From: gtaylor@vme.heurikon.com (Gregory Taylor)
Newsgroups: comp.music
Subject: Re: just-tuned synth
Date: 8 Jan 93 15:34:01 GMT
Reply-To: gtaylor@vme.heurikon.com (Gregory Taylor)
Organization: Heurikon Corporation, Madison, WI

I've gotten several notes via e-mail from comp.music readers who
wondered about how different the Pythagorean tunings could be 
different from a simple Just tuning. This is the kind of thing
that we could be here for days and weeks writing out. In addition,
this newsgroup is populated by learned persons whose sandals I'm
not even qualified to untie. However, *I'm* the poor geek who got
the mail, so I'll elaborate briefly in the hope that my colleagues
will correct and add to what's here:

>>I've done software that tunes a synth to a just ( Pythagorean) scale
>>thru an MP401 board on a pc. I'm not a keyboardist, but a couple of
>>rounds and chorales I've sequenced for it sound great! They have the
>>purity of sound of an excellent choir singing a capella ( sp?) or
>>with a string accompaniment.

Pythagoras (like a number of Chinese theorists) derived his thirds
by tuning up in successive fifths. Divide a string into three parts,
fret the string two-thirds of the way up, and pluck. The frequency
ratio is 3/2, or 1.5 times the original frequency. To get the next
note, one simply repeats the process with a string 2/3 the length
of the original, up in a cycle of fifths. Of course, when you've
supposedly come back "around" the cycle of fifths, you'll find that
your original pitch has "moved" by an amount (called a Pythagorean
comma). Anyway, the major third is close enough to 1/1 that Mr. P.
just tuned up a fifth four times in a row.

Strictly speaking, this doesn't really work as Just Intonation when you
get around to deriving an entire scale. Just Intonation tends to favor
small, whole-number ratios. Many just scales are identified as N-limit
scales, where N is the smallest number used as a denominator in the 
frequency ratio. 

The third that Pythagoras comes up with by successive fifths is:
1/1 * 3/2 * 3/2 * 3/2 * 3/2
1/1   3/2   9/8  27/16 81/64

or a frequency ratio of 1 to 1.2656

Whereas the most common derivation for a Just third is a frequency 
ratio of 5/4

2/1 - octave
3/2 - fifth
4/3 - fourth
5/4 - third
6/5 - minor third
7/6 - the "septimal" minor - a keen interval not on your keyboard.
etc.

or 1 to 1.25

Trust me on this; the difference is audible, with P's 1.2656 being an
acquired taste - kinda like moose tenderloin in a caramel sauce with chiles.

All this is to be compared with the equal temperament system, which
conjures with the amazing "twelfth root of two" value (which, although
an irrational number, we can tune to *precisely* :)  ) to come up with
the location of the major third. Suffice it to say that it's not very
close to the street where Mr. Pythagorean Third lives - to my ears,
anyhow. Finally, the 5/4 value works real well within the whole 
overtone series thing, too....it's the 4th harmonic (which is 1.25
when expressed within a single octave compass).

Incidentally, there are a whole bunch of possible major thirds out 
there. That's the nice thing about abandoning equal temperament.

One poster asked about a quick and easy source of information about
tunings. I'd normally tend to recommend Harry Partch's "The Genesis 
of a Music", but the polemics in the book might get a little thick for
newbies. J. Murray Barbour's "Tuning and Temperament" has lots of 
information, but it's also such a transparent polemic for the view that
ET descended directly from the gods that I find it obnoxious. Failing
this, do any of you have any feelings about the New Grove articles on
tunings/temperaments? I've not seen a newer edition.

With regards,
Gregory Taylor
--
The leveling weight of music and weather/closes the stores, drives the people
underground/into cool rooms under ducts and water pipes./There are ways to
prove the logic of whatever happens./In a movie, the darkness between each 
frame proves it./Gregory Taylor/Heurikon Corporation/Madison, WI/608-828-3385
Newsgroups: comp.music,rec.music.synth,comp.sys.ibm.pc.hardware