Yahoo Tuning Groups Ultimate Backup tuning Hypothesis, and clarification for Monz.

Hypothesis: If you temper out all but one of the unison vectors in a
periodicity block, you get a distributionally even scale.

The terms "commatic" and "chromatic" unison vectors came from Paul
Hahn, and their meaning is completely dependent on the context of the
system in which they are used. My hypothesis concerns many such
systems.

The unison vectors you temper out are called "commatic" unison
vectors. In my paper I don't assume that they are necessarily tempered
out, or if they are, whether adaptive JI is used to do so, but in any
case notes separated by a commatic unison vector get the same name.

The unison vector you don't temper out is called a "chromatic" unison
vector. The meaning of this is clear by analogy with the usual meaning
of chromatic alterations in common-practice Western diatonic 5-limit
music.

Examples:

1. Common-practice Western diatonic 5-limit music.
The diatonic scale is the periodicity block defined by the two unison
vectors 81:80 (the syntonic comma) and 25:24 (the chromatic semitone).
Tempering out the syntonic comma (that is, treating 81:80 as the
commatic unison vector) but not the chromatic semitone (that is,
treating the 25:24 as the chromatic unison vector) leads to meantone
temperament. The diatonic scale is distributionally even in meantone
temperament:
L s L L L s L (and rotations thereof).
135:128 can substitute for 25:24 as the chromatic unison vector if you
wish.

2. My symmetrical decatonic system for 7-limit music.
The symmetrical decatonic scale is the periodicity block defined by
the three unison vectors 64:63, 50:49, and 49:48. Tempering out the
64:63 and 50:49 (that is, treating those as the commatic unison
vectors) but not the 49:48 (that is, treating that as the chromatic
unison vector) leads to "paultone" temperament (close to 22-tET). The
decatonic scale is distributionally even in "paultone" temperament:
s s s s L s s s s L (and rotations thereof).
225:224 can substitute for either 64:63 or 50:49 as the commatic
unison vectors if you wish, and 25:24 or 28:27 can substitute for
49:48 as the chromatic unison vector if you wish.

3. Dave Keenan's 31-tone MIRACLE scale for 11-limit music.
The 31-tone scale is the periodicity block defined by the four unison
vectors (if I recall correctly) 441:440, 385:384, 225:224, and 81:80.
Treating the first three as commatic unison vectors and the last one
as a chromatic unison vector leads to the distibutionally even scale
(that is especially well-represented in 72-tET) that Dave described.
Many substitutions are possible for the commatic and chromatic unison
vectors.

Other examples abound in Dave Keenan's work.

So, can anyone prove this hypothesis?

P.S. If you leave two of the unison vectors un-tempered-out, you get
the best candidate (I think) for what Carl Lumma was reaching for in
suggesting a 2-dimensional generalization of an MOS. A periodicity
block in strict JI with n unison vectors (or, equivalently, n distinct
prime factors) would be the n-dimensional generalization of an MOS.

🔗paul@stretch-music.com

🔗David J. Finnamore <daeron@bellsouth.net>

Paul Erlich wrote:

> 3. Dave Keenan's 31-tone MIRACLE scale for 11-limit music.
> The 31-tone scale is the periodicity block defined by the four unison
> vectors (if I recall correctly) 441:440, 385:384, 225:224, and 81:80.

I don't understand unison vectors yet so the following might be obvious to the others following
this thread. (Or, OTOH, I may be crawling out too far on a limb.) But it was interesting to me to
see how those commas break down into their constituent primes in a way that reveals, I think, some
level of balance and organization.

81 = 3^4 : 80 = 5 (* 2^4)
225 = 5^2 * 3^2 : 224 = 7 (* 2^5)
385 = 11 * 7 * 5 : 384 = 3 (* 2^7)
441 = 7^2 * 3^2 : 440 = 11 * 5 (* 2^3)

Comma Contains To this power
primes | 3 | 5 | 7 |11 |
------------------------------------
81:80 | 3,5 | 4 | 1 | | |
225:224 | 3,5,7 | 2 | 2 | 1 | |
385:384 | 3,5,7,11 | 1 | 1 | 1 | 1 |
441:440 | 3,5,7,11 | 2 | 1 | 2 | 1 |
------------------------------------
TOTALS | 9 | 5 | 4 | 2 |

Ain't that purdy? Perhaps if one started with a "balanced" set of "unison vector generators," so
to speak, it would lead to more scales with remarkable properties. Maybe that's how Dave Keenan
came up with it in the first place?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_22135.html#22135

> Hypothesis: If you temper out all but one of the unison vectors
> in a periodicity block, you get a distributionally even scale.
>
> The terms "commatic" and "chromatic" unison vectors came from
> Paul Hahn, and their meaning is completely dependent on the
> context of the system in which they are used. My hypothesis
> concerns many such systems.
>
> <etc.>

Right - I had a very strong suspicion that you chose to
use those two terms the way you did because of the context
in which they were embedded.

Thanks very much for this post, Paul. I think it made things
a lot clearer than if I had simply "read your paper again".

I'm catching up on two days of not reading the list, so I see
there are some responses to this that may clarify even more.
But I wanted to respond quickly to this post.

(And at last, my DSL is working! Now I can cram 13 times as
much downloading into my internet addiction time!)

-monz
http://www.monz.org
"All roads lead to n^0"

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "David J. Finnamore" <daeron@b...> wrote:
> Comma Contains To this power
> primes | 3 | 5 | 7 |11 |
> ------------------------------------
> 81:80 | 3,5 | 4 | 1 | | |
> 225:224 | 3,5,7 | 2 | 2 | 1 | |
> 385:384 | 3,5,7,11 | 1 | 1 | 1 | 1 |
> 441:440 | 3,5,7,11 | 2 | 1 | 2 | 1 |
> ------------------------------------
> TOTALS | 9 | 5 | 4 | 2 |
>
> Ain't that purdy?

Yes.

> Perhaps if one started with a "balanced" set of
"unison vector generators," so
> to speak, it would lead to more scales with remarkable properties.

Perhaps. But my intuition says there's no real connection between the
two kinds of balance. Mainly because there is no single matrix that
uniquely corresponds to the tuning. There are other sets of unison
vectors that correspond, with different cumulative powers of the 4
primes.

> Maybe that's how Dave Keenan
> came up with it in the first place?

No. I extracted a set of unison vectors from the tuning after the
fact. Actually, I knew I was distributing the 224:225 and 384:385
(which happen to point in approximately the same direction) and I knew
I was avoiding comma steps (80:81), but I didn't have any candidates
for the fourth vector until I'd finished. In fact I still didn't have
them in a form that explicitly contained 384:385 until Kees rearranged
them.

The path leading to this tuning started way back with Carl Lumma
presenting a particular 12-tone 7-limit just scale (as a lattice) and
asking whether anyone thought there was a better way to "use" the
224:225 septimal kleisma in 12 notes (there wasn't). Both Paul and I
suggested distributing the kleisma (Paul in 72-EDO, me in a planar
temperament). I noticed that distributing the 224:225 also tended to
distribute the 384:385 reasonably well. I wanted to see how this
11-limit micro-temperament fared with more notes. I went to 22. Paul
Erlich said "it's uneven, why stop there". I went to 31 and found it
quite even (3 step sizes).

Then a long time passed until Jo Pehrson asked for some good 19-tone
subsets of 72-EDO. I realised that some would come from the 72-EDO
version of that 31 tone tuning. I posted the full 31 of 72. Paul
Erlich realised that it was a MOS with a 7/72 octave generator and
would therefore have a 21 tone MOS subset which was improper but
turned out to be CS. Graham Breed pointed out that the 21-tone scale
contained three 7-tone neutral third MOS and showed us how to draw
good lattices of it. etc.

🔗paul@stretch-music.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "David J. Finnamore" <daeron@b...> wrote:
> > Comma Contains To this power
> > primes | 3 | 5 | 7 |11 |
> > ------------------------------------
> > 81:80 | 3,5 | 4 | 1 | | |
> > 225:224 | 3,5,7 | 2 | 2 | 1 | |
> > 385:384 | 3,5,7,11 | 1 | 1 | 1 | 1 |
> > 441:440 | 3,5,7,11 | 2 | 1 | 2 | 1 |
> > ------------------------------------
> > TOTALS | 9 | 5 | 4 | 2 |
> >
> > Ain't that purdy?
>
> Yes.
>
> > Perhaps if one started with a "balanced" set of
> "unison vector generators," so
> > to speak, it would lead to more scales with remarkable properties.
>
> Perhaps. But my intuition says there's no real connection between
the
> two kinds of balance. Mainly because there is no single matrix that
> uniquely corresponds to the tuning. There are other sets of unison
> vectors that correspond, with different cumulative powers of the 4
> primes.
>
> > Maybe that's how Dave Keenan
> > came up with it in the first place?
>
> No. I extracted a set of unison vectors from the tuning after the
> fact. Actually, I knew I was distributing the 224:225 and 384:385
> (which happen to point in approximately the same direction) and I
knew
> I was avoiding comma steps (80:81), but I didn't have any
candidates
> for the fourth vector until I'd finished. In fact I still didn't
have
> them in a form that explicitly contained 384:385 until Kees
rearranged
> them.
>
> The path leading to this tuning started way back with Carl Lumma
> presenting a particular 12-tone 7-limit just scale (as a lattice)
and
> asking whether anyone thought there was a better way to "use" the
> 224:225 septimal kleisma in 12 notes (there wasn't). Both Paul and
I
> suggested distributing the kleisma (Paul in 72-EDO, me in a planar
> temperament). I noticed that distributing the 224:225 also tended
to
> distribute the 384:385 reasonably well. I wanted to see how this
> 11-limit micro-temperament fared with more notes. I went to 22.
Paul
> Erlich said "it's uneven, why stop there". I went to 31 and found
it
> quite even (3 step sizes).
>
> Then a long time passed until Jo Pehrson asked for some good 19-
tone
> subsets of 72-EDO. I realised that some would come from the 72-EDO
> version of that 31 tone tuning. I posted the full 31 of 72. Paul
> Erlich realised that it was a MOS with a 7/72 octave generator and
> would therefore have a 21 tone MOS subset which was improper but
> turned out to be CS. Graham Breed pointed out that the 21-tone
scale
> contained three 7-tone neutral third MOS and showed us how to draw
> good lattices of it. etc.

What a great story! This has truly been one of the most wonderful
collaborative developments of this list!

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > The path leading to this [MIRACLE] tuning started way back with
Carl Lumma
...
>
> What a great story! This has truly been one of the most wonderful
> collaborative developments of this list!

Yes indeed. But so far it's only theory. We could use some more
collaboration from composers or performers.

We need some music that could only be played in this tuning, either
the 31-tone or preferably the 21-tone. Even if it is only a medly of
existing pieces showing off its neutral scales, hexanies, dekanies,
quasi-JI majors and minors etc, subsets which have probably never
before appeared together in so few notes to such great accuracy.

Is this a job for you, Joseph Pehrson? Forget 19. Go for 21.

By the way, Fokker discovered the 31-tone scale's 7-limit periodicity
block, but apparently not the quasi-just linear temperament of it that
we have found. See fokker-m.scl in the Scala archive. The description
reads:
"Fokker-M 7-limit periodicity block 81/80 & 225/224 & 1029/1024, KNAW
B72, 1969"

Regards,
-- Dave Keenan

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22135.html#22348

> --- In tuning@y..., paul@s... wrote:
> >
> > What a great story! This has truly been one of the most
> > wonderful collaborative developments of this list!
>
> Yes indeed. But so far it's only theory. We could use some
> more collaboration from composers or performers.
>
> We need some music that could only be played in this tuning,
> either the 31-tone or preferably the 21-tone. Even if it is
> only a medly of existing pieces showing off its neutral scales,
> hexanies, dekanies, quasi-JI majors and minors etc, subsets
> which have probably never before appeared together in so few
> notes to such great accuracy.

Hey, I'm too busy/lazy to search for it right now, so someone
post a Scala file (or degrees-of-72 or cents or whatever) of
the 21-tone, and I'll tune something up to it and start composing.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗jpehrson@rcn.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22135.html#22348

>
> We need some music that could only be played in this tuning, either
> the 31-tone or preferably the 21-tone. Even if it is only a medly
of existing pieces showing off its neutral scales, hexanies,
dekanies,
> quasi-JI majors and minors etc, subsets which have probably never
> before appeared together in so few notes to such great accuracy.
>
> Is this a job for you, Joseph Pehrson? Forget 19. Go for 21.
>

Hi Dave!

Well, I haven't started my piece for trombone and synth since my
computer broke down... and probably it will be another week before
everything is "together" again... so I have some time to reconsider...

But I was all ready to use the 19-tone scale that Paul Erlich found
for me in just...

And I thought this "miraculous" scale was, basically, in 31...out of
72...

OK... so you can see I'm getting mightily confused, so if somebody
can run this scale by me again in 21 tones with cents values, maybe I
will try it out rather than the 19.

Could you please explain, again, why it would be so much superior to
the 19-tone just scale (#3 for those keeping track of such things)
that Paul Erlich outlined for me??

If I can get this info. within the next few days, I will try out the
scale, and perhaps use it rather than the 19...

Trombone players won't know the difference, anyway. [That's just a
JOKE players...]

_________ ______ _____
Joseph Pehrson

🔗nanom3@home.com

--- > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> /tuning/topicId_22135.html#22348
>
> >
> > We need some music that could only be played in this tuning,
either
> > the 31-tone or preferably the 21-tone. Even if it is only a medly
> of existing pieces showing off its neutral scales, hexanies,
> dekanies,
> > quasi-JI majors and minors etc...

I don't have much to contribute regarding the mathematical
construction of scales but I have been playing with the Miracle scale
since Dave Keenan first published his full 31 tone of 72 EDO to try
and figure out what everyone was so excited about. What I find
musically, without purporting to understand all of its intricacies,
is that it contains an awful lot of really neat harmonies. So here
is my small contribution to this collaboration , a piece entitled,
appropriately for me, Nada Clue (Miracle of 72). It uses Graham
Breed's "high" scale, and some of the drones also use the orginal
Keenan 31 tone scale. The tunings were realized using Robert Walkers
FTS , Symbolic Composer, a Lisp program for the MAC and MetaSynth.
You can find it on my web site at
http://www.elucida.com/DNA_Music1.html, along with some work using
Jackies PhiMOS and Schumann Resonance tunings. Beware - they are
long downloads if you don't have high speed internet access.

Although I have little to say about the mathematics of 72 I do know
some interesting Numerological facts about it. It is related to
precession (25920/360 =72), it is of course half of the 144 choosen
souls, and in Gematria Chokmah, the sphere of wisdom, reduces to 72.
SOmehow I doubt any of these factoids relates to the scales inherent
properties but for us numerologists the appearance of coincidence is
a cause for joy:-)O

Mary
http://www.elucida.com

🔗Graham Breed <graham@microtonal.co.uk>

"monz" wrote:

> Hey, I'm too busy/lazy to search for it right now, so someone
> post a Scala file (or degrees-of-72 or cents or whatever) of
> the 21-tone, and I'll tune something up to it and start composing.

Hi Joe!

I'm going to need a word for the difference between 7 semitones and an
octave in Miracle tuning. So let's use "quomma" for somewhere between
a quartertone and a comma. Traditional names always refer to keys,
not pitches.

I've posted another copy of one of the files I posted in
</tuning/topicId_22183.html#22183>. The black notes
starting on C# above middle C (depending on your Scala setup) are the
nominals 0 to 9. So there's a semitone between each pair of black
notes, except for every other Bb-C# which is a quomma wider. You
could try a non-octave tuning where all semitones are equal. I found
this easy to get used to, but expect having octave repetition will be
better long-term.

Anyway, with this tuning each black note can be raised one quomma by
moving a key to the right. This change is what my "^" symbol refers
to. So the key to the right of the first C# is 0^. Where there are
two white notes between a pair of black notes, you can also move down
a quomma to get 0v, 2v, 5v and 7v.

The "high" means that when in doubt a key is a quomma higher then the
key to the left, rather than a quomma lower than the key to the right.
This also means the ambiguous notes are lower then the "low" tuning
so "high" could also be called "minor".

The 21 note Blackjack scale is all black notes plus all white notes a
quomma higher than a black note, plus 0v, a quomma lower than every
other C#.

You can check <http://x31eq.com/decimal_lattice.htm> for
clues on 11-limit harmony in decimal notation. Everything between
nominals approximates an 11-limit interval, except 4 semitones which
is the wolf in 31-equal but I find sounds okay in 72-equal. It
approximates a 21:16.

Have I mentioned neutral third scales before? 0 anti-Dorian is
0-1^-3-4^-6-7^-9-0 which fits the high tuning. 0 Rast is
0-2v-3-4^-6-8v-9-0 which doesn't. Both are subsets of the 10-note
neutral-third MOS 0-1^-2v-3-4^-5v-6-7^-8v-9-0 which almost fits.
These are good starting points, but you'll only cover a third of the
Miracle scale with neutral thirds.

------------miracle_24hi.scl----------------
24 note mapping for Erlich/Keenan Miracle scale
!
! high version, tuned to 72-equal
!
24
!
33.3333333333333333
66.6666666666666667
150.0
183.333333333333333
233.333333333333333
266.666666666666667
300.0
383.333333333333333
416.666666666666667
500.0
533.333333333333333
583.333333333333333
616.666666666666667
650.0
733.333333333333333
766.666666666666667
816.666666666666667
850.0
883.333333333333333
966.666666666666667
1000.0
1083.33333333333333
1116.66666666666667
1200.0
----------------end file--------------------

🔗paul@stretch-music.com

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > Is this a job for you, Joseph Pehrson? Forget 19. Go for 21.
> >
That's exactly what I've been telling Joseph.
>
> Hi Dave!
>
> Well, I haven't started my piece for trombone and synth since my
> computer broke down... and probably it will be another week before
> everything is "together" again... so I have some time to
reconsider...
>
> But I was all ready to use the 19-tone scale that Paul Erlich found
> for me in just...
>
> And I thought this "miraculous" scale was, basically, in 31...out
of
> 72...

Joseph, I know you have more important things in your life, but you
must have blacked out about a week of discussions, in which you
yourself were involved. Remember the lattice I posted that you called
a "hexany Mobius strip"? Remember all the talk about the "blackjack
scale"? In fact, I even have an Excel spreadsheet called "Pehrson"
with the scale on it.

Basically, once I realized that the MIRACLE scale had a single
generator, I saw that a 21-tone MOS based on the same generator would
take advantage of the same features of 72-tET that give the MIRACLE
scale so many "extra" consonances.
>
> OK... so you can see I'm getting mightily confused, so if somebody
> can run this scale by me again in 21 tones with cents values, maybe
I
> will try it out rather than the 19.

0.000
33.333
116.667
150.000
233.333
266.667
350.000
383.333
466.667
500.000
583.333
616.667
700.000
733.333
816.667
850.000
933.333
966.667
1050.000
1083.333
1166.667

>
> Could you please explain, again, why it would be so much superior
to
> the 19-tone just scale (#3 for those keeping track of such things)
> that Paul Erlich outlined for me??

Umm . . . you have a lot of posts to reread (or read for the first
time). Look for "21-tone" or "blackjack" in the subject line. But
firstly, when you did your 19-tET vs. 19-tJI (#3) comparison, I
remarked (off-list) that you should try a comparison with the
blackjack scale -- it would beat both.

🔗paul@stretch-music.com

🔗Graham Breed <graham@microtonal.co.uk>

Paul wrote:

> How did I miss this?
>
> http://x31eq.com/decimal_lattice.htm
>
> When did that go up?

At the weekend I think. And I mentioned it in a post you replied to.
Have you "blacked out" a week of discussions?

The lattice as it stands might make a good harmonically-based keyboard
layout, if only we had a keyboard to lay it out on. As an
improvement, how about shifting the "seven-limit" rows to the right,
so instead of

0^ 3^ 6^
2^ 5^ 8^ 1^
7^ 0 3 6
2 5 8 1

we have

0^ 3^ 6^
2^ 5^ 8^
0v 3 6
2 5 8

which fills out to

0^
2^ 5^ 8^
1^ 4^ 7^ 0
0^ 3^ 6^ 9^ 2 5 8
2^ 5^ 8^ 1 4 7 0v
0 3 6 9 2v 5v 8v
2 5 8 1v 4v 7v 0w
0v 3v 6v 9v
2v 5v 8v
0w

That means some melodies (hopefully common 11-limit ones ;) should be
easier. The harmonic template (numbers have a different meaning) is:

5
7
1 3 9 11

although it looks like it can't be consistently applied -- drat!. It
also doesn't naturally repeat over octaves. I'm not sure if the
layout could be made to fit a Wilson-Bosanquet keyboard comfortably
(Kraig?), but it should be better than

0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ 8^ 9^ 0
0 1 2 3 4 5 6 7 8 9 0v
0v 1v 2v 3v 4v 5v 6v 7v 8v 9v 0w
0w

Hmm, how about

0^ 3^ 6^
2^ 5^ 8^ 1^
7^ 0 3 6
2 5 8 1

he says, thinking as he types.

Graham

🔗ligonj@northstate.net

--- In tuning@y..., nanom3@h... wrote:
> --- > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > /tuning/topicId_22135.html#22348
> >
We need some music that could only be played in this tuning, either
the 31-tone or preferably the 21-tone. Even if it is only a medly of
existing pieces showing off its neutral scales, hexanies, dekanies,
quasi-JI majors and minors etc...
>
> So here
> is my small contribution to this collaboration , a piece entitled,
> appropriately for me, Nada Clue (Miracle of 72). It uses Graham
> Breed's "high" scale, and some of the drones also use the orginal
> Keenan 31 tone scale. The tunings were realized using Robert
Walkers
> FTS , Symbolic Composer, a Lisp program for the MAC and MetaSynth.
> You can find it on my web site at
> http://www.elucida.com/DNA_Music1.html, along with some work using
> Jacky's PhiMOS and Schumann Resonance tunings. Beware - they are
> long downloads if you don't have high speed internet access.

Mary,

As always - this is just incredible! Beautiful amibient piece! Love
it - and you!

Jacky Ligon

>
> Although I have little to say about the mathematics of 72 I do know
> some interesting Numerological facts about it. It is related to
> precession (25920/360 =72), it is of course half of the 144 choosen
> souls, and in Gematria Chokmah, the sphere of wisdom, reduces to
72.
> SOmehow I doubt any of these factoids relates to the scales
inherent
> properties but for us numerologists the appearance of coincidence
is
> a cause for joy:-)O
>
> Mary
> http://www.elucida.com

🔗paul@stretch-music.com

In the 7-limit, the MIRACLE scale is the periodicity block defined by
the unison vectors

2401:2400
225:224
81:80

with the 2401:2400 and 225:224 tempered out, but the 81:80 remaining
intact.

Graham's lattices show the periodicity at the 225:224 but not at the
2401:2400. Thus they're missing out on a lot of their possibilities.
A better 3-D orientation would remedy this by orienting these two
vectors approximately along the X and Y axes. I won't try that now
(Dave?).

The MIRACLE scale in 72-tET is
0 2 5 7 9 12 14 16 19
21 23 26 28 30 33 35 37
39 42 44 46 49 51 53 56
58 60 63 65 67 70 (72)

Here's a 7-limit lattice (standard orientation, unfortunately) that
uses symbols to show one member of each of six 2401:2400s:

46
/|\
60 / | \ .
39------/--9 \ .
/|\`. |/,'/|\`.\ .
53--/-|-\-23--/-|-\-65
/|\/. 2/,\/|\/.44 ,\/|\
67--/-|/,'/*\`.\|/,'/!\`.\| \
46--------16--/-|-\-58--/-|-\-28 \
9\-/-|/,'/T\/.37/,\/+\/.\7/,\/ `.\
60------\-30--/-|/,'-0-`.\|/,'42-`.\----12
\`. /,\/I\/.51----H---21--------63
\ 44--/\|/,'14-`.\|/,'56-`.\|/--26
\ |\/ 65--------35---------5
\|/,'28-`.\|/,'70 `.\|/
7--------49--------19
. \`.\|/,'/|`. \|/
. \ 63--/-----33
. \ | / 12
\|/
26

*: 37 hidden beneath
!: 7 hidden beneath
T: 51 hidden beneath
+: 21 hidden beneath
I: 65 hidden beneath
H: 35 hidden beneath

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_22135.html#22365

> Joseph, I know you have more important things in your life,

Not true! There is nothing more important in my life than Tuning
List! Doesn't everybody feel similarly?! Of course, it might be
nice to have a working computer so I can access it...

>but you
> must have blacked out about a week of discussions, in which you
> yourself were involved.

Usually I "black out" for much longer periods than this... and, after
all, the scale is calle the "blackjack" so why wouldn't I!

(I missed a good portion of the '60s for this reason...)

Remember the lattice I posted that you called
> a "hexany Mobius strip"? Remember all the talk about the "blackjack
> scale"? In fact, I even have an Excel spreadsheet called "Pehrson"
> with the scale on it.

Well, I'm happy again to have instigated something that I'm only
vaguely conscious about... It isn't the first time!

>
> Basically, once I realized that the MIRACLE scale had a single
> generator, I saw that a 21-tone MOS based on the same generator
would
> take advantage of the same features of 72-tET that give the MIRACLE
> scale so many "extra" consonances.
> >
> > OK... so you can see I'm getting mightily confused, so if
somebody
> > can run this scale by me again in 21 tones with cents values,
maybe
> I
> > will try it out rather than the 19.
>
> 0.000
> 33.333
> 116.667
> 150.000
> 233.333
> 266.667
> 350.000
> 383.333
> 466.667
> 500.000
> 583.333
> 616.667
> 700.000
> 733.333
> 816.667
> 850.000
> 933.333
> 966.667
> 1050.000
> 1083.333
> 1166.667
>
> >
> > Could you please explain, again, why it would be so much superior
> to
> > the 19-tone just scale (#3 for those keeping track of such
things)
> > that Paul Erlich outlined for me??
>
> Umm . . . you have a lot of posts to reread (or read for the first
> time).

Unfortunately, I *did* read all of them... the "absorption" was,
obviously, a different matter!

Look for "21-tone" or "blackjack" in the subject line. But
> firstly, when you did your 19-tET vs. 19-tJI (#3) comparison, I
> remarked (off-list) that you should try a comparison with the
> blackjack scale -- it would beat both.

Hmmm. Well, I will reread this stuff. Everybody seems so
enthusiastic about this scale, I *have* to try it!

However, it looks as though this is not an "octave repeating" scale,
correct??

Also, I was all ready to use 19-tone "meantone" notation for my new
piece... so this might entail a different approach, yes??

Thanks for the help!!!

_________ _______ _____
Joseph Pehrson

🔗paul@stretch-music.com

🔗Graham Breed <graham@microtonal.co.uk>

> Joseph Pehrson:
> >
> > However, it looks as though this is not an "octave repeating"
> scale,
> > correct??

Paul Erlich:

> It _is_ octave repeating.

Paul and I have both speculated about related non-octave scales. That
might be where the confusion comes from.

> > Also, I was all ready to use 19-tone "meantone" notation for my
new
> > piece...
>
> This would be a mistake for the 19-tJI (#3) scale. I thought you
were
> going to use 72-tET notation, or at least HEW . . . ?
>
> >so this might entail a different approach, yes??
>
> You'd use 72-tET notation, as blackjack and MIRACLE are subsets of
72-
> tET.

Well, you could do that. Or, as Blackjack can also be expressed as a
subset of 31-equal, you could still use meantone notation. Whatever
you or the performers find most convenient.

Say, is Miracle the temperament or the subset now?

Miracle tuning would ideally use a decimal notation, which a lot of my
posts have been working towards. But if you're about to write the
piece, you don't want to learn a new system of notation and teach it
to the players, do you?

Graham

🔗paul@stretch-music.com

🔗monz <joemonz@yahoo.com>

🔗jpehrson@rcn.com

🔗paul@stretch-music.com

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_22135.html#22382

> > Would you mind please posting the 72-tET ASCII notation
> > for "blackjack" for me??
>
> Didn't Monz do that already?

I think I gave the 72-EDO notation for the 31-out-of-72-EDO
"miracle" scale (can we stop using all capitals now?).

I just search the archives breifly and didn't find a notation
for "blackjack" (except for Graham's decimal notation).

I'll do it... hang on.

> Wait 'till you see this new blackjack lattice, folks . . .

I'm looking forward to it, Paul. I was going to send a post
suggesting that you expand the distance between lattice points
on your last lattice, so that you could fit in more notes.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗paul@stretch-music.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> I'm looking forward to it, Paul. I was going to send a post
> suggesting that you expand the distance between lattice points
> on your last lattice, so that you could fit in more notes.

I had a better idea. In the orientation Graham and I were using:

5
/|\
/ | \
/ 7 \
/,' `.\
1---------3

not only did the notes get too close together, but a whole dimension
of infinite extent got truncated because of overlapping notes getting
stacked on top of one another in the direction perpendicular to the
screen/page. So I changed the lattice orientation to this:

7
/|\
/ | \
/ 5 \
/,' `.\
1---------3

and now the lattice fills up the 2-D plane, continuing infinitely in
both directions, while overlaps are avoided. Tetrads and hexanies, of
course, are the same shape they were before.

I think Graham's otherwise excellent page misses out on a lot of the
power of the MIRACLE and blackjack scales by ignoring this extra
dimension.

In other words, Graham shows how the lattice repeats itself at the
225:224 but doesn't show how it repeats itself at the 2400:2401.

Stay tuned . . .

🔗monz <joemonz@yahoo.com>

🔗Graham Breed <graham@microtonal.co.uk>

Paul wrote:
> --- In tuning@y..., "Graham Breed" <graham@m...> wrote:
>
> > Paul and I have both speculated about related non-octave scales.
>
> I did?

Your first post titled "DAVE KEENAN'S MIRACLE SCALE":

"It looks like the generator itself would make for a really
interesting non-octave ET, with nearly pure 5:7:8:12:18:22 hexads."

> > Well, you could do that. Or, as Blackjack can also be expressed
as
> a
> > subset of 31-equal,
>
> Umm . . . I'm thinking of these scales as "quasi-just", having a
> maximum 3-4 cent deviation from JI in the consonances. So 31-equal
> won't do.

No, but 31-equal notation to be adjusted by performers might.

> > Say, is Miracle the temperament or the subset now?
>
> I thought it was always the 31-tone subset. Miraculous because it
has
> more 7-limit tetrads than notes.

So what do we call the temperament?

That feature isn't so special. All you need is an MOS with a lot more
notes than it takes to hold the tetrad. Say you have an n note scale
that only contains 2 tetrads: major and minor. Add a note on the
chain, and you get another pair of tetrads -- the original pair
transposed by the generator. Add n notes and you have a 2n note scale
with 2n+2 tetrads. So in this case it takes 8 steps to get a pair of
tetrads and 31 notes will have (31-8)*2=46 tetrads. Or something like
that. Any MOS with sufficient notes would do.

The temperament is miraculous in the efficiency with which it
approximates 11-limit JI.

Graham

🔗Graham Breed <graham@microtonal.co.uk>

🔗jpehrson@rcn.com

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:
>
> No, but 31-equal notation to be adjusted by performers might.

In 31-equal, the scale would have too many consonances, since the
81:80 vanishes.

>
> That feature isn't so special. All you need is an MOS with a lot
more
> notes than it takes to hold the tetrad. Say you have an n note
scale
> that only contains 2 tetrads: major and minor. Add a note on the
> chain, and you get another pair of tetrads -- the original pair
> transposed by the generator. Add n notes and you have a 2n note
scale
> with 2n+2 tetrads. So in this case it takes 8 steps to get a pair
of
> tetrads and 31 notes will have (31-8)*2=46 tetrads. Or something
like
> that.

There are "only" 36 tetrads.

>Any MOS with sufficient notes would do.
>
> The temperament is miraculous in the efficiency with which it
> approximates 11-limit JI.
>
And 7-limit JI! Just try to find something comparable. Then you'll
realize how special it is.

🔗Graham Breed <graham@microtonal.co.uk>

Paul wrote:

> not only did the notes get too close together, but a whole
dimension
> of infinite extent got truncated because of overlapping notes
getting
> stacked on top of one another in the direction perpendicular to the
> screen/page. So I changed the lattice orientation to this:
>
> 7
> /|\
> / | \
> / 5 \
> /,' `.\
> 1---------3
>
> and now the lattice fills up the 2-D plane, continuing infinitely in
> both directions, while overlaps are avoided. Tetrads and hexanies,
of
> course, are the same shape they were before.

You'll find dimensions of infinite extent get cluttered however you
try to get them into a finite diagram.

> I think Graham's otherwise excellent page misses out on a lot of the
> power of the MIRACLE and blackjack scales by ignoring this extra
> dimension.

I don't *ignore* this dimension! It's built right into the lattice,
before you even place any notes on it.

> In other words, Graham shows how the lattice repeats itself at the
> 225:224 but doesn't show how it repeats itself at the 2400:2401.

All the chords you show, I show. That's the same power, but with less
complexity. If I had a cylindrical screen, I'd wrap the 225:224 as
well.

BTW, if it helps anybody, 2401:2400 is (-5 -1 -2 4)H.

Graham

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:
>
> You'll find dimensions of infinite extent get cluttered however you
> try to get them into a finite diagram.

You'll find that's not the case, once you see the new lattice. It
should be up by the time you read this.
>
> > I think Graham's otherwise excellent page misses out on a lot of
the
> > power of the MIRACLE and blackjack scales by ignoring this extra
> > dimension.
>
> I don't *ignore* this dimension! It's built right into the lattice,
> before you even place any notes on it.

Well, I mean you can't _see_ how the _scale_ (i.e., subset) repeats
in one of the two dimensions of infinite extent.
>
> > In other words, Graham shows how the lattice repeats itself at the
> > 225:224 but doesn't show how it repeats itself at the 2400:2401.
>
> All the chords you show, I show. That's the same power, but with
less
> complexity.

Someone might not see certain progressions if they look at your
lattice.

🔗Graham Breed <graham@microtonal.co.uk>

Paul wrote:

> > No, but 31-equal notation to be adjusted by performers might.
>
> In 31-equal, the scale would have too many consonances, since the
> 81:80 vanishes.

A few minor thirds. They shouldn't get in the way -- unless you want
them to.

> So in this case it takes 8 steps to get a pair
> of
> > tetrads and 31 notes will have (31-8)*2=46 tetrads. Or something
> like
> > that.
>
> There are "only" 36 tetrads.

That should be 13 steps for a tetrad -- I was reading the wrong table.
So the calculation almost works. (31-(13-1)*2=36 because 13 noes
would give 2 not 0.

> >Any MOS with sufficient notes would do.
> >
> > The temperament is miraculous in the efficiency with which it
> > approximates 11-limit JI.
> >
> And 7-limit JI! Just try to find something comparable. Then you'll
> realize how special it is.

I'm fully aware of how special it is. Fortunately, as it's still tops
in the 7-limit, we don't need to sell 11-limit harmony to sell the
tuning.

I have been looking at some more of Dave Keenan's 11-limit generators,
and I'll post more on that sometime. The only way 10+1 can be beaten
is in the number of notes it takes for a complete chord. Meantone
only needs 10 for a 7-limit tetrad and only 11 (instead of 19) for a
9-limit pentad. A scale generated from 4 equal divisions of 15:8 (or
7 equal divisions of 3:1, or 271.3 cents) only needs 17 notes (instead
of 22) for an 11-limit hexad. 3 generators to a 4:3 (or 2 to a 6:5)
only needs 16 notes for a hexad, but it isn't in Dave's list and so is
probably a terrible approximation.

Graham

🔗monz <joemonz@yahoo.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

I've shown below, various subsets against the 31 notes arranged as a
chain of 7/72 oct generators. This way, you can slide any of the
subsets left or right along the chain and they will still work.
Unfortunately this means that it's up to you to take a particular
subset, in a particular position and sort the notes into pitch order
and convert them to cents. Let me know if you want me to do that (and
for which subsets in which positions).

You will need to view this in a wide window (about 95 characters
across), in a monospaced font, to make any sense of it.

Degrees of 72-EDO as a chain of 7-step (116.67c) generators
39 46 53 60 67 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49
56 63 70 5 12 19 26 33
+--------------------------------------31 note
MOS-proper---------------------------------+
+-----------------------21 note
MOS-improper----------------+
+--------11 note MOS-improper-+
+-----10 note MOS-proper---+
+-neutral+-triad--+
+-neutral+-tetrad-+--------+

+--------+-7-note-+-neutral+---3rd--+--MOS---+--------+ (Mohajira)

+-----+-----------+-----+-----+-----------+-----+
(Arist_diatdor)

+-----------------+-----+-----+-----+-----+-----+ (Arist_diatred)
+--+--------------+--+--+--------------+--+
(Xenakis_schrom)
5*7 1*5 1*7 3*5 3*7
1*3

+-----+--Hexany-1----+--+--------------+-----+
39 46 53 60 67 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49
56 63 70 5 12 19 26 33

+-----------------+--------+-Arabic-+--------+--------+---------------
--+

+-----------------+--+-----------+-----+----Bagpipe-4----+------------
-----+

+-----------------+--+----JI-major--+--+-----------------+------------
-----+

+-----------------+-----------------+--+--JI-minor----+--+------------
-----+

+-----------------+-----Pythagorean-+-pentatonic------+---------------
--+
5*7 1*5 1*7 3*5 3*7 5*9 1*3 7*9 1*9
3*9

+-----+--------------+--+---Dekany-1---+--+--+-----------+-----+------
-----------+
39 46 53 60 67 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49
56 63 70 5 12 19 26 33

5--------------7-----1----Harmonics----3-----------------9-------11
10 14 2 6
4 12
8
16

I'm sorry I haven't had time to write much explanation. Does the above
make any sense at all? The names in parenthesis are scala file names.
I've repeated the 72-EDO degree numbers on 3 lines to make it easier
to refer to them. Various 11-limit otonalities can be constructed by
referring to the "Harmonics" section at the bottom. Other hexanies and
dekanies are possible by choosing from different sets of 4 or 5
factors.

Regards,
-- Dave Keenan

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_22135.html#22382

> > > >
> > > > Also, I was all ready to use 19-tone "meantone" notation for
my new piece...
> > >
> > > This would be a mistake for the 19-tJI (#3) scale.
> >
> > Why is that?? At least I had all the "notes" and could
> > just "deviate" with cents values...
>
> Oh, if you're using cents values . . . OK.
>

Paul! I may be dense, but you must suspect that I'm EXCEEDINGLY
dense!

> > >
> > > You'd use 72-tET notation, as blackjack and MIRACLE are subsets
> of
> > 72-tET.
> > > >

You know... I'm getting excited about this now. I may entirely throw
out the 19 just idea and go with the new "miracles."

I think the 72-tET notation is eminently PLAYABLE... at least as easy
as the "cents" stuff I was going to "fiddle" with...

I think they can do it...

_________ _____ ____ ___
Joseph Pehrson

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> ...here's a pitch-height graph labelled with all the info I
> gave in message 22385:
>
> /tuning/files/monz/blackjack-pitch-
> height.jpg
>
>
> (Copy the link, paste it into an editor, remove the line-break,
> then copy that and paste it into your browser.)

The decatonic nature of this scale is quite clearly shown
on this graph. The entire structure is a series of 1/3-
and 5/6 semitone intervals. This can be thought of as
pairs of notes separated by 1/3-semitone, each pair
5/6-semitone apart, the pitch-height periodicity ocurring
10 times at every 7/6-semitone.

If this cycle were to be continued, the "leftover" single note
at the top (C<) begins a new cycle, 1/3-semitone below the
original one. The new pitches would fill in the spaces and
give the 31-tone subset. Continuing the process further
would give the 41-tone subset. And all of these remain within
the 72-EDO superset.

This information should be a good starting point for
exploration for those here who want to find out what's so
special about this scale, who are uncomfortable thinking
in terms of ratio-space, but have no problem thinking in
terms of pitch-height. It should help to uncover any
mystification about the properties of this scale in terms
of MOS, maximal evenness, etc.

Paul's latest lattice (message 22394) pretty much shows
what's so special about this scale in 11-limit ratio-space.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗PERLICH@ACADIAN-ASSET.COM

🔗monz <joemonz@yahoo.com>

🔗graham@microtonal.co.uk

Paul wrote:

> > You'll find dimensions of infinite extent get cluttered however you
> > try to get them into a finite diagram.
>
> You'll find that's not the case, once you see the new lattice. It
> should be up by the time you read this.

Ah, I see it's periodic in both directions. That's clever, but it's still
a mess and I can't make sense of it. Are you sure it's lined up right?
Perhaps fewer connections would help. Or a toroidal monitor.

At least Joseph Pehrson likes it.

Try connecting all the 11-limit intervals on my graph, and then see how
powerful it is!

> > > I think Graham's otherwise excellent page misses out on a lot of
> the
> > > power of the MIRACLE and blackjack scales by ignoring this extra
> > > dimension.
> >
> > I don't *ignore* this dimension! It's built right into the lattice,
> > before you even place any notes on it.
>
> Well, I mean you can't _see_ how the _scale_ (i.e., subset) repeats
> in one of the two dimensions of infinite extent.

The scale doesn't repeat. It is. The just approximation repeats, which
you can see by going up four 7:4 steps, down two 5:4s and back a 3:2 and
ending up where you started.

There are three dimensions of infinite extent: you're forgetting octaves.
And that'd be four in the 11-limit.

> > > In other words, Graham shows how the lattice repeats itself at the
> > > 225:224 but doesn't show how it repeats itself at the 2400:2401.
> >
> > All the chords you show, I show. That's the same power, but with
> less
> > complexity.
>
> Someone might not see certain progressions if they look at your
> lattice.

How so? They should see all progressions, they might not realise how
clever they are.

Graham

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., graham@m... wrote:
>
> Ah, I see it's periodic in both directions. That's clever, but
it's still
> a mess and I can't make sense of it.

Really? Are you seeing a fixed-width font, like Courier?

>Are you sure it's lined up right?

It is for me. Maybe you're getting some extra line-feeds?

> Perhaps fewer connections would help.

Never. Gotta see all the 7-limit consonances.

> Or a toroidal monitor.

Yeah, one that you can twist so that whatever's on the inner hole can
be turned toward the outside.
>
> At least Joseph Pehrson likes it.
>
> Try connecting all the 11-limit intervals on my graph,

Now that _would_ be a mess.

We could split up the fifths into neutral thirds on my graph just as
you did on your graph . . . and there would _still_ be no overlaps on
mine.
>
> How so? They should see all progressions,

What about a 2400:2401 pump?

🔗graham@microtonal.co.uk

Paul wrote:

> --- In tuning@y..., graham@m... wrote:
> >
> > Ah, I see it's periodic in both directions. That's clever, but
> it's still
> > a mess and I can't make sense of it.
>
> Really? Are you seeing a fixed-width font, like Courier?

Doh... I know what it is. My e-mail program automatically converts
anything within slashes into italics. So it was losing pairs of slashes.

> > Perhaps fewer connections would help.
>
> Never. Gotta see all the 7-limit consonances.

It looks okay now, but it's still over-complex. Good to make a point.
It'd still work for the Miracle scale, but the 41-note MOS would give
overlaps again, wouldn't it?

The trick is to notice how small a section of the lattice is actually
repeating.

I think the 5-dimension should be behind the 7-3 plane, so it looks like
the old lattice from a different perspective.

> > How so? They should see all progressions,
>
> What about a 2400:2401 pump?

What about it?

Graham

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., graham@m... wrote:

> It looks okay now, but it's still over-complex. Good to make a
point.
> It'd still work for the Miracle scale,

I'll try that.

> but the 41-note MOS would give
> overlaps again, wouldn't it?

Not if I increase the distance between notes. Still only a 2-d
manifold of unison vectors vanishing in the 7-limit.
>
> The trick is to notice how small a section of the lattice is
actually
> repeating.

Yes, good to notice that.
>
> I think the 5-dimension should be behind the 7-3 plane,
> so it looks like
> the old lattice from a different perspective.

I like preserving the appearance of otonal tetrads and utonal tetrads.
>
> > > How so? They should see all progressions,
> >
> > What about a 2400:2401 pump?
>
> What about it?

You wouldn't be able to see it on your lattice, if you wanted to see
all the consonances it entails.

🔗jpehrson@rcn.com

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:

/tuning/topicId_22135.html#22378

>
> Well, you could do that. Or, as Blackjack can also be expressed as
a subset of 31-equal, you could still use meantone notation.

How could "blackjack" be notated in meantone notation again?? I
guess that means putting in two other enharmonics then in 19... yes??

What do you do, add Cb and Fb (??)

Of course, for the players to learn the scale, I believe there would
have to be some kind of "reference point" to 12-tET in cents or
whatever...

That's why the 72-tET notation seems like it might be so nice... the
12-tET is already in there!

It looks pretty practical to me...

Whatever you or the performers find most convenient.
>
> Say, is Miracle the temperament or the subset now?
>
> Miracle tuning would ideally use a decimal notation, which a lot of
my posts have been working towards.

How would that be done again, Graham??

But if you're about to write the
> piece, you don't want to learn a new system of notation and teach it
> to the players, do you?
>

Actually, the 72-tET notation seem so "transparent"... all they have
to do is practice 1/6 tone, 1/3 tone... they ALREADY know
quartertones... and the 12-tET is already there for a reference...

It really seems right now to be the "easiest" to me... (??)

_________ ______ _______
Joseph Pehrson

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Graham Breed" <graham@m...> wrote:
>
> /tuning/topicId_22135.html#22378
>
> >
> > Well, you could do that. Or, as Blackjack can also be expressed
as
> a subset of 31-equal, you could still use meantone notation.
>
> How could "blackjack" be notated in meantone notation again?? I
> guess that means putting in two other enharmonics then in 19...
yes??

When blackjack is stuffed into a meantone, the errors of the
intervals relative to JI go up quite a bit, but you get a few extra
consonances. Here's what the notation would look like in meantone,
and the 31-tET degree numbers:

note 31-tET degree
B# 30
B 28
Ax 27
A# 25
Bbb 24
Gx 22
Ab 21
Abb 19
G 18
Gb 16
F# 15
F 13
E# 12
E 10
Dx 9
D# 7
Ebb 6
Cx 4
Db 3
Dbb 1
C 0
>
> What do you do, add Cb and Fb (??)

Afraid not. That would just be a chain of 20 fifths, not a chain of
20 miracle generators (which is what the blackjack scale is). The
miracle generator is a "minor second" (in meantone).
>
> Of course, for the players to learn the scale, I believe there
would
> have to be some kind of "reference point" to 12-tET in cents or
> whatever...
>
> That's why the 72-tET notation seems like it might be so nice...
the
> 12-tET is already in there!
>
> It looks pretty practical to me...

It's going to work out a lot better than meantone notation, because
(1) meantone is farther from just and (2) who really knows how to
play meantone these days anyway? -- while quarter tones and even
sixth tones are not so rare anymore (at least in Boston).
>
> Actually, the 72-tET notation seem so "transparent"... all they
have
> to do is practice 1/6 tone, 1/3 tone... they ALREADY know
> quartertones... and the 12-tET is already there for a reference...

1/12 tones can be learned by practicing just 5:4s or 5:3s from 12-tET
pitches.

1/6 tones can be learned by practicing just 7:4s or 7:3s from 12-tET
pitches)

1/4 tones can be learned by practicing just 11:4s or 11:3s from 12-
tET.
>
> It really seems right now to be the "easiest" to me... (??)
>
There are a whole bunch of 72-tET performers (and anyone who's played
Ezra Sims's music knows how to think of it as JI).

🔗graham@microtonal.co.uk

Paul wrote:

> When blackjack is stuffed into a meantone, the errors of the
> intervals relative to JI go up quite a bit, but you get a few extra
> consonances. Here's what the notation would look like in meantone,
> and the 31-tET degree numbers:
>
> note 31-tET degree
> B# 30
> B 28
> Ax 27
> A# 25
> Bbb 24
> Gx 22
> Ab 21
> Abb 19
> G 18
> Gb 16
> F# 15
> F 13
> E# 12
> E 10
> Dx 9
> D# 7
> Ebb 6
> Cx 4
> Db 3
> Dbb 1
> C 0

That looks right, although I'd use ^ and v for fifthtones instead of
double sharps and flats. The best way would be for all single steps to be
written as such, to cut down on the guesswork.

> > Of course, for the players to learn the scale, I believe there
> would
> > have to be some kind of "reference point" to 12-tET in cents or
> > whatever...
> >
> > That's why the 72-tET notation seems like it might be so nice...
> the
> > 12-tET is already in there!
> >
> > It looks pretty practical to me...
>
> It's going to work out a lot better than meantone notation, because
> (1) meantone is farther from just and (2) who really knows how to
> play meantone these days anyway? -- while quarter tones and even
> sixth tones are not so rare anymore (at least in Boston).

You'd have to write cents along the top, but that shouldn't be difficult
if you're only using 21 notes. The main thing's that it'd be simpler to
work with, because you wouldn't have so many symbols to work with in the
sketches.

It's up to you. Ultimately I think the decimal notation's best -- for
11-limit music in general. But you need to compromise to get the music
written and performed, so use whatever you're comfortable with.

Graham

🔗graham@microtonal.co.uk

Paul wrote:

> > > > How so? They should see all progressions,
> > >
> > > What about a 2400:2401 pump?
> >
> > What about it?
>
> You wouldn't be able to see it on your lattice, if you wanted to see
> all the consonances it entails.

Sure you can. We're only using Blackjack and 7-limit are we?

4-7^-0v-2
2-5^-8-0
0-3^-6-8^
8^-1^-4-6^
6^-9-2-4
3-6^-9-1
7^-0v-3-5
4-7^-0v-2

All 7-limit tetrads, all sharing a note with adjacent chords, ends up
where it started. I think it involves a 2400:2401 pump, but the lattice
doesn't make that obvious. All the consonances are there. On your
lattice,

F<-Av-C<-D>
D>-Gb}-A>-C
C-E{-G-Bb<
Bb<-Db-F<-G>
G>-Bv-D>-F<
Ev-G>-Bb-Db}
Av-C<-Ev-F#{
F<-Av-C<-D>

Looks like it drifts somewhere.

Now, you said something about neutral thirds on your lattice which I
didn't reply to because I didn't quite understand it. Well, my lattice
shows 0-3-5-8 as the primary consonance. It sounds quite good with 3-8 in
the lower octave and 0-5 in the higher. The intervals 3-8 and 0-5 are
both 7:5. 0-3 and 5-8 are 11:9. 3-5 is 8:7 and 0-8 is 7-6. So all
intervals are 11-limit consonances. However, you can't write it in JI, so
I don't think your lattice will show it as all consonances.

Graham

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_22135.html#22547

> You'd have to write cents along the top, but that shouldn't
> be difficult if you're only using 21 notes. The main thing's
> that it'd be simpler to work with, because you wouldn't have
> so many symbols to work with in the sketches.
>
> It's up to you. Ultimately I think the decimal notation's
> best -- for 11-limit music in general. But you need to
> compromise to get the music written and performed, so use
> whatever you're comfortable with.

Graham brings up an excellent point here.

In the same way that we may use cursive for writing down
personal messages quickly, and a more formal print-like writing
style for more formal utterances, there's no reason why a
composer couldn't get used to using a very simplified notation
for writing sketches down quickly, and then "translate" that
into a totally different notation for score and performance
purposes.

This is not unlike what Partch apparently did during composition
of his _Li-Po Lyrics_, sketching the vocal lines using 1/3-tone
and 1/4-tone accidentals (implying 36-EDO), and writing the
actual score in JI and rational notation.

Nor is it dissimilar from the (still uncompleted) procedure
I used myself in composing _A Noiseless Patient Spider_, where
I wrote the sketch-score (except the "Meditation" section)
starting in 72-EDO then switching to 144-EDO, whereas the final
score will be printed in HEWM + 72 (or possibly 144)-EDO.

I suppose a lot of microtonal (especially JI?) composers
work this way, at least some of the time.

I think decimal notation would be a great sketch notation,
and would use HEWM + 72-EDO for the finished score.

One thing that I find really lovely about the blackjack /
miracle / 72-EDO family of tunings is how so many *different*
layers of tuning structure are embedded the relatively simple
72-EDO superset. So it's no surprise to me that 72-EDO has
caught on to the degree it has so far.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗paul@stretch-music.com

--- In tuning@y..., graham@m... wrote:
> Paul wrote:
>
> > > > > How so? They should see all progressions,
> > > >
> > > > What about a 2400:2401 pump?
> > >
> > > What about it?
> >
> > You wouldn't be able to see it on your lattice, if you wanted to see
> > all the consonances it entails.
>
> Sure you can. We're only using Blackjack and 7-limit are we?
>
> 4-7^-0v-2
> 2-5^-8-0
> 0-3^-6-8^
> 8^-1^-4-6^
> 6^-9-2-4
> 3-6^-9-1
> 7^-0v-3-5
> 4-7^-0v-2
>
> All 7-limit tetrads, all sharing a note with adjacent chords, ends up
> where it started. I think it involves a 2400:2401 pump, but the lattice
> doesn't make that obvious.

I don't think that's possible.

All the consonances are there. On your
> lattice,
>
> F<-Av-C<-D>
> D>-Gb}-A>-C
> C-E{-G-Bb<
> Bb<-Db-F<-G>
> G>-Bv-D>-F<
> Ev-G>-Bb-Db}
> Av-C<-Ev-F#{
> F<-Av-C<-D>
>
> Looks like it drifts somewhere.

How could that not be the case on your lattice? Is it because 2400:2401 is an exact overlap on
your lattice? That sure would be a great advantage to your lattice (if you could show all the
consonant intervals too . . .)!
>
> Now, you said something about neutral thirds on your lattice which I
> didn't reply to because I didn't quite understand it. Well, my lattice
> shows 0-3-5-8 as the primary consonance. It sounds quite good with 3-8 in
> the lower octave and 0-5 in the higher. The intervals 3-8 and 0-5 are
> both 7:5. 0-3 and 5-8 are 11:9. 3-5 is 8:7 and 0-8 is 7-6. So all
> intervals are 11-limit consonances. However, you can't write it in JI, so
> I don't think your lattice will show it as all consonances.

OK -- but at least I have the strongest consonant intervals there.

🔗graham@microtonal.co.uk

paul@stretch-music.com () wrote:

> --- In tuning@y..., graham@m... wrote:
>
> > 4-7^-0v-2
> > 2-5^-8-0
> > 0-3^-6-8^
> > 8^-1^-4-6^
> > 6^-9-2-4
> > 3-6^-9-1
> > 7^-0v-3-5
> > 4-7^-0v-2
> >
> > All 7-limit tetrads, all sharing a note with adjacent chords, ends up
> > where it started. I think it involves a 2400:2401 pump, but the
> > lattice doesn't make that obvious.
>
> I don't think that's possible.

It must be possible, because it happens. Here's the lattice I worked
with:

1^----4^-/--7^
/ \ / /
/ \ / /
0^----3^----6^----9^----2
/ \/ \/ /
/ /\ /\ /
2^----5^-/--8^-\--1--/--4-----7
/ \ / / \ /
/ \ / / \ /
4^----7^----0-----3-----6-----9
\/ \/ /
/\ /\ /
9^ \--2--/--5-----8
\ /
\ /
4-----7-----0v

The progression goes up four 7:4 steps and comes back by minor sixths, so
I think it shows the unison vector you wanted. You can check it on your
lattice. If you can't set fixed-width, check the Blackjack lattice on my
website <http://x31eq.com/decimal_lattice.htm>

> > F<-Av-C<-D>
> > D>-Gb}-A>-C
> > C-E{-G-Bb<
> > Bb<-Db-F<-G>
> > G>-Bv-D>-F<
> > Ev-G>-Bb-Db}
> > Av-C<-Ev-F#{
> > F<-Av-C<-D>
> >
> > Looks like it drifts somewhere.
>
> How could that not be the case on your lattice? Is it because 2400:2401
> is an exact overlap on your lattice? That sure would be a great
> advantage to your lattice (if you could show all the consonant
> intervals too . . .)!

That's the idea, two of the three unison vectors are built into the
lattice. The only visible drift is the kleisma. I think it's a small
miracle in itself, but you said it was a defect before.

All 7-limit intervals would be shown if you connected them up. I think
that makes the diagram look cluttered. For the rest of the 11-limit, they
aren't always direct but they are consistent. You have to remember the
template like for a generalised keyboard.

Graham

🔗paul@stretch-music.com

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_22135.html#22552

> > > > What about a 2400:2401 pump?
> > >
> > > What about it?
> >
> > You wouldn't be able to see it on your lattice, if you wanted to
see
> > all the consonances it entails.
>
> Sure you can. We're only using Blackjack and 7-limit are we?
>
> 4-7^-0v-2
> 2-5^-8-0
> 0-3^-6-8^
> 8^-1^-4-6^
> 6^-9-2-4
> 3-6^-9-1
> 7^-0v-3-5
> 4-7^-0v-2
>
> <etc. - snip>

Graham, I'd like to make some MIDI-files to go along with this
post, but am unclear about some things. I thought we were all
referring to Blackjack, Miracle, and 41-tone as subsets of 72-EDO,
but your wording implies to me that there's a different tuning
involved.

I know that Paul is using 72-EDO, but what exactly is the tuning
you're using to go along with this decimal notation? Is it
31-EDO, or something else? Please explain.

Also, is the generator of this tuning really exactly 116.7 cents,
or is that an approximation? If so, what is a more exact value?

Enough reading and being confused... it's time to hear this stuff...

-monz
http://www.monz.org
"All roads lead to n^0"

🔗monz <joemonz@yahoo.com>

🔗jpehrson@rcn.com

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_22135.html#22512

> > Actually, the 72-tET notation seem so "transparent"... all they
> have to do is practice 1/6 tone, 1/3 tone... they ALREADY know
> > quartertones... and the 12-tET is already there for a reference...
>
> 1/12 tones can be learned by practicing just 5:4s or 5:3s from 12-
tET pitches.
>
> 1/6 tones can be learned by practicing just 7:4s or 7:3s from 12-
tET pitches)
>
> 1/4 tones can be learned by practicing just 11:4s or 11:3s from 12-
> tET.
> >

This is a very important post, which I have saved.

So, in other words, a performer, in realizing just or *no beating*
intervals from 12-tET is actually practicing the 72-tET system!

That's incredible!

Of course, the 1/4 tones are the hardest to learn that way (no
beating), but most performers can easily do them, anyway, in
reference to 12-tET.

I'm quite convinced... no doubt about it...

_______ _______ ______
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_22135.html#22560

> >
> > It's up to you. Ultimately I think the decimal notation's
> > best -- for 11-limit music in general. But you need to
> > compromise to get the music written and performed, so use
> > whatever you're comfortable with.
>
>
> Graham brings up an excellent point here.
>
> In the same way that we may use cursive for writing down
> personal messages quickly, and a more formal print-like writing
> style for more formal utterances, there's no reason why a
> composer couldn't get used to using a very simplified notation
> for writing sketches down quickly, and then "translate" that
> into a totally different notation for score and performance
> purposes.

Huh? Perhaps this has been explained further down the road and, if
so, please point me to the pertinent post...

BUT, is this suggesting that people write down 72-tET NUMBERS rather
than "notes" at all??

That doesn't seem very "musicianly" to me! The "traditional" 72-tET
notation at least uses "real" notes!

One can also use a kind of "Halberstadt template" notation, wherein
one just writes down the notes on the keyboard as a template and
then "translates" into the correct notes of 72-tET.

But just numbers?? I don't get it. It doesn't seem like writing
music to me... (??)

_________ _______ _______
Joseph Pehrson

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_22135.html#22802

[regarding Graham Breed's decimal notation for MIRACLE scales]

> Huh? Perhaps this has been explained further down the road
> and, if so, please point me to the pertinent post...
>
> BUT, is this suggesting that people write down 72-tET NUMBERS
> rather than "notes" at all??
>
> That doesn't seem very "musicianly" to me! The "traditional"
> 72-tET notation at least uses "real" notes!

No, Joe... you're confused. I think you missed too much of
the MIRACLE discussion while your computer was down.

Graham's numbers have nothing to do with 72-EDO _per se_,
except that they do represent notes which are a subset
of it.

The numbers 0 thru 9 are simply an easy way to designate
the 10 nominals of the simplest MIRACLE MOS, and then Graham
uses ^ and v (and ^^ and vv and beyond) to designate the
alterations of those pitches which occur as one keeps cycling
by means of the generator. Just like they way regular
notation uses A thru G and then adds # and b (and x and bb,
etc.) where necessary.

There's an improper MOS at 11 tones, but then it just so
happens with this weirdly amazing family of tunings that
further proper MOS's occur at 21, 31, and 41 tones. More
MOS's occur again, expectedly, at 51 and 61, but we've
only taken a small look at 21 (blackjack), barely scratched
the surface of 31 (canasta), and hardly discussed 41 at all.
(See, it doesn't even have a card-game-name yet...)

The theorists most involved in this - Keenan, Erlich, and Breed
- all concur that if one is going to use more tones than 41
then the entire 72-EDO superset might as well be pressed into
service, but they also concur that canasta is such a MIRACLE
that even 41 isn't all that exciting, so 72 is hardly necessary.

Graham's notation is just a way of vastly simplifying the
representation of this family of tunings. I like it a lot,
but it is a whole new way of notating the pitches that
doesn't relate it in any way to 12-EDO. If that's important,
as it probably is for you with your live performers but isn't
for Graham and his electronics, then keep the ASCII 72-EDO
that you're familiar with. But Graham's decimal notation
is worth learning for the different light it sheds on the
structure of the family.

> One can also use a kind of "Halberstadt template" notation, wherein
> one just writes down the notes on the keyboard as a template and
> then "translates" into the correct notes of 72-tET.
>
> But just numbers?? I don't get it. It doesn't seem like writing
> music to me... (??)

Joe, back around 900 AD someone (usually claimed to have
been Hucbald, but he's been discredited by now) had a sudden
brillant flash of insight in the struggle to notate music,
that elusive time-bound sonic art: since there were only 7
pitches, why not simply use the first 7 letters of the alphabet,
with their built-in order, to represent the successive diatonic
pitches? Bingo! Modern standard musical notation was born.

Along with this, there's the standard musical lingo of
"2nd", "3rd", "4th", "5th", "6th", 7th", "8ve" (shorthand
for "octave", which is Latin for "8th"), "9th", "10th", "11th",
"12th", and "13th". The ones beyond "8ve" are certainly
used to designate those intervals, but the basic system
starts at 1 and stops at 7, with the "8ve" being the special
case that's equivalent to 1. The only thing different here
is that they refer more to intervals than to pitches (altho
indirectly they refer to the pitches too), so *ordinal* numbers
are used to designated the *cardinality* of the scale.

Graham's numbers work in exactly the same way, with the
exact analogue being the letter notation.

Another method like this which you should be familiar with
is Allen Forte's use of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, t, e
(t being a single digit which = 10, e = 11) to designate the
12 pitch-classes of 12-EDO. This has by now become standard
in all the academic theory journals.

But did you ever stop to consider how incongruous it is that
if the "root" = 0, then a "3rd" = 4, a "5th" = 7, a "7th" = 10,
etc.? My guess is that you probably haven't, simply because
you've grown up with this wierd amalgamation and never gave
it a second thought. (If you have thought of it, then good
for you! Most musicians don't.)

I just posted this before I read yours; it should help:
/tuning/topicId_22793.html#22805

-monz
http://www.monz.org
"All roads lead to n^0"

🔗paul@stretch-music.com

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_22135.html#22826

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > But did you ever stop to consider how incongruous it is that
> > if the "root" = 0, then a "3rd" = 4, a "5th" = 7, a "7th" = 10,
> > etc.? My guess is that you probably haven't, simply because
> > you've grown up with this wierd amalgamation and never gave
> > it a second thought. (If you have thought of it, then good
> > for you! Most musicians don't.)
>
> Hmm . . . any guitarist would know it as a way of finding notes
> on a single string.
>
> We had to sing from numbers like these on the first day of
> Musicianship 318.
>
> You really think most musicians don't?

Well, of I course I know that musicians come across these
numerical relationships all the time. What I meant was
that most of them have such familiarity with these relationships
that they don't *ponder* or question the juxtaposition of
the two different ordering systems. They just accept it
unthinkingly, because that's what they learned.

I thought that poining out this juxtaposition to Joe would
help him understand what Graham is doing with his decimal
notation.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗graham@microtonal.co.uk

monz wrote:

>
> Graham's numbers have nothing to do with 72-EDO _per se_,
> except that they do represent notes which are a subset
> of it.
>
> The numbers 0 thru 9 are simply an easy way to designate
> the 10 nominals of the simplest MIRACLE MOS, and then Graham
> uses ^ and v (and ^^ and vv and beyond) to designate the
> alterations of those pitches which occur as one keeps cycling
> by means of the generator. Just like they way regular
> notation uses A thru G and then adds # and b (and x and bb,
> etc.) where necessary.

That's it. I also show at
<http://x31eq.com/decimal_notation.htm> how it can be written
on the staff.

> The theorists most involved in this - Keenan, Erlich, and Breed
> - all concur that if one is going to use more tones than 41
> then the entire 72-EDO superset might as well be pressed into
> service, but they also concur that canasta is such a MIRACLE
> that even 41 isn't all that exciting, so 72 is hardly necessary.

No, I think 45 notes are useful as well. You can include one version of
Partch's 43 note scale and Wilson's D'alessandro in it. I don't mean to
fetishize these scales, but they show what resources have been found
useful before. If you think in Miracle terms, 31 notes give enormous
potential. If you prefer to think in free 11-limit, you may still find 45
are enough.

> Graham's notation is just a way of vastly simplifying the
> representation of this family of tunings. I like it a lot,
> but it is a whole new way of notating the pitches that
> doesn't relate it in any way to 12-EDO. If that's important,
> as it probably is for you with your live performers but isn't
> for Graham and his electronics, then keep the ASCII 72-EDO
> that you're familiar with. But Graham's decimal notation
> is worth learning for the different light it sheds on the
> structure of the family.

Yes, but get the music written first!

I should also point out that, in the keyboard mappings I posted here
before, the numerals refer to the black notes, and the ^ and v to
"chromatic" shifts from there. So it's easier to think with this mapping
and decimal notation than always be converting to and from 72-equal.

I must have miscounted on the Blackjack mapping before: 12 notes is a
fifth. So getting the scale to repeat every fifth might work, and you
don't have to remember which bits of the keyboard are different.

Graham

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

On 5/15/01 1:47 AM, "paul@stretch-music.com" <paul@stretch-music.com> wrote:

>> But did you ever stop to consider how incongruous it is that
>> if the "root" = 0, then a "3rd" = 4, a "5th" = 7, a "7th" = 10,
>> etc.? My guess is that you probably haven't, simply because
>> you've grown up with this wierd amalgamation and never gave
>> it a second thought. (If you have thought of it, then good
>> for you! Most musicians don't.)
>
> Hmm . . . any guitarist would know it as a way of finding notes on a
> single string.

Agree... mucho...
that's the whole idea of tablature.
Very intuitive to guitarists, especially helpful
in greater than 12 situations.
Not always so much to know what note's where,
but at least for soloing on guitar, We've found,
it's easiest to know what interval *distances* you can work with.

But I can sympathize with Monz...
it's agnonizing to look at, at first. ISNT IT?!?!?
The idea of a 0-based point coordinate system
trying to coexist with a 1-based box coordinate system
usually smacks the brain halves together
like you're staying after school and clapping erasers.
In other words, it might have been less painful
if instead of the word "octave" coming out of a 1-based system,
starting with 1 and ending with 8,
If you're mapping 7-equal, AAAAAA!!!
the root (1) is 0, the 2nd is 1, the 5th is 4, the 7th is 6!?

It's always messy un-confusing numbers numbers and numbers...
especially, I've found, in music theory:
"The 3rd harmonic is called a fifth.
The 5th harmonic is called the third.
The 7th harmonic is near the seventh, but not exactly."

, comma ,

...What's on second, I Don't Know's on third.

or if someone mumbles "yeah, 19... minor third, huh..."
What? The minor third in 19 is close to the 6:5?
Or the 19th harmonic as a minor third?
There's always going to be a way that a number is used
that will confuse it with another use of that number.
THIS IS WHY NUMEROLOGY CAN NEVER BE MONOLITHIC!!!

Addam

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_22135.html#22818

> No, Joe... you're confused. I think you missed too much of
> the MIRACLE discussion while your computer was down.
>

You are right.... it became a lot clear after I had read the rest of
the posts!

> Graham's numbers have nothing to do with 72-EDO _per se_,
> except that they do represent notes which are a subset
> of it.
>
> The numbers 0 thru 9 are simply an easy way to designate
> the 10 nominals of the simplest MIRACLE MOS, and then Graham
> uses ^ and v (and ^^ and vv and beyond) to designate the
> alterations of those pitches which occur as one keeps cycling
> by means of the generator. Just like they way regular
> notation uses A thru G and then adds # and b (and x and bb,
> etc.) where necessary.
>
> There's an improper MOS at 11 tones, but then it just so
> happens with this weirdly amazing family of tunings that
> further proper MOS's occur at 21, 31, and 41 tones. More
> MOS's occur again, expectedly, at 51 and 61, but we've
> only taken a small look at 21 (blackjack), barely scratched
> the surface of 31 (canasta), and hardly discussed 41 at all.
> (See, it doesn't even have a card-game-name yet...)
>

Yes, I've got it. Well, I really do believe the path I want to
follow now is with 72-tET notation and all the various MIRACLE
subsets. There's a lifetime of practical work in that...

And, of course, that's my MAIN opposition, and the opposition, most
probably, of EVERY performing musician....

> Graham's numbers work in exactly the same way, with the
> exact analogue being the letter notation.
>
> Another method like this which you should be familiar with
> is Allen Forte's use of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, t, e
> (t being a single digit which = 10, e = 11) to designate the
> 12 pitch-classes of 12-EDO. This has by now become standard
> in all the academic theory journals.

Of course I learned this stuff. One can't get through music school
without it... However, it was ALWAYS used in thoretical studies.

Nobody every PLAYED from it, and people PLAY from notation,
supposedly...

>
> But did you ever stop to consider how incongruous it is that
> if the "root" = 0, then a "3rd" = 4, a "5th" = 7, a "7th" = 10,
> etc.? My guess is that you probably haven't, simply because
> you've grown up with this wierd amalgamation and never gave
> it a second thought. (If you have thought of it, then good
> for you! Most musicians don't.)
>

Of course I have thought of that, just in comparing it with
traditional set theory.

But the point is, we have to work and build on what we already HAVE,
or they whole thing is a theoretical or academic exercise... in or
out of a school!

____________ _______ ______ ___
Joseph Pehrson