Re: Representing Meantone Tunings with "NNN" [TD641.5]

Hi Paul,

Regarding my webpage entitled
"Representing Meantone Tunings with Nowitzkian Note Names"
at
http://nowitzky.hypermart.net/justint/nnnmt.htm
thanks for checking out the webpage!

("BELATED thanks" I should say, since your comments were from a month ago!)

On Wed, 17 May 2000 15:06:04 -0400 [TD641.5],
Paul H. Erlich <perlich@acadian-asset.com> wrote:
>
>Hey Mark -- nice work. My only comment is on this:
>
>>5LJI (5-Limit Just Intonation):
>>The Comma digits used here show the values that the subsequent Meantone
>>Temperament attempts to approximate.
>
>There is no single 5-limit JI system that meantone attempts to approximate
>-- rather, it approximates a larger set of JI relationships than any single
>JI system could achieve. A reasonable simplification would be that meantone
>simultaneously approximates several JI systems. "Approximation" in this
>sense should be evaluted not on a pitch-by-pitch, but on a consonant
>interval-by-consonant interval basis.

By "approximate", I meant, for example, that
1/4 CMT: 5 C, 4 E, 4 3/4 G
is an approximation of
5LJI: 5 C, 4 E, 5 G

But I see what you're getting at too. If "JI systems" are confined to
12-notes-per-octave, then I agree with you. Extending the idea further,
you could say that 12-notes-per-octave "1/4 CMT systems" are subsets of the
"1/4 CMT universe". For example, since I used 6Ab in the 5LJI and 1/4 CMT
systems on the webpage, I couldn't also have 3G#.

>Also, I noticed that the 5-limit system you do propose is not one of the
>"famous" ones, though any of the latter would suffer the same problem. Out
>of curiousity, where does your 5-limit system come from?

Well, for that table entitled "Nowitzkian Note Names for a
12-Notes-Per-Octave Keyboard", here's how I arrived at the 5LJI and 1/4 CMT
tunings:

- The 1/4 CMT tuning corresponds to a tuning shown in an article entitled
"Mean-Tone Temperament, The Classical System of Instrumental Tuning", by
William Braid White. This article appeared in the Journal of the
Acoustical Society of America, Volume 15, Number 1, way back in 1943.
Fellow tuning list member Dave Hill sent the article to me a couple months
ago.

- To get the comma values for 5LJI, I took the 1/4 CMT values and rounded
them to the nearest integer.

So these tunings put the "wolf" interval at C# to Ab.

Hey thanks again for your comments,

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

P.S.: I delayed my reply to your comments, partially 'cuz I've been
working on a new webpage that I thought would be relevant to the
discussion. But now that the new page is done, I realize it has no direct
relevance. Anyway, the new page is at
<http://nowitzky.hypermart.net/justint/optimal.htm>, and I'll make a bigger
deal out of announcing it in a subsequent post to the Tuning List, so it
isn't lost in this "P.S.". :)

+-------------------------------------------------------
| Mark Nowitzky
| email: nowitzky@alum.mit.edu
| www: http://nowitzky.hypermart.net
| "If you haven't visited Mark Nowitzky's home
| page recently, you haven't missed much..."
+-------------------------------------------------------
_____________________________________________
NetZero - Defenders of the Free World
Click here for FREE Internet Access and Email
http://www.netzero.net/download/index.html

Mark Nowitzky wrote (regarding
http://nowitzky.hypermart.net/justint/nnnmt.htm),

>- To get the comma values for 5LJI, I took the 1/4 CMT values and rounded
>them to the nearest integer.

>So these tunings put the "wolf" interval at C# to Ab.

Of course, your 5LJI tuning also has "wolves" between F# and C#, between D
and A, and between Bb and F. My comment was merely that the famous 12-tone
JI schemes put the wolves in different places, but they're all pretty much
equally "wolfy", which is why they all got very little use in practice. Only
in the 15th century did something like a 5-limit JI tuning with 12 fixed
pitches get any real use, and it was really a Pythagorean tuning from Gb to
B, where the triads D-Gb-A, A-Db-E, and E-Ab-B are very nearly 5-limit just
major triads, and the triads Gb-A-Db, Db-E-Ab, and Ab-B-Eb are very nearly
5-limit just minor triads. This is called schismatic tuning, after the
2-cent interval known as a schisma, because, in terms of NNN, 5Gb is only a
schisma away from 4F#, 5Db is only a schisma away from 4C#, and 5Ab is only
a schisma away from 4G#. This, however, was before the 5-limit consonant
standard for triads fully came into vogue, and long before the system of
modes had been reduced to the "major" and "minor" forms of tonality that
characterize the music of Handel, Mozart, etc.

_________________________________________________

We are Moving!

As of June 26, 2000, Acadian Asset Management will be at a
new location in Boston's financial district.

Please contact us at:
Acadian Asset Management
Ten Post Office Square, 8th Floor
Boston, MA 02109.

All phone, fax and email remain the same.