Re: Representing Meantone Tunings with "NNN" [TD642.2]

Hi Margo!

Belated thanks to you too for taking the time to check out my "Meantone"
webpage.

On Thu, 18 May 2000 09:55:37 -0700 (PDT) [TD642.2],
Margo Schulter <mschulter@value.net> wrote:
>>
>> Date: Wed, 17 May 2000 09:06:36 -0700
>> From: Mark Nowitzky <mnowitzky@yahoo.com>
>>
>> "Representing Meantone Tunings with Nowitzkian Note Names"
>>
>> http://nowitzky.hypermart.net/justint/nnnmt.htm
...
>Incidentally -- and this may just be my personal quirk -- on first reading
>your notation, I noticed a curious coincidence. If the default comma
>number is taken as "5", then a pure major third like 5C5 4E5 (with the E a
>comma number lower than the C, or in other words a syntonic comma smaller
>than Pythagorean) will suggest by the comma numbers the ratio 5:4. Another
>way of stating this is that the 5th partial of 5C5 will coincide with the
>4th partial of 4E5. Of course, this is just a fortuitous consequence of
>choosing a default comma number of "5", but nevertheless could have
>mnenomic value. (Here I'm counting the fundamental as the "first
>partial.")
>
>Likewise with a pure minor third such as 5E5 6G5 -- suggesting the ratio
>of 5:6, although here it's the _fifth_ partial of G which will coincide
>with the _sixth_ partial of E.
>
>Of course, one might say that with 5C5 4E5, the "5:4" represents the
>string-ratio (or organ-pipe ratio, etc.), while with 5E5 6G5 the
>"5:6" represents the frequency-ratio.

Actually, my notation aside, the coincidences you site remind me of the
confusion I've had with the following:
The Third partial coincides with a (Perfect) Fifth (plus an octave), and
the Fifth partial coincides with a (Major) Third (plus two octaves).
Fortunately the Seventh partial coincides with a (Minor) Seventh (plus two
octaves).

...
>An aside: I've experimented with Pythagorean on two 12-note keyboards, a
>"Xeno-Gothic" tuning providing the usual 3-limit intervals plus various
>5-limit and 7-limit approximations for intervals generally treated as
>unstable in a Gothic or "neo-Gothic" stylistic setting. I wonder if a
>notation based on Pythagorean rather than syntonic comma numbers might fit
>this system. For example, using your "5" as a default comma number,
>
>5D4 6F#4
>
>would be a major third a Pythagorean comma wider than the usual 81:64,
>possibly the kind of wide cadential interval Marchettus of Padua
>(1318) may have intended for a progression such as
>
>6F#4 5G4
>5D4 5C4
>
>Here we have a new "isotope" of the standard M3-5 resolution by stepwise
>contrary motion, with the major third a comma wider than usual, and the
>melodic semitone in the upper part a comma narrower than the usual limma
>of 256:243 (or ~90.22 cents) -- or about 66.76 cents, a kind of
>"third-tone" interval.

Okay, I'm assuming these definitions when I read the above:

Pythagorean comma (AKA Ditonic comma) = 531441/524288
Syntonic comma (AKA Comma of Didymus) = 81/80

Then to represent the tunings you want above, I'd use "5EX4" instead of
"6F#4" (where "X" means double-sharp, as it does in conventional music
notation). Or "5E##4" if you prefer to represent double-sharps that way.

I.e., (octave digits omitted):

ratio PCN NNN
------------------- --- ---
3/2 5G 5G
387420489/268435456 6F# 5EX
729/512 5F# 5F#
9/8 5D 5D
1/1 5C 5C

where PCN = using your suggested "Pythagorean Comma Numbers", and
NNN = using "Nowitzkian Note Names"

As a matter of fact, with NNN, you wouldn't even need the comma numbers if
you stick to strictly Pythagorean tuning. (But the down side is that you
gotta contend with double-sharps and double-flats.)

By the way, do you really think Marchettus of Padua would have meant such
an interval (387420489/268435456, unless I screwed up the math)? Could
they count that high back then? :) Seriously, I've been puzzled by that
"D + F#, C + G" cadence for awhile.

Anyway, thanks again,

--Mark Nowitzky
nowitzky@alum.mit.edu, AKA tuning-owner@egroups.com

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| Mark Nowitzky
| email: nowitzky@alum.mit.edu
| www: http://nowitzky.hypermart.net
| "If you haven't visited Mark Nowitzky's home
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