> to you)?
>

I understand a little, but I feel neither of us is really there.
I do want to start having commas more apparent.

My idea for something to be taught to actual singers is something like
this:

How about this? a notation that indicates the fundamental of each
chord in such a way that is simple and clear, and if it is a
pythagorean note, then indicate what power of 3 that it is (in other
words, like the old time barbershopper's "clock" system that says how
many fifths away on the circle of fifths we are). Then use different
note heads to simply indicate a 3, 5, 7 etc identity in relation to
the fundamental. If that could be combined with a very simple
indication of melodic comma shifts (no need to specify different
commas), that should cover everything. What do you think?

-Aaron

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Messages in this topic (10)
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 8:59 pm (PST)
> My idea for something to be taught to actual singers is something
> like this:
>
> How about this? a notation that indicates the fundamental of
> each chord in such a way that is simple and clear, and if it is
> a pythagorean note, then indicate what power of 3 that it is
> (in other words, like the old time barbershopper's "clock"
> system that says how many fifths away on the circle of fifths
> we are). Then use different note heads to simply indicate a
> 3, 5, 7 etc identity in relation to the fundamental. If that
> could be combined with a very simple indication of melodic
> comma shifts (no need to specify different commas), that should
> cover everything. What do you think?

Sounds good, except I don't like the part about the fundamental's
absolute pitch. First off, what happens if it isn't a power
of 3? Secondly, I don't think performers (or composers) need
to know this. There's no way a bass is going to be able to say,
"Oh, 3129/8080, I was a bit flat there, wasn't I?"

Performers are more likely to be interested in which note
remain unchanged or nearly unchanged between chords. They
can then tune pure to it/them. For nearly unchanged notes,
yes I agree plenty of mileage could be gotten out of only one
type of "nearly". I like drawing lines to make the common
tones obvious. Then one doesn't even have to read music (or
both clefs) to know that he's got to match the bari's previous
pitch. Dotted lines make a good nearly, with perhaps a plus
or minus sign above them to indicate direction.

That takes care of shifts. Drift I think is best shown as a
cumulative cents offset from concert pitch. Every time the
offset changes direction -- say it's been going flat and
starts going sharp -- the current offset should be printed
above the barline. Say we start out at A=440, go 20 cents
flat every other bar for 5 bars, then go 20 cents sharp every
other bar for 5 bars and end on A=440. The notation would
show "-100" above the 6th barline and "+-0" above the last
barline. The point is that this is something that should
happen naturally if one tunes pure and respects the shifts
notation (above). It's just there as a check so you can
troubleshoot whether you're going flat 'cause you're tired
or because the composer/arranger intended it.

-Carl

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Messages in this topic (10)
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 10:06 pm (PST)
[I wrote...]
> Just take this example:
>
> http://lumma.org/music/score/Retrofit_JI.pdf
> (this is 4MB, might take a while to load)
>
> It's a formative showing of the notation I suggested in my
> previous message, but I haven't gone through and made sure
> the commas don't accumulate. I really should get around to
> doing that one day.

There are 24 root changes in the piece:

9/8
3/2
7/4
9/8
6/5
9/8
16/9
8/7
28/15
60/49
10/7
3/2
4/3
9/8
27/16
3/2
64/35
7/4
9/8
21/16
4/3
9/8
8/5
9/8

This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
or about 94 cents. I don't know if that's up or down, but
I don't think it'll be a problem in a 3-minute piece with
24 root changes.

-Carl

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Messages in this topic (10)
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 11:40 pm (PST)
> This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
> or about 94 cents. I don't know if that's up or down, but
> I don't think it'll be a problem in a 3-minute piece with
> 24 root changes.

One *can* write a neoclassical piece in 12-tET, fancifully
assign it an 11-limit adaptive JI tuning, and come out OK.

-Carl

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Messages in this topic (10)
2. Re: Kirkwood gaps scale
Posted by: "George D. Secor" gdsecor@yahoo.com gdsecor
Tue Oct 24, 2006 2:13 pm (PST)
--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Robert walker" <robertwalker@>
wrote:
>
> > I had a go at retuning it to 17-et - a similarly sized semitone
> though not otherwise very close, anyway, this is the mode I used,
> closest to your scale:
> > 0 3 4 6 7 10 13 14 17
> > and here is the result:
>
> Here are retunings to 31, 34 (the sharp 7 version), 46, 68, and 99,
> as well as the JI original. I'm fond of the extra shimmer 99 adds
on,
> though it generally sounds a lot like the JI version. Anyway, people
> can compare to 31 and see if sharp fifths are really melodically
> better as George claims.
>
> http://bahamas.eshockhost.com/~xenharmo/midi/examples/kirkwood/

Hi Gene & Robert,

I got a chance to listen to the Kirkwood gaps (original & all of the
retunings) last night (several times) to decide if I had any
preferences. I observed that 34 is the only tuning that has both a
good 5/4 and 5/3 and also has the Archytas comma tempered out in the
upward jump of the fifth between 7/4 and 4/3 (at 0:27-0:28). But I
can't say that it's clearly my favorite for that reason, because I
also liked 46 and 68 better than the others.

In the paper, I didn't mean to give the impression that wide fifths
are necessarily better melodically under all circumstances -- I was
writing specifically about a diatonic scale, where wide fifths result
in significantly narrower-than-12ET diatonic semitones. Several
times in the paper (pp. 59, 74, 75), I made statements to the effect
that chromatic semitones in 31-ET could be used to enhance the
melodic effect of that (narrow-fifth) tuning, so the size of the
fifths is, under more general circumstances, beside the point.

If your tuning has dozens of tones/octave, you then have
opportunities to alter the mood on the fly by making subtle
substitutions in your scale subset, so it's not wise to draw
conclusions too quickly about one of these tunings vs. another.

--George

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Messages in this topic (9)
3. Re: History again - first mention of 55-division?
Posted by: "threesixesinarow" CACCOLA@NET1PLUS.COM threesixesinarow
Tue Oct 24, 2006 4:33 pm (PST)
--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> ...
> What Sauveur said in 1701 would be good to know too, though I'm
> waiting for that article on order.

http://gallica.bnf.fr/ark:/12148/bpt6k3503q/f6.item
"Système générale des Intervalles des Sons" 1701, page 299
http://gallica.bnf.fr/ark:/12148/bpt6k3489j/f420.table
"Table générale des Systemes temperés de Musique." 1707, article page
203, commentary page 117
http://gallica.bnf.fr/ark:/12148/bpt6k35149
"Table generale des Systemes temperés de Musique " 1711, article page
307, commentary page 80
http://gallica.bnf.fr/ark:/12148/bpt6k3516x/f449.table
"Rapport des Sons des Cordes d'Instruments de Musique aux Fléches des
Cordes; Et nouvelle détermination des Sons fixes." 1713, page 324

all in _Histoire de l'Académie royale des sciences avec les mémoires
de mathématique et de physique tirés des registres de cette Académie_

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"Robert walker" <robertwalker@robertinventor.com> writes:

> It also turned up a number of modes of et scales
> as trivalent, and they were all with even
> number of notes, so I wonder if it goes both
> ways, that all et trivalent scales have an
> even number of notes

Clearly not: consider this subset of 12et: C D F# .

Unless I'm misunderstanding you.

> and that all rational
> trivalent scales have an odd number,
> at least the data seems to suggest
> that as a conjecture.

- Rich Holmes

Welcome to the Alternate Tunings Mailing List.Hi Rich,

Yes you are right. It would be nice to have an example with more than
three notes.

In fact, thinking it over, I realised, you can probably take any
just trivalent scale and if you retune it to the nearest et values in a high enough
numbered et to distinguish the step sizes, there is a good
chance of getting a trivalent scale, if the et. is appropriate (probably Gene
can advise about what that means here).

So anyway just tried out 53 tone "Just" major from the Scala archives and
that's a seven note trivalent scale:

203.774 384.906 498.113 701.887 883.019 1086.792 2/1

(which one can check in Scala using Show Intervals then check
to see that the highest number and the average number of different
intervals per interval class are both 3).

Another one was 29 "just" major:
206.897 372.414 496.552 703.448 868.966 1075.862 2/1

22 tone "just" major is quadrivalent BTW, four intervals in each interval class:
218.182 381.818 490.909 709.091 872.727 1090.909 2/1

I'm not sure why they didn't turn up in my search of the archives for trivalent
modes six years ago.

Robert

--- In tuning@yahoogroups.com, "Robert walker" <robertwalker@...> wrote:

> As I remember it, in fact Dan's
> method here seems to be a rather general one involving
> creating a trivalent scale given a choice of 3 numbers,
> and some constants.

Given its dependence on sqrt(2)+1, I don't see how it can be general.
How would it even manage to get 5-limit trivalent scales?

> Also do you have any ideas about his conjecture that
> the only completely rational trivalent scales are the
> ones with an odd number of notes?

I don't see a proof offhand; I suppose the first thing to look at is
whether the n/2 class can ever have exactly three intervals if the
rest do. It might be another reason to do a brute force search.

> Just did a search of my desktop for "trivalent" and found a note
> about it where I mention that the search turned up
> only one trivalent scale with an even number of notes
> which was:
>
> arabic1.scl | From Fortuna. Try C or G major

Welcome to the Alternate Tunings Mailing List.Hi Gene,

I meant general in the sense that it constructs a family of trivalent scales in much the same way that the golden ratio is used to construct a family of MOS scales. Which is an interesting result because as I understood it, unlike what I was doing which was hit or miss, every scale in his family was trivalent.

Dan would need to come in on this - I was under the impression that his scales were irrational ones but may be wrong. However, once one has a family of trivalent scales like that, one can try converting them into rational scales with the same patterns of L, M and S steps and roughly the same step sizes, like the conversion of the diatonic into a trivalent et scale done in reverse.

So for example the diatonic:

1/1 9/8 5/4 45/32 3/2 27/16 15/8 2/1
= 9/8 10/9 9/8 16/15 9/8 10/9 16/15

L M - L S - L M - S

L M L S L M S
M L S L M S L
L S L M S L M
S L M S L M L
L M S L M L S
M S L M L S L
S L M L S L M

one step intervals are L, M, S
two step intervals are
LM, ML, LS, and re-orderings

three step intervals are
LML, MLS, LSL, and re-orderings

so any scale with this sequence of L, M and S steps is trivalent
so long as the three step sizes are unequal, and none of those
six two or three step sizes are equal to each other.
.
So having come up with an L M S step pattern that works, ET, or otherwise, one can then create many rational scales from it.

Maybe one could begin to see why even numbered trivalent scales are impossible in some way by looking at the patterns of step sizes in even trivalent scales and to see what it is about how they work that means that they can't convert into rationals. Those are just a couple of ideas I'm throwing out which I might follow up if I have time later.

Here:
http://f1.grp.yahoofs.com/v1/cK5BRZEHgUbcTrVSXNq_Gf312XP2b5Wsli335JCOjaZBCpvKNE4wstbrq48cU59yKjBwupgMVSp0zkwT4xNJqvV7qjilOu-DOTHW/Robert%20Walker/n-valent_et_modes.txt
(notation explained here:
http://f1.grp.yahoofs.com/v1/cK5BReOWKVbcTrVSfGxGbRvxMcNMPsBuHxqdMDyvfuRmP4t-5UlqZ_lZ04VshnXEPIW1GYgKHR5gU1TS5bpQrJ8XS-74NO6l4Ml8/Robert%20Walker/n-valent_modes_info.txt
)
the first trivalent scale is:
2 1 4 1 4 V___3_n__5_alt_class_1_z_2_a_1_[_2]_b_4_[_2]_abab_class_2_z_6_a_5_[_3]_b_3_aaba_class_3_z_9_a_7_[_3]_b_6_aaba_alt_class_4_z10_a_8_[_2]_b11_[_2]_abab_ 29 Hira-joshi: Japan

BTW it has 5 steps so I'm just mis-remembering when I thought that the et scales had only even numbers of step sizes. It is just that the only even numbered ones were the n-et ones.

The other intriguing observation, forgot to mention, is that every even numbered trivalent scale includes the midpoint in one of its interval classes (marked m in that list there). If one could prove that as a property of even numbered trivalent scales, then it would follow that they have to be irrational.

The first one of those is:
1 3 2 3 1 2 v___3m_n__6_ Prometheus Neapolitan
pattern:
S L M L S M
rotates as:
S L M L S M
L M L S M S
M L S M S L
L S M S L M
S M S L M L
M S L M L S

There the problem is that in one of its rotations it is:
L S M S L M
so has to contain the midpoint so can't be rational.

So one can try to prove that will always happen for trivalent scales with an even number of notes

I wonder whether trivalent scales are in some way ones that people tend to favour during scale construction, or whether they are somehow an outcome of some of the construction methods used to make scales. It would be interesting to know more about why they are so common.

BTW another thread in our discussion was the idea of "tribonacci scales"

/tuning/topicId_16054.html#16054

Robert
----- Original Message -----
From: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Sent: Thursday, October 26, 2006 11:43 AM
Subject: [tuning] Digest Number 4178

Welcome to the Alternate Tunings Mailing List.
Messages In This Digest (13 Messages)
1a. Hyper-MOS question From: c.m.bryan
1b. Hyper-MOS question From: Robert walker
1c. Re: Hyper-MOS question From: Gene Ward Smith
2a. Re: towards a hyper MOS From: Carl Lumma
2b. Re: towards a hyper MOS From: Gene Ward Smith
2c. Re: towards a hyper MOS From: Gene Ward Smith
3a. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Aaron Wolf
3b. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
4a. Re: everybody loves microtones....................... From: misterbobro
4b. Re: everybody loves microtones....................... From: Ozan Yarman
5a. Towards a hyper MOS From: Robert walker
5b. Towards a hyper MOS From: Robert walker
5c. Towards a hyper MOS From: Robert walker
View All Topics | Create New Topic Messages
1a. Hyper-MOS question
Posted by: "c.m.bryan" chrismbryan@gmail.com chrisbryan82
Wed Oct 25, 2006 5:33 am (PST)
Hello everyone, many of you may already know me from MMM, this is my
first post to tuning...

I have a question about the generation of hyper-MOS scales, which I
saw decribed on the tunesmithy website. Whereas the generation of
non-hyper MOSes is straightforward (repeated iterations of the
generator), I can't figure out an algorithm for hyper-MOS. For
instance, the one referenced on that website is:

1/1 9/8 5/4 45/32 3/2 27/16 15/8 (2/1)

or in terms of the generator:

(3/2)^0*(5/4)^0
(3/2)^2*(5/4)^0
(3/2)^0*(5/4)^1
(3/2)^2*(5/4)^1
(3/2)^1*(5/4)^0
(3/2)^3*(5/4)^0
(3/2)^1*(5/4)^1

and rearranged to show (ir)regularity:

(3/2)^0*(5/4)^0

(3/2)^1*(5/4)^0
(3/2)^0*(5/4)^1
(3/2)^1*(5/4)^1

(3/2)^2*(5/4)^0
(3/2)^2*(5/4)^1

(3/2)^3*(5/4)^0

Is there some pattern that I'm missing?

Thanks,

Chris Bryan

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Messages in this topic (3)
1b. Hyper-MOS question
Posted by: "Robert walker" robertwalker@robertinventor.com robert_inventor5
Wed Oct 25, 2006 8:20 am (PST)
Welcome to the Alternate Tunings Mailing List.Hi Chris,

Dan Stearns and I had a long discussion a while back about the generation of hyper-mos scales which should be trivalent, three interval sizes for each possible number of scale degrees, by analogy with a MOS which has two interval sizes for each possible number of scale degrees. It was partly motivated by the observation that many scales in the SCALA archive are trivalent, and many are also quadrivalent, or more.

I had the idea of a generator involving two intervals such as 5/4 6/5 alternating,
1/1 5/4 3/2,
which you then repeat as:
1/1 5/4 3/2 15//8 9/8 45/32 27/16
which is where the scale you found on the tunesmithy site comes from.
That is trivalent, as you can check in Scala.

If you continue a bit further you get another trivalent "moment of symmetry" at seventeen notes.

So the method produces a few trivalent scales, but is only moderately successful.

You can read more about my alternating generators method here:
http://www.tunesmithy.netfirms.com/fts_help/More_scales.htm#mos

Dan followed another line of investigation which was far more successful and he could produce many trivalent scales to order. However I got busy with programming and he took a long break not long after so the discussion got rather broken off, and I can't say I understand his method quite yet. Also since then I have been so pre-occupied with programming FTS and particularly the upcoming FTS 3.0 release that I have done little by way of further scales explorations.

I added that programming to FTS as part of the process of investigation. I see that II didn't explain the situation properly in the help. I just explained how the option in FTS works. To give the context properly, I should have explainedhat what I was describing is just one of two current candidates for a notion of Hyper MOS, and give the reader some pointer to Dan's more successful method

Anyway, he is the one to ask, as I think his version is probably a better candidate for the label "Hyper MOS" as it is more successful at generating trivalent scales.

This is Dan Stearn's Eureka post about his Hyper MOS method:

/tuning/topicId_16061.html#16585
/tuning/topicId_19322.html#19361

Then it got taken up again on the tuning-math list in a discussion of "Tribonacci scales". But I hadn't got the time to follow the discussion there either.

I hope I get a chance to get back to this some time. It would be great to program Dan's method as well as mine.

He also had an interesting conjecture that every trivalent scale with an even number of notes has to contain an irrational interval, and later I think he concluded that it has to be a subset of an n-et as well, so if that is so, the rational hyper-mos's have to have an odd number of notes.

Unfortunately I haven't go tthe time to join in discussing it at present either. But if anyone feels they could summarize the conclusions and describe it to me - the construction algorithm for his scales - sufficiently so that I can program it, then I'm interested to program it in FTS, and should do to complement my own less successful method. Or later once I have the time to read it all up and get up to date with it. Also I wonder if it might be in Scala by now?

Meanwhile anyway I'll add a note to the help and to the window in FTS to make it clear that Dan Stearns method exists and is probably a better candidate for the term "hyper MOS"

Robert

----- Original Message -----
From: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Sent: Wednesday, October 25, 2006 11:50 AM
Subject: [tuning] Digest Number 4177

Welcome to the Alternate Tunings Mailing List.
Messages In This Digest (6 Messages)
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Aaron Wolf
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
2. Re: Kirkwood gaps scale From: George D. Secor
3. Re: History again - first mention of 55-division? From: threesixesinarow
View All Topics | Create New Topic Messages
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Aaron Wolf" backfromthesilo@yahoo.com backfromthesilo
Tue Oct 24, 2006 8:32 am (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Thanks for the enthusiastic encouragement. I wasn't going to
> > bother, but fine... here it is:
> >
> > http://ozmusic.com/aaron/mp3/WMeetAgain/WMA.pdf
> >
> > I wrote in pencil and I scanned it quickly, it isn't perfect.
>
> Thanks! I'll try to set aside time this weekend to crunch on
> this.
>
> > I've been trying to figure out what notation will be easiest
> > for me to read musically, easiest to use when editing on the
> > computer, and easiest to explain to others...
>
> From my point of view, you aren't there yet. :)
>
> What did you think of my suggestion (did it make any sense
> to you)?
>

I understand a little, but I feel neither of us is really there.
I do want to start having commas more apparent.

My idea for something to be taught to actual singers is something like
this:

How about this? a notation that indicates the fundamental of each
chord in such a way that is simple and clear, and if it is a
pythagorean note, then indicate what power of 3 that it is (in other
words, like the old time barbershopper's "clock" system that says how
many fifths away on the circle of fifths we are). Then use different
note heads to simply indicate a 3, 5, 7 etc identity in relation to
the fundamental. If that could be combined with a very simple
indication of melodic comma shifts (no need to specify different
commas), that should cover everything. What do you think?

-Aaron

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Messages in this topic (10)
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 8:59 pm (PST)
> My idea for something to be taught to actual singers is something
> like this:
>
> How about this? a notation that indicates the fundamental of
> each chord in such a way that is simple and clear, and if it is
> a pythagorean note, then indicate what power of 3 that it is
> (in other words, like the old time barbershopper's "clock"
> system that says how many fifths away on the circle of fifths
> we are). Then use different note heads to simply indicate a
> 3, 5, 7 etc identity in relation to the fundamental. If that
> could be combined with a very simple indication of melodic
> comma shifts (no need to specify different commas), that should
> cover everything. What do you think?

Sounds good, except I don't like the part about the fundamental's
absolute pitch. First off, what happens if it isn't a power
of 3? Secondly, I don't think performers (or composers) need
to know this. There's no way a bass is going to be able to say,
"Oh, 3129/8080, I was a bit flat there, wasn't I?"

Performers are more likely to be interested in which note
remain unchanged or nearly unchanged between chords. They
can then tune pure to it/them. For nearly unchanged notes,
yes I agree plenty of mileage could be gotten out of only one
type of "nearly". I like drawing lines to make the common
tones obvious. Then one doesn't even have to read music (or
both clefs) to know that he's got to match the bari's previous
pitch. Dotted lines make a good nearly, with perhaps a plus
or minus sign above them to indicate direction.

That takes care of shifts. Drift I think is best shown as a
cumulative cents offset from concert pitch. Every time the
offset changes direction -- say it's been going flat and
starts going sharp -- the current offset should be printed
above the barline. Say we start out at A=440, go 20 cents
flat every other bar for 5 bars, then go 20 cents sharp every
other bar for 5 bars and end on A=440. The notation would
show "-100" above the 6th barline and "+-0" above the last
barline. The point is that this is something that should
happen naturally if one tunes pure and respects the shifts
notation (above). It's just there as a check so you can
troubleshoot whether you're going flat 'cause you're tired
or because the composer/arranger intended it.

-Carl

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Messages in this topic (10)
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 10:06 pm (PST)
[I wrote...]
> Just take this example:
>
> http://lumma.org/music/score/Retrofit_JI.pdf
> (this is 4MB, might take a while to load)
>
> It's a formative showing of the notation I suggested in my
> previous message, but I haven't gone through and made sure
> the commas don't accumulate. I really should get around to
> doing that one day.

There are 24 root changes in the piece:

9/8
3/2
7/4
9/8
6/5
9/8
16/9
8/7
28/15
60/49
10/7
3/2
4/3
9/8
27/16
3/2
64/35
7/4
9/8
21/16
4/3
9/8
8/5
9/8

This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
or about 94 cents. I don't know if that's up or down, but
I don't think it'll be a problem in a 3-minute piece with
24 root changes.

-Carl

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Messages in this topic (10)
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 11:40 pm (PST)
> This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
> or about 94 cents. I don't know if that's up or down, but
> I don't think it'll be a problem in a 3-minute piece with
> 24 root changes.

One *can* write a neoclassical piece in 12-tET, fancifully
assign it an 11-limit adaptive JI tuning, and come out OK.

-Carl

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Messages in this topic (10)
2. Re: Kirkwood gaps scale
Posted by: "George D. Secor" gdsecor@yahoo.com gdsecor
Tue Oct 24, 2006 2:13 pm (PST)
--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Robert walker" <robertwalker@>
wrote:
>
> > I had a go at retuning it to 17-et - a similarly sized semitone
> though not otherwise very close, anyway, this is the mode I used,
> closest to your scale:
> > 0 3 4 6 7 10 13 14 17
> > and here is the result:
>
> Here are retunings to 31, 34 (the sharp 7 version), 46, 68, and 99,
> as well as the JI original. I'm fond of the extra shimmer 99 adds
on,
> though it generally sounds a lot like the JI version. Anyway, people
> can compare to 31 and see if sharp fifths are really melodically
> better as George claims.
>
> http://bahamas.eshockhost.com/~xenharmo/midi/examples/kirkwood/

Hi Gene & Robert,

I got a chance to listen to the Kirkwood gaps (original & all of the
retunings) last night (several times) to decide if I had any
preferences. I observed that 34 is the only tuning that has both a
good 5/4 and 5/3 and also has the Archytas comma tempered out in the
upward jump of the fifth between 7/4 and 4/3 (at 0:27-0:28). But I
can't say that it's clearly my favorite for that reason, because I
also liked 46 and 68 better than the others.

In the paper, I didn't mean to give the impression that wide fifths
are necessarily better melodically under all circumstances -- I was
writing specifically about a diatonic scale, where wide fifths result
in significantly narrower-than-12ET diatonic semitones. Several
times in the paper (pp. 59, 74, 75), I made statements to the effect
that chromatic semitones in 31-ET could be used to enhance the
melodic effect of that (narrow-fifth) tuning, so the size of the
fifths is, under more general circumstances, beside the point.

If your tuning has dozens of tones/octave, you then have
opportunities to alter the mood on the fly by making subtle
substitutions in your scale subset, so it's not wise to draw
conclusions too quickly about one of these tunings vs. another.

--George

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Messages in this topic (9)
3. Re: History again - first mention of 55-division?
Posted by: "threesixesinarow" CACCOLA@NET1PLUS.COM threesixesinarow
Tue Oct 24, 2006 4:33 pm (PST)
--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> ...
> What Sauveur said in 1701 would be good to know too, though I'm
> waiting for that article on order.

http://gallica.bnf.fr/ark:/12148/bpt6k3503q/f6.item
"Système générale des Intervalles des Sons" 1701, page 299
http://gallica.bnf.fr/ark:/12148/bpt6k3489j/f420.table
"Table générale des Systemes temperés de Musique." 1707, article page
203, commentary page 117
http://gallica.bnf.fr/ark:/12148/bpt6k35149
"Table generale des Systemes temperés de Musique " 1711, article page
307, commentary page 80
http://gallica.bnf.fr/ark:/12148/bpt6k3516x/f449.table
"Rapport des Sons des Cordes d'Instruments de Musique aux Fléches des
Cordes; Et nouvelle détermination des Sons fixes." 1713, page 324

all in _Histoire de l'Académie royale des sciences avec les mémoires
de mathématique et de physique tirés des registres de cette Académie_

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Messages in this topic (3)
1c. Re: Hyper-MOS question
Posted by: "Gene Ward Smith" genewardsmith@coolgoose.com genewardsmith
Wed Oct 25, 2006 2:20 pm (PST)
--- In tuning@yahoogroups.com, c.m.bryan <chrismbryan@...> wrote:

> Is there some pattern that I'm missing?

Robert means it has the trivalence property; for every interval class
besides octaves, there are exactly three members. That is, for single
scale steps you get 16/15, 10/9 and 9/8, for four scale steps you get
64/45, 40/27 and 3/2, and so forth.

There are other ways to generalize the MOS property--for instance, any
Fokker block or order-preserving homomorphic image of a Fokker block.
No doubt a great deal of work could be expended on the question of the
relationships between these sorts of things.

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Messages in this topic (3)
2a. Re: towards a hyper MOS
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Wed Oct 25, 2006 10:23 am (PST)
Wow, I wish I understood this. Dan, any chance you could write
up a version for someone who doesn't know what an [a,b] index
or silver-weighted generator are? What framework are you
working in here... for exmaple, how do you know the thirds
of "[1,6] = [1,5,1]" will be 420 cents?

-Carl

--- In tuning@yahoogroups.com, "D.Stearns" <STEARNS@...> wrote:
> OK, I finally got it...
>
> EUREKA:
>
> By using a one- to two-dimensional conversion it is possible to
> convert any given [a,b] index into an [a,b,c] index that is a
> three-term "MOS"/"trivalence" within a given periodicity.
>
> SILVER-WEIGHTED:
>
> In the past I've referred to the scales created by a silver-weighted
> generator as a sort of generalized Pythagorean scale. I think this
> is a useful frame of reference for converting [a,b] indexes into
> [a,b,c] indexes.
>
> Here's the generalized formula for deriving a silver-weighted
> generator for any given [a,b] index within a given periodicity.
>
> X = P/((a+S*b))*(z+S*y)
>
> Where:
>
> "P" = any given periodicity
>
> "S" = the sqrt(2)+1 silver constant which can be expressed by the
> series 1/2, 2/5, 5/12, 12/29, 29/70, ...
>
> "z"/"a", "y"/"b" = the adjacent fractions of a given [a,b] index
>
> and "X" = the resulting silver-weighted generator
>
> THE SCHISMATIC SILVER FOLD:
>
> Multiplying "X" by the forth term of the two-term [a,b] index and
> scaling it by "P" gives a silver comma -- or by way of the analogy,
> a generalized Pythagorean comma.
>
> Converting the one-dimensional silver chain into a two-dimensional
> schismatic silver fold will give the two to three stepsize
> cardinality.
>
> SOME "SIMPLE" EXAMPLES:
>
> These are all the 7-tone scale index conversions:
>
> [1,6] = [1,5,1]
> [2,5] = [2,2,3]
> [3,4] = [3,3,1]
> [4,3] = [3,1,3]
> [5,2] = [2,3,2]
> [6,1] = [1,1,5]
>
> Here are the 7-tone silver schismic two-dimensional, two generator,
> three-term MOS conversions.
>
> [1,6] = [1,5,1]
>
> 420---232----45
> / \ / \ / \
> / \ / \ / \
> 0--1013---826---639
>
> 0 45 232 420 639 826 1013 1200
> 0 187 374 593 780 968 1155 1200
> 0 187 406 593 780 968 1013 1200
> 0 219 406 593 780 826 1013 1200
> 0 187 374 561 607 794 981 1200
> 0 187 374 420 607 794 1013 1200
> 0 187 232 420 607 826 1013 1200
>
> [2,5] = [2,2,3]
>
> 376--1079---582
> / \ / \ / \
> / \ / \ / \
> 0---703---206---909
>
> 0 206 376 582 703 909 1079 1200
> 0 171 376 497 703 874 994 1200
> 0 206 326 532 703 824 1029 1200
> 0 121 326 497 618 824 994 1200
> 0 206 376 497 703 874 1079 1200
> 0 171 291 497 668 874 994 1200
> 0 121 326 497 703 824 1029 1200
>
> [3,4] = [3,3,1]
>
> 1066---742---419
> / \ / \ / \
> / \ / \ / \
> 0---876---553---229
>
> 0 229 419 553 742 876 1066 1200
> 0 190 324 513 647 837 971 1200
> 0 134 324 458 647 781 1010 1200
> 0 190 324 513 647 876 1066 1200
> 0 134 324 458 687 876 1010 1200
> 0 190 324 553 742 876 1066 1200
> 0 134 363 553 687 876 1010 1200
>
> [4,3] = [3,1,3]
>
> 898---534---169
> / \ / \ / \
> / \ / \ / \
> 0---836---471---107
>
> 0 107 169 471 534 836 898 1200
> 0 63 364 427 729 791 1093 1200
> 0 302 364 666 729 1031 1137 1200
> 0 63 364 427 729 836 898 1200
> 0 302 364 666 773 836 1137 1200
> 0 63 364 471 534 836 898 1200
> 0 302 409 471 773 836 1137 1200
>
> [5,2] = [2,3,2]
>
> 194---855---316
> / \ / \ / \
> / \ / \ / \
> 0---661---122---783
>
> 0 122 194 316 661 783 855 1200
> 0 72 194 539 661 733 1078 1200
> 0 122 467 590 661 1006 1128 1200
> 0 345 467 539 884 1006 1078 1200
> 0 122 194 539 661 733 855 1200
> 0 72 417 539 610 733 1078 1200
> 0 345 467 539 661 1006 1128 1200
>
> [6,1] = [1,1,5]
>
> 570---428---285
> / \ / \ / \
> / \ / \ / \
> 0--1057---915---772
>
> 0 285 428 570 772 915 1057 1200
> 0 143 285 487 630 772 915 1200
> 0 143 344 487 630 772 1057 1200
> 0 202 344 487 630 915 1057 1200
> 0 143 285 428 713 856 998 1200
> 0 143 285 570 713 856 1057 1200
> 0 143 428 570 713 915 1057 1200
>
> SOME "EXOTIC" EXAMPLES:
>
> The Bohlen-Pierce Lambda scale can be seen as [5,4] index where P =
> 1:3. Converting this into a silver schismic two-dimensional, two
> generator, three-term MOS results in a [5,4] = [4,1,4] index
> conversion and the following scale:
>
> 1535--1092---649---206
> / \ / \ / \ / \
> / \ / \ / \ / \
> 0--1459--1016---573---130
>
> 0 130 206 573 649 1016 1092 1459 1535 1902
> 0 76 443 519 886 962 1329 1405 1772 1902
> 0 367 443 810 886 1253 1329 1696 1826 1902
> 0 76 443 519 886 962 1329 1459 1535 1902
> 0 367 443 810 886 1253 1383 1459 1826 1902
> 0 76 443 519 886 1016 1092 1459 1535 1902
> 0 367 443 810 940 1016 1383 1459 1826 1902
> 0 76 443 573 649 1016 1092 1459 1535 1902
> 0 367 497 573 940 1016 1383 1459 1826 1902
>
> Using Olivier Messiaen's "concept of limited transposability" as a
> umbrella term under which I group all symmetric type scales, let's
> look at a couple of silver schismic "Trihill" (after David Clampitt)
> conversions within a given P.
>
> Here's the [8,2] static symmetrical decatonic (after Paul Erlich)
> where P = 1:2^(1/2):
>
> [4,1] = [3,1,1]
>
> 281---187
> / \ / \
> / \ / \
> 0---506---413
>
> 0 187 281 413 506 600 787 881 1013 1106 1200
> 0 94 226 319 413 600 694 826 919 1013 1200
> 0 132 226 319 506 600 732 826 919 1106 1200
> 0 94 187 374 468 600 694 787 974 1068 1200
> 0 94 281 374 506 600 694 881 974 1106 1200
>
> Here's the [6,3] "Tcherepnin scale" where P = 1:2^(1/3):
>
> [2,1] = [1,1,1]
>
> 181
> / \
> / \
> 0---309
>
> 0 181 309 400 581 709 800 981 1109 1200
> 0 128 219 400 528 619 800 928 1019 1200
> 0 91 272 400 491 672 800 891 1072 1200
>
> THE WRONG COMMA -- ETC:
>
> So there it is... a generalized one- to two-dimensional Myhill/MOS
> conversion formula!
>
> Last night I was attempting to generalize the syntonic comma, turns
> out that this was the right idea but the wrong comma... it was the
> generalization of the Pythagorean comma that allowed any given [a,b]
> index to morph into a "Trihill"/"MOS" [a,b,c] index.
>
> While the c/b, b/a proportions could perhaps be better addressed, I
> think the silver-weighting works well enough as it is for now...
>
> --Dan Stearns

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Messages in this topic (11)
2b. Re: towards a hyper MOS
Posted by: "Gene Ward Smith" genewardsmith@coolgoose.com genewardsmith
Wed Oct 25, 2006 2:39 pm (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Wow, I wish I understood this. Dan, any chance you could write
> up a version for someone who doesn't know what an [a,b] index
> or silver-weighted generator are? What framework are you
> working in here... for exmaple, how do you know the thirds
> of "[1,6] = [1,5,1]" will be 420 cents?

Seems like a rather severely specialized way of coming up with
trivalent scales. It would be nice if Scala could sort though a
directory and obtain a list of all scales with a given property.

I might try to see how far brute force gets me with given triples of
generators.

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Messages in this topic (11)
2c. Re: towards a hyper MOS
Posted by: "Gene Ward Smith" genewardsmith@coolgoose.com genewardsmith
Wed Oct 25, 2006 2:46 pm (PST)
--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> I might try to see how far brute force gets me with given triples of
> generators.

I started going through my catalog of 5-limit Fokker blocks, and found
that trivalent scales are extremely common among these. For larger
scales, we can derive 7-limit trivalent scales by marvel tempering.

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Messages in this topic (11)
3a. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Aaron Wolf" backfromthesilo@yahoo.com backfromthesilo
Wed Oct 25, 2006 2:53 pm (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > My idea for something to be taught to actual singers is something
> > like this:
> >
> > How about this? a notation that indicates the fundamental of
> > each chord in such a way that is simple and clear, and if it is
> > a pythagorean note, then indicate what power of 3 that it is
> > (in other words, like the old time barbershopper's "clock"
> > system that says how many fifths away on the circle of fifths
> > we are). Then use different note heads to simply indicate a
> > 3, 5, 7 etc identity in relation to the fundamental. If that
> > could be combined with a very simple indication of melodic
> > comma shifts (no need to specify different commas), that should
> > cover everything. What do you think?
>
> Sounds good, except I don't like the part about the fundamental's
> absolute pitch. First off, what happens if it isn't a power
> of 3? Secondly, I don't think performers (or composers) need
> to know this. There's no way a bass is going to be able to say,
> "Oh, 3129/8080, I was a bit flat there, wasn't I?"
>
> Performers are more likely to be interested in which note
> remain unchanged or nearly unchanged between chords. They
> can then tune pure to it/them. For nearly unchanged notes,
> yes I agree plenty of mileage could be gotten out of only one
> type of "nearly". I like drawing lines to make the common
> tones obvious. Then one doesn't even have to read music (or
> both clefs) to know that he's got to match the bari's previous
> pitch. Dotted lines make a good nearly, with perhaps a plus
> or minus sign above them to indicate direction.
>

Carl,

I like a lot of your ideas, but we always need to remember what the
mindset of a stylistic performer is. Everything you are talking about
comes from the world of POLYPHONY. When everyone is concerned about
what their particular part is, then your points make sense. And even
in barbershop there is definitely a bit of that. However,
barbershoppers do best when they hear themselves as part of a whole.

Put further, woodshedders historically were concerned first with what
the root harmonic movement was and secondly with simply whether they
were staying on a note or moving. But whether they move a third or
fifth isn't important, but what place they have in the new chord.

Watch Dave Steven's "What Are We Trying To Preserve" at barbershop.org
for a very wonderful entertaining presentation:

<a
href="http://stellent.spebsqsa.org/web/groups/public/documents/native/cb_00083.ram">What
Are We Trying To Preserve</a>

He discusses a system used by the old folks who did not read music:

They basically used a system to say that the chord was "home" or 1, 2,
3, or 4 away on the 5ths-system (meaning 3 identities). They just
held up 1 or 2 or 3 or 4 fingers, and everyone knew what that chord
sounded like and found a note that fit into that chord.

That is a great way to work when working in a style which is totally
based around those progressions.

My concept is to get away from everyone treating their own part as the
focus and go toward listening to the unified result of all the parts.

Anyway, recognizing that different sorts of music work differently, I
feel pretty confident that identifying the root of each chord is
extremely useful in barbershop specifically. Realize that 95% of all
barbershop roots fit a very small number of different roots.

Main point is I'm not imposing my ideas of what people should know.
The style is built on the tradition of knowing certain basic chord
progressions and then knowing how to sing the different identities of
each chord. So I'm trying to reflect that more clearly. Just because
classical singers are so polyphonically focused and don't listen to
each other doesn't mean I should compromise my notation to fit that
mindset.

-Aaron

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Messages in this topic (12)
3b. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Wed Oct 25, 2006 8:53 pm (PST)
>>> How about this? a notation that indicates the fundamental of
>>> each chord in such a way that is simple and clear, and if it
>>> is a pythagorean note, then indicate what power of 3 that it
>>> is (in other words, like the old time barbershopper's "clock"
>>> system that says how many fifths away on the circle of fifths
>>> we are). Then use different note heads to simply indicate a
>>> 3, 5, 7 etc identity in relation to the fundamental. If that
>>> could be combined with a very simple indication of melodic
>>> comma shifts (no need to specify different commas), that
>>> should cover everything. What do you think?
>>
>> Sounds good, except I don't like the part about the fundamental's
>> absolute pitch. First off, what happens if it isn't a power
>> of 3? Secondly, I don't think performers (or composers) need
>> to know this. There's no way a bass is going to be able to say,
>> "Oh, 3129/8080, I was a bit flat there, wasn't I?"
>>
>> Performers are more likely to be interested in which note
>> remain unchanged or nearly unchanged between chords. They
>> can then tune pure to it/them. For nearly unchanged notes,
>> yes I agree plenty of mileage could be gotten out of only one
>> type of "nearly". I like drawing lines to make the common
>> tones obvious. Then one doesn't even have to read music (or
>> both clefs) to know that he's got to match the bari's previous
>> pitch. Dotted lines make a good nearly, with perhaps a plus
>> or minus sign above them to indicate direction.
>
> Carl,
>
> I like a lot of your ideas, but we always need to remember
> what the mindset of a stylistic performer is. ... When
> everyone is concerned about what their particular part is,
> then your points make sense. ... woodshedders historically
> were concerned first with what the root harmonic movement
> was and secondly with simply whether they were staying on
> a note or moving. But whether they move a third or fifth
> isn't important, but what place they have in the new chord.

Hi Aaron,

There's obviously a disconnect somewhere, because you seem
to be agreeing with what I said!

> He discusses a system used by the old folks who did not read
> music:
>
> They basically used a system to say that the chord was "home"
> or 1, 2, 3, or 4 away on the 5ths-system (meaning 3 identities).
//
> That is a great way to work when working in a style which is totally
> based around those progressions.

Exactly. And not a good way for music that isn't.

> My concept is to get away from everyone treating their own part
> as the focus and go toward listening to the unified result of all
> the parts.

Such a focus would tend to come from a notation that emphasized
absolute pitch, would it not? And be discouraged by one that
emphasized vertical harmony?

> Anyway, recognizing that different sorts of music work
> differently, I feel pretty confident that identifying the
> root of each chord is extremely useful in barbershop
> specifically.

That's why the notation I suggested labels all the roots in
the music.

> Just because
> classical singers are so polyphonically focused and don't listen to
> each other doesn't mean I should compromise my notation to fit that
> mindset.

Where did you get this from?

-Carl

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Messages in this topic (12)
4a. Re: everybody loves microtones.......................
Posted by: "misterbobro" misterbobro@yahoo.com misterbobro
Wed Oct 25, 2006 5:45 pm (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > That's nice to know Cameron. Say, is that a feminine name? If so,
> > you are the second woman I get to meet on the tuning list.
> >
> > Cordially,
> > Oz.
>
> Hey Cameron, I googled you and found:
>
> http://cdbaby.com/cd/kosmolith2
>
> I didn't know you had this! Do tell (this is you, right?).
> The kosmolith website is apparently down at the moment.
>
> -Carl
>

Haha! or YoHoHo! I had almost forgotten about that, recorded in 2001-
after that I got fed up with 12-tET. come to think of it, there's
some freeform intonation (aka deliberately out of tune) on there
that I liked a lot.

Ozan- sorry I'm just a stinky hairy old guy like most of us here,
probably. :-)

-Cameron Bobro

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Messages in this topic (33)
4b. Re: everybody loves microtones.......................
Posted by: "Ozan Yarman" ozanyarman@ozanyarman.com ozanyarman
Thu Oct 26, 2006 12:30 am (PST)
Stinky hairy ol' guy you say? Would that perchance make me the bigfoot? LOL

Interesting CD on 'slave trade', if I got that right. Which piece would you
recommend as best?

Oz.

----- Original Message -----
From: "misterbobro" <misterbobro@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 26 Ekim 2006 Perþembe 3:33
Subject: [tuning] Re: everybody loves microtones.......................

> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> > >
> > > That's nice to know Cameron. Say, is that a feminine name? If so,
> > > you are the second woman I get to meet on the tuning list.
> > >
> > > Cordially,
> > > Oz.
> >
> > Hey Cameron, I googled you and found:
> >
> > http://cdbaby.com/cd/kosmolith2
> >
> > I didn't know you had this! Do tell (this is you, right?).
> > The kosmolith website is apparently down at the moment.
> >
> > -Carl
> >
>
> Haha! or YoHoHo! I had almost forgotten about that, recorded in 2001-
> after that I got fed up with 12-tET. come to think of it, there's
> some freeform intonation (aka deliberately out of tune) on there
> that I liked a lot.
>
> Ozan- sorry I'm just a stinky hairy old guy like most of us here,
> probably. :-)
>
> -Cameron Bobro
>

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Messages in this topic (33)
5a. Towards a hyper MOS
Posted by: "Robert walker" robertwalker@robertinventor.com robert_inventor5
Wed Oct 25, 2006 9:12 pm (PST)
Welcome to the Alternate Tunings Mailing List.Hi Gene,

As I remember it, in fact Dan's
method here seems to be a rather general one involving
creating a trivalent scale given a choice of 3 numbers,
and some constants. At least as general as the normal
way of making a MOS in terms of the family of
scales you can make with it.

I vaguely remember that there was some connection
with n-ets maybe, using those numbers to select
particular subsets of a larger n-et, or maybe that was
some other closely related topic, we are talking about
a discussion that took place over five years ago now
and my memory of it isn't that clear. At any rate
there is a particular construction that
he is describing there which generates those
scales in a recursive way.

It took a lot of explaining and I came very close to understanding
it when we had to break off.

In that post, he uses a number
of shorthands like silver weighting etc and his a b c notation,
all of which have to be explained before one can know what they mean
- they do refer to particular very specific constructions
that need to be described and understood, and there
are many assumed details that one can't
know from reading the post, just
like a mathematical paper relying on
results and definitions in earlier papers.

Also do you have any ideas about his conjecture that
the only completely rational trivalent scales are the
ones with an odd number of notes? He may even
have proved it or got very close
to doing so, I remember discussing that with
him too, and it is rather intriguing.

In our discussions, we found that it
takes time on both sides, for the
mathematician listener to figure out
what he is saying, also for him to work out
how to say it and to answer all the questions.
I think what it is is that he has figured out some of
the mathematical techniques he needed
largely for himself, so has got used to rather
individual ways of presenting it, but whenever I have had the
opportunity to follow it through, the reasoning
has been sound, the results interesting, and
often coming at the ideas with a novel slant.
There is a lot more to what he says than
often appears at first sight when one reads it.
It is well worth the effort on both sides if
one has the time to do it.

If no-one else can describe his technique to me, I
hope to come back to the discussion
when I have more time.

BTW I did a comprehensive search of the
SCALA archive for trivalent scales and higher
valency scales ,and you will still find it somewhere in the
files area under my name probably.
I can't remember how I did it, except that it
involved some programming and I have a vague
idea I may have somehow used FTS to prepare
the list of scales, but can't remember how.

Just did a search of my desktop for "trivalent" and found a note
about it where I mention that the search turned up
only one trivalent scale with an even number of notes
which was:

arabic1.scl | From Fortuna. Try C or G major
>
> 1/1 100 cents 200 cents 300 cents 350 cents 500 cents 600 cents 700
cents 800 cents 900
> cents 1000 cents 1050 cents 2/1
>
> Or in 24-tet:
> 0 2 4 6 7 10 12 14 16 18 20 21 24
> steps:
> 2 2 2 1 3 2 2 2 2 2 1 3

It also turned up a number of modes of et scales
as trivalent, and they were all with even
number of notes, so I wonder if it goes both
ways, that all et trivalent scales have an
even number of notes and that all rational
trivalent scales have an odd number,
at least the data seems to suggest
that as a conjecture.

Thanks,

Robert

----- Original Message -----
From: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Sent: Wednesday, October 25, 2006 11:50 AM
Subject: [tuning] Digest Number 4177

Welcome to the Alternate Tunings Mailing List.
Messages In This Digest (6 Messages)
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Aaron Wolf
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track) From: Carl Lumma
2. Re: Kirkwood gaps scale From: George D. Secor
3. Re: History again - first mention of 55-division? From: threesixesinarow
View All Topics | Create New Topic Messages
1a. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Aaron Wolf" backfromthesilo@yahoo.com backfromthesilo
Tue Oct 24, 2006 8:32 am (PST)
--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Thanks for the enthusiastic encouragement. I wasn't going to
> > bother, but fine... here it is:
> >
> > http://ozmusic.com/aaron/mp3/WMeetAgain/WMA.pdf
> >
> > I wrote in pencil and I scanned it quickly, it isn't perfect.
>
> Thanks! I'll try to set aside time this weekend to crunch on
> this.
>
> > I've been trying to figure out what notation will be easiest
> > for me to read musically, easiest to use when editing on the
> > computer, and easiest to explain to others...
>
> From my point of view, you aren't there yet. :)
>
> What did you think of my suggestion (did it make any sense
> to you)?
>

I understand a little, but I feel neither of us is really there.
I do want to start having commas more apparent.

My idea for something to be taught to actual singers is something like
this:

How about this? a notation that indicates the fundamental of each
chord in such a way that is simple and clear, and if it is a
pythagorean note, then indicate what power of 3 that it is (in other
words, like the old time barbershopper's "clock" system that says how
many fifths away on the circle of fifths we are). Then use different
note heads to simply indicate a 3, 5, 7 etc identity in relation to
the fundamental. If that could be combined with a very simple
indication of melodic comma shifts (no need to specify different
commas), that should cover everything. What do you think?

-Aaron

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Messages in this topic (10)
1b. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 8:59 pm (PST)
> My idea for something to be taught to actual singers is something
> like this:
>
> How about this? a notation that indicates the fundamental of
> each chord in such a way that is simple and clear, and if it is
> a pythagorean note, then indicate what power of 3 that it is
> (in other words, like the old time barbershopper's "clock"
> system that says how many fifths away on the circle of fifths
> we are). Then use different note heads to simply indicate a
> 3, 5, 7 etc identity in relation to the fundamental. If that
> could be combined with a very simple indication of melodic
> comma shifts (no need to specify different commas), that should
> cover everything. What do you think?

Sounds good, except I don't like the part about the fundamental's
absolute pitch. First off, what happens if it isn't a power
of 3? Secondly, I don't think performers (or composers) need
to know this. There's no way a bass is going to be able to say,
"Oh, 3129/8080, I was a bit flat there, wasn't I?"

Performers are more likely to be interested in which note
remain unchanged or nearly unchanged between chords. They
can then tune pure to it/them. For nearly unchanged notes,
yes I agree plenty of mileage could be gotten out of only one
type of "nearly". I like drawing lines to make the common
tones obvious. Then one doesn't even have to read music (or
both clefs) to know that he's got to match the bari's previous
pitch. Dotted lines make a good nearly, with perhaps a plus
or minus sign above them to indicate direction.

That takes care of shifts. Drift I think is best shown as a
cumulative cents offset from concert pitch. Every time the
offset changes direction -- say it's been going flat and
starts going sharp -- the current offset should be printed
above the barline. Say we start out at A=440, go 20 cents
flat every other bar for 5 bars, then go 20 cents sharp every
other bar for 5 bars and end on A=440. The notation would
show "-100" above the 6th barline and "+-0" above the last
barline. The point is that this is something that should
happen naturally if one tunes pure and respects the shifts
notation (above). It's just there as a check so you can
troubleshoot whether you're going flat 'cause you're tired
or because the composer/arranger intended it.

-Carl

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Messages in this topic (10)
1c. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 10:06 pm (PST)
[I wrote...]
> Just take this example:
>
> http://lumma.org/music/score/Retrofit_JI.pdf
> (this is 4MB, might take a while to load)
>
> It's a formative showing of the notation I suggested in my
> previous message, but I haven't gone through and made sure
> the commas don't accumulate. I really should get around to
> doing that one day.

There are 24 root changes in the piece:

9/8
3/2
7/4
9/8
6/5
9/8
16/9
8/7
28/15
60/49
10/7
3/2
4/3
9/8
27/16
3/2
64/35
7/4
9/8
21/16
4/3
9/8
8/5
9/8

This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
or about 94 cents. I don't know if that's up or down, but
I don't think it'll be a problem in a 3-minute piece with
24 root changes.

-Carl

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Messages in this topic (10)
1d. Re: Score PDF for barbershop (was: re: One more barbershop JI track)
Posted by: "Carl Lumma" clumma@yahoo.com clumma
Tue Oct 24, 2006 11:40 pm (PST)
> This adds up to 3^18 * 5^-2 * 7^-1, or 387420489/367001600,
> or about 94 cents. I don't know if that's up or down, but
> I don't think it'll be a problem in a 3-minute piece with
> 24 root changes.

One *can* write a neoclassical piece in 12-tET, fancifully
assign it an 11-limit adaptive JI tuning, and come out OK.

-Carl

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Messages in this topic (10)
2. Re: Kirkwood gaps scale
Posted by: "George D. Secor" gdsecor@yahoo.com gdsecor
Tue Oct 24, 2006 2:13 pm (PST)
--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Robert walker" <robertwalker@>
wrote:
>
> > I had a go at retuning it to 17-et - a similarly sized semitone
> though not otherwise very close, anyway, this is the mode I used,
> closest to your scale:
> > 0 3 4 6 7 10 13 14 17
> > and here is the result:
>
> Here are retunings to 31, 34 (the sharp 7 version), 46, 68, and 99,
> as well as the JI original. I'm fond of the extra shimmer 99 adds
on,
> though it generally sounds a lot like the JI version. Anyway, people
> can compare to 31 and see if sharp fifths are really melodically
> better as George claims.
>
> http://bahamas.eshockhost.com/~xenharmo/midi/examples/kirkwood/

Hi Gene & Robert,

I got a chance to listen to the Kirkwood gaps (original & all of the
retunings) last night (several times) to decide if I had any
preferences. I observed that 34 is the only tuning that has both a
good 5/4 and 5/3 and also has the Archytas comma tempered out in the
upward jump of the fifth between 7/4 and 4/3 (at 0:27-0:28). But I
can't say that it's clearly my favorite for that reason, because I
also liked 46 and 68 better than the others.

In the paper, I didn't mean to give the impression that wide fifths
are necessarily better melodically under all circumstances -- I was
writing specifically about a diatonic scale, where wide fifths result
in significantly narrower-than-12ET diatonic semitones. Several
times in the paper (pp. 59, 74, 75), I made statements to the effect
that chromatic semitones in 31-ET could be used to enhance the
melodic effect of that (narrow-fifth) tuning, so the size of the
fifths is, under more general circumstances, beside the point.

If your tuning has dozens of tones/octave, you then have
opportunities to alter the mood on the fly by making subtle
substitutions in your scale subset, so it's not wise to draw
conclusions too quickly about one of these tunings vs. another.

--George

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Messages in this topic (9)
3. Re: History again - first mention of 55-division?
Posted by: "threesixesinarow" CACCOLA@NET1PLUS.COM threesixesinarow
Tue Oct 24, 2006 4:33 pm (PST)
--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> ...
> What Sauveur said in 1701 would be good to know too, though I'm
> waiting for that article on order.

http://gallica.bnf.fr/ark:/12148/bpt6k3503q/f6.item
"Système générale des Intervalles des Sons" 1701, page 299
http://gallica.bnf.fr/ark:/12148/bpt6k3489j/f420.table
"Table générale des Systemes temperés de Musique." 1707, article page
203, commentary page 117
http://gallica.bnf.fr/ark:/12148/bpt6k35149
"Table generale des Systemes temperés de Musique " 1711, article page
307, commentary page 80
http://gallica.bnf.fr/ark:/12148/bpt6k3516x/f449.table
"Rapport des Sons des Cordes d'Instruments de Musique aux Fléches des
Cordes; Et nouvelle détermination des Sons fixes." 1713, page 324

all in _Histoire de l'Académie royale des sciences avec les mémoires
de mathématique et de physique tirés des registres de cette Académie_

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Messages in this topic (8)
5b. Towards a hyper MOS
Posted by: "Robert walker" robertwalker@robertinventor.com robert_inventor5
Wed Oct 25, 2006 9:13 pm (PST)
Welcome to the Alternate Tunings Mailing List.Hi Gene,

As I remember it, in fact Dan's
method here seems to be a rather general one involving
creating a trivalent scale given a choice of 3 numbers,
and some constants. At least as general as the normal
way of making a MOS in terms of the family of
scales you can make with it.

I vaguely remember that there was some connection
with n-ets maybe, using those numbers to select
particular subsets of a larger n-et, or maybe that was
some other closely related topic, we are talking about
a discussion that took place over five years ago now
and my memory of it isn't that clear. At any rate
there is a particular construction that
he is describing there which generates those
scales in a recursive way.

It took a lot of explaining and I came very close to understanding
it when we had to break off.

In that post, he uses a number
of shorthands like silver weighting etc and his a b c notation,
all of which have to be explained before one can know what they mean
- they do refer to particular very specific constructions
that need to be described and understood, and there
are many assumed details that one can't
know from reading the post, just
like a mathematical paper relying on
results and definitions in earlier papers.

Also do you have any ideas about his conjecture that
the only completely rational trivalent scales
(Message over 64 KB, truncated)

Robert walker escreveu:
> Welcome to the Alternate Tunings Mailing List.Hi Gene,
[...]

Huh! A message of 252kB!!!
Please remove the old messages when replying!
Thanks,
Hudson

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