[tuning] explanation of Graham Breed's "decimal" notation

> ----- Original Message -----
> From: <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, June 21, 2001 6:11 AM
> Subject: [tuning] Re: updated "MIRACLE" definition webpage
>
>
> > Now, could I please ask you to elaborate a bit on Graham Breed's
> decimal notation, so I can understand it??

Here's a real quick stab at it, Joe... but it should help.

First of all, realize that Graham's use of numbers here is
exactly equivalent to the way we currently use the first seven
letters of the alphabet to describe the 7-tone diatonic cardinality.
So the digits 0 ... 9 are the 10-tone cardinality of the basic
MIRACLE scale.

Secondly, realize that Graham's notation is generalized in
that it is applicable to 31- and 41-EDO as well as 72-EDO.
I'm going to discuss it here only in terms of 72-EDO.

OK, now look at the graph of the 11-tone MIRACLE scale
(the one with the green line) on my webpage:

http://www.ixpres.com/interval/dict/miracle.htm#11-tone

and keep this window open or print it out... you'll be
using my two recent graphs a lot in following what I say here.

Now look at the "melodic" description of Graham's notation at
http://x31eq.com/decimal_notation.htm

Let's start the scale at the -5 generator, giving the pitch
which is 6&1/6 Semitones, which we will call "0". We are
considering Graham's *10*-tone scale here, not the 11-tone one
I illustrate. So here the +5 generator, giving the pitch at
5&5/6 Semitones, which is the one close to the origin, will be
omitted. So we'll use as our bounding notes the -5 and +4 generators.

Here's a list of the basic 10-tone MIRACLE scale, in descending
order (as usual with me... sorry Dave Keenan...), with Graham's
"s" and "q" interval-spacing:

Generator Graham's Semitones interval between
"note" Semitones 72-EDO-degrees

-5 0 6&1/6
> 1&1/2 9 s+q
+4 9 4&2/3
> 1&1/6 7 s
+3 8 3&1/2
> 1&1/6 7 s
+2 7 2&1/3
> 1&1/6 7 s
+1 6 1&1/6
> 1&1/6 7 s
0 5 0
> 1&1/6 7 s
-1 4 10&5/6
> 1&1/6 7 s
-2 3 9&2/3
> 1&1/6 7 s
-3 2 8&1/2
> 1&1/6 7 s
-4 1 7&1/3
> 1&1/6 7 s
-5 0 6&1/6

So as Graham explains, "s" in 72-EDO MIRACLE is 2^(7/72),
and "q" is the new interval we find at 11 tones, which is
2^(2/72). So the large interval at the end of the 10-tone
cycle is "s+q" = 2^(9/72).

Graham's notation uses "v" and "^" as accidentals, in which
"the ^ and v symbols denote upward and downward shifts of q".

So if we decide to include that 11th tone of 5&5/6 Semitones,
which is the +5 generator, it would be called 0v, which indicates
that the origin, 6&1/6 Semitones, has been flattened by "q",
which is 2^(2/72) = 33&1/3 cents = 1/3 Semitone = "1/6-tone":

"0" - "q" = "0v"
6&1/6 Semitones - 1/3 Semitone = 5&5/6 Semitones

So now look at the 32-tone MIRACLE graph on my webpage
(the one with the green and red lines):
http://www.ixpres.com/interval/dict/miracle.htm#32-tone

All subsequent pitches added to our gamut will be described
in terms of the basic 10-tone scale tabulated above.

So if we continue the cycle on the positive side, adding
the +6 generator = 7 Semitones, it would be "1v", which is:

"1" - "q" = "1v"
7&1/3 Semitones - 1/3 Semitone = 7 Semitones

and so on.

If we continue on the negative side, adding the -6 generator
= 5 Semitones, it would be "9^", which is:

"9" + "q" = "9^"
4&2/3 Semitones + 1/3 Semitone = 5 Semitones

and so on.

The -15 generator = 6&1/2 Semitones would be "0^", because
it is one "q" higher than our origin note "0":

"0" + "q" = "0^"
6&1/6 Semitones + 1/3 Semitone = 6&1/2 Semitones

The -16 generator would be "9^^", because it is one "q" higher
than "9^".

On the positive side, the +15 generator = 5&1/2 Semitones
would be "0vv", because it is one "q" lower than "Ov":

"0v" - "q" = "0v"
5&5/6 Semitones - 1/3 Semitone = 5&1/2 Semitones

The +16 generator = 6&2/3 Semitones would be "1vv", because
it is one "q" lower than "1v":

"1v" - "q" = "1vv"
7 Semitones - 1/3 Semitone = 6&2/3 Semitones

Note, however, that this "1vv" is 1/2 "q" higher than "0^".
We've run up against the next level of enharmonicity, which
delimits the Canasta scale.

Sheesh... this is really starting to look like secret code...
I can see why you were so confused. But if you follow the
above explanation by consulting my MIRACLE graphs it should
be clear. When I have time I'll make a webpage putting
Graham's notation onto those graphs.

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

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I've made a Dictionary entry webpage for Graham's
"decimal notation":

http://www.ixpres.com/interval/dict/decimal.htm

It includes graphs of the 11-tone and 32-tone MIRACLE
scales in my MIRACLE webpage, but with Graham's notation
instead of Semitones.

It should make this much easier to understand.

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

_________________________________________________________
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Joe!
When do you think the one on "Public Education" will go up. :)

monz wrote:

> I've made a Dictionary entry webpage for Graham's
> "decimal notation":
>
> http://www.ixpres.com/interval/dict/decimal.htm
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

> ----- Original Message -----
> From: Kraig Grady <kraiggrady@anaphoria.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, June 21, 2001 11:46 AM
> Subject: Re: [tuning] explanation of Graham Breed's "decimal" notation
>
>
>
> Joe!
> When do you think the one on "Public Education" will go up. :)
>
> monz wrote:
>
> > I've made a Dictionary entry webpage for Graham's
> > "decimal notation":
> >
> > http://www.ixpres.com/interval/dict/decimal.htm

Sorry, Kraig... I make it a policy to stay out of the flame wars
in every possible way, shape, or form.

And if I *do* ever decide to make a webpage about "Public Education"
(not likely, but you never know), it will *not* be in my Tuning Dictionary!
It will be in my "Other stuff" area... but I'm more likely to discuss
the "public education" of the Sumerians than of present-day America...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

Sorry for the joke joe-

monz wrote:

> Sorry, Kraig... I make it a policy to stay out of the flame wars
> in every possible way, shape, or form.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

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

/tuning/topicId_25435.html#25435

>
>
> Sheesh... this is really starting to look like secret code...
> I can see why you were so confused. But if you follow the
> above explanation by consulting my MIRACLE graphs it should
> be clear. When I have time I'll make a webpage putting
> Graham's notation onto those graphs.
>

Thanks so much, Monz, for your help with this. I'm still puzzling
through it, but this helps a lot.

Please don't "gum up" your great MIRACLE webpages with lots of
extraneous stuff, though... They are entirely CLEAR as they stand,
in my opinion...

Thanks!

_________ ______ _______
Joseph Pehrson

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

/tuning/topicId_25435.html#25438

> I've made a Dictionary entry webpage for Graham's
> "decimal notation":
>
> http://www.ixpres.com/interval/dict/decimal.htm
>
>
> It includes graphs of the 11-tone and 32-tone MIRACLE
> scales in my MIRACLE webpage, but with Graham's notation
> instead of Semitones.
>
> It should make this much easier to understand.
>

Thanks Monz, ... er... for keeping this on a SEPARATE page from the
other MIRACLE pages. :)

Actually, it makes it quite easy to understand... I could see
the "accumulation" of commas and the whole thing...

Thanks!

________ ______ ______
Joseph Pehrson

> ----- Original Message -----
> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, June 21, 2001 10:56 AM
> Subject: [tuning] explanation of Graham Breed's "decimal" notation
>
>
> On the positive side, the +15 generator = 5&1/2 Semitones
> would be "0vv", because it is one "q" lower than "Ov":
>
> "0v" - "q" = "0v"
> 5&5/6 Semitones - 1/3 Semitone = 5&1/2 Semitones

Duh... oops, my bad. There's a typo here.

The last (resulting) note should obviously be "0vv"
and not "0v".

The error was copied into the webpage I made from this
post, and has now been fixed, so if you've bookmarked it,
go back and refresh/reload.
http://www.ixpres.com/interval/dict/decimal.htm

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

_________________________________________________________
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