Just Intervals?

[Susan Kline]

---------------------- multipart/alternative attachment
Hello, Bradley. I'm glad to see that great minds are (finally)
working alike. As you were writing this post, I was hauling out
Jorgensen's tuning book to go over just intervals.

I took "just" to mean simple ratios, which expressed themselves
as unbeating intervals, but found I was very rusty on the subject
of just thirds. Wouldn't it be cute if I were simply wrong, and
just thirds and sixths DID beat, and they WERE used in string
playing?

Well, the glossary was helpful (page 773).

"Some commonly known just intervals are the ratios 1:2 octave of
1200 cents, 2:3 fifth of 702 cents, 3:4 fourth of 498 cents,
4:5 major third of 386 cents, 5:6 minor third of 316 cents, and
6:7 small minor third of 267 cents."

 From this one can see that octaves, fifths, and fourths are
extremely close to the tempered intervals, and therefore a string
player can play them "just" and still agree quite well with a
piano. But the just major third -- 14 cents narrow; the minor
third -- 16 cents wide; and the "other" minor third, a useless
33 cents narrow; how to reconcile them? I really don't think
that a string player will be willing to bend everything that
far out of shape, just to get a non-beating interval, when they
are vibrating and usually playing the notes in succession
instead of simultaneously most of the time, anyway.

I also found page 18, where very early Pythagorean temperaments
were discussed. They were derived by tuning octaves, fourths,
and fifths just, which yielded wide Pythagorean major thirds.

Jorgensen talked about a tempering system used by Boulliau,
for fretted instruments. It was a Pythagorean temperament, and
gave major thirds of differing sizes, listed in Table 7-2,
on page 20. It dates from the 14th century.

Jorgensen says about it: "As a just intonation it was the
most amazing system in history because one could modulate through
all twenty-four major and minor keys in it without encountering
wolf intervals, though there were two very harsh fifths. Musically,
it served as a substitute for equal temperament. By comparison
with equal temperament, it was extremely easy to calculate
on paper and transfer to the frets of lutes and viols."

It sounds to me like this system may be about as just as a
fixed pitch instrument can manage, but the major thirds,
as given on page 20, seem to come in three flavors; 6.77
cents wide from just; 12.78 cents wide from just; and
21.5 cents wide from just.

I did check out Hilary Hahn, by the way. I went to
Amazon and listened to sound clips of her playing Bach.
Good playing. Thank you.

 From what you were saying, I expected playing with very
little vibrato, and strangely narrow (from tempered)
major thirds, and wide minor thirds, but I didn't find
them. Plenty of vibrato, and I thought she showed a
strong bias toward wide major thirds, and toward
sharping as she reached for major thirds and sixths, sort
of an aggressive reaching for them, which I found very
pleasing. Also small semitones when resolving leading
tones. And a tendency to sharp the upper register a little,
but not to excess. Very tasteful, really.

Maybe you are hearing something different, and it would
be interesting to find out what it is. Here's the link to
get the sound clips. The most telling one for interval
size I found to be number five, the Minuet II from the
third partita (in E.)

http://www.amazon.com/exec/obidos/ASIN/B0000029UH/qid%3D1020641792/sr%3D2-2/ref%3Dsr%5F2%5F2/002-2730038-9330451

A score would have helped, but I went to a piano and
found out the note names. The same double-stopped line
occurs again and again, which helps to hear it. The
lower line is moving, against the stationary upper note (B.)
The lower line is G# - A - G# - F# - E - F# - G# - A

Then it moves lower, so that E - D# - E can be easily
heard. The G# and D# she plays very high, and she
resolves them by a very small semitone, smaller than
tempered by a good bit. Her G# - B doublestop is a
very pleasing (to me) quite narrow minor third. It
is smaller than tempered by a good lick; but it is
smaller than just by a humongous amount.

Now, what were we arguing about, again?

Regards,

Susan Kline

At 05:59 PM 5/5/2002 -0700, you wrote:
>I am trying to work on a good definition for just intervals, but I am kind 
>of stuck . . .
>
>I have all the just intervals worked out (i.e. smallest ratios):
>P5  3/2
>P4  4/3
>M6  5/3
>M3  5/4
>m7  7/4
>m3  6/5
>TT  7/5
>m6 8/5
>M7 9/5
>M2  7/6
>m2  8/7
>
>This takes care of the simple ratio aspect, but it doesn't really address 
>the psychoacoustical aspect of beating. You only get beating if you have 
>overtones (sine tones will not create beats even if they are set 
>out-of-tune; you will only hear very, very slight flutterings). Since 
>overtones on the piano grow sharp because of inharmonicity, it's possible 
>to set an interval just (non-beating) but then the fundamental ratios are 
>slightly expanded.
>
>Does anyone have any suggestions?
>
>Bradley M. Snook

---------------------- multipart/alternative attachment
An HTML attachment was scrubbed...
URL: https://www.moypiano.com/ptg/pianotech.php/attachments/65/89/1f/e8/attachment.htm

---------------------- multipart/alternative attachment--