TUNING digest 554

To answer some questions for clarity and pose a couple more:
>From marc sabat:

> Wondering if you've heard Tenney's Harmonium #5 for string trio (on
> ARRAYMUSIC's CD "Strange City/Ville Etrange", Artifact Records
> (http://www.io.org/~artifac/).
>
> I find this piece has fairly non-static harmonic motion.

No, I haven't. I was referring (about static music) to the cornucopia of
harmonic-series works of his like "In a Large Open Space", "Spectral
Canon", "Saxony", etc.
-----
from Johnny Reinhard:
> If you define JI in 12 tones per octave and they are based on the the
> better resultion of 12 ET than you should not be surprised if a listener
> hears very little difference.

"Big Gulp"'s tuning is 11-limit: 1/1, 33/16, 9/8, 28/12, 5/4, 21/16,
11/8, 3/2, 99/64, 27/16, 7/4, 15/8. It should be a good scale for piano
music which sounds convincing yet distinctly xenharmonic.

> Surely a restriction of only 12 keys is still a restriction and this can
> be overcome by writing for another instrument (or more) playing with the
> restricted keyboard. The additional instruments can play additional
> microtones. Wyschnegradsky's *Meditation* has the cello (or in my case
> bassoon) playing quartertones and sixthtones against a conventionally
> tuned piano extremely effectively.

Of course it is nice to write for an ensemble, but when it comes to
analyzing a tuning purely, a solo keyboard may be ideal. Adding
instruments, especially ones without fixed-pitch, would muddy up the sound
and act as a "cover" for hearing the true relationships. I also find
that the more action is involved (varied instruments, harmonic motion,
etc.), the less xenharmonic a piece sounds. This is taken from comments
of non-JI musicians who I have listen to it.

A couple of questions:
-Who is "Twinings"? I haven't heard his music.
-Is the consensus that harmonic "pull" towards another tonality is the
same in a just-equivalent of 12TET? Could the dominant triad not be more
effective in 12TET because the triadic third is more dissonant (the
sharper quality may lead the ear more strongly to the tonic).

Thanks for the discussion and keep it coming.
Adam B. Silverman


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The situation you described when you play both sides of a string while moving
the partition (slide) between the two strings, is what is called a "monochord",
obviously meaning "one string", but clearly implying split into two vibrating
sections.

As for the relationship of the pitches in this situation, here's the scoop:
As we know from how frets work, ONE of the two vibrating sections of the string
ALONE while moving the slide can illustrate simple harmonic relationships. If
you don't play both sides of the string, you have a definite nut and bridge, the
sound you're considering vibrating between the bridge and the slide (that would
define the bridge).

As you perhaps know, the frequency of a vibrating string is inversely
proportional to its length. So if you place the slide half-way up the string,
you will have a string half the length, and thus twice the frequency of the
pitch of the open string (i.e., no slide on the string at all). If you were to
place the slide 1/3 of the entire string's length from the bridge, the vibrating
length of the string would be 1/3, so the frequency would be 3 times that of the
string if you remove the slide from the string completely. Similarly for 1/4 of
the open string length, and so forth.

But what if you DO sound the other side as well. Let's place the slide in
those same places and see what happens to the other part of the string. Well,
if you place the slide in the middle, you have 1/2 on one side, and 1/2 on the
other, so the pitches are the same, as you correctly pointed out. If you place
the slide 1/3 of the way to one side as before, one part of the string will be
at 3-times the open-string frequency. What about the other side? Well, it has
2/3 of the open-string length, so it will sound at 3/2 of the open-string
frequency.

So, if you place the slide 1/3 of the open string length from one end, you
will have pitches at 3 times and 3/2 times the frequency of the string if the
slide were removed completely. Clearly, 3 is two times 3/2, so the two
frequencies are octaves apart.

Similarly, if you place the slide at 1/4 of the open string length from one
end, you will have string lengths of 1/4 and 3/4. Those will then sound at
frequencies of 4 and 4/3 times the frequency of the open string. Clearly 4 is 3
times 4/3, so the two frequencies will be a factor of three apart.

But you asked what are the relationships between all of these frequencies?
Well, let's look at those three cases:

In terms of lengths: In terms of frequency: In terms of Pitch:

1/1 1/1 P1
|-----------------------| |-----------------------| |-----------------------|

1/2 1/2 2/1 2/1 P8 P8
|-----------0-----------| |-----------0-----------| |-----------0-----------|

1/3 2/3 3/1 3/2 P12 P5
|-------0---------------| |-------0---------------| |-------0---------------|

1/4 3/4 4/1 4/3 dbl 8va P4
|-----0-----------------| |-----0-----------------| |-----0-----------------|
(Note: P1 = perfect uni.
P8 = perf. octave
P12 = perf. 12th
P5 = perf. fifth
(etc.) )

The frequencies of the shorter parts of the strings are harmonics (1 times, 2
times, 3 times, and 4 times) of the frequency of the open string. The others
are somewhat more complexly related to the each other and to the open string.
If I recall correctly, they are called "superparticulars".

But here's what's neat about the frequencies sounded by the larger parts of
the string when you make the play harmonics on the smaller parts: The are as
far above the common open-string root pitch as the current harmonic is above the
previous harmonic.

So first time around, on the shorter half (left) of the string, we went from
1/1 times whatever the frequency of the open string may be, to 2/1 times that
frequency, a jump of an octave, which is as far above the open-string root pitch
as the harmonic side jumped up. The next time around, the pitch of the harmonic
side jumped from an octave above the root pitch to a twelfth above the root
pitch, an increase of a perfect fifth, and by golly, the longer end is a perfect
fifth above that common root pitch. Similarly, the third time, the pitch of the
harmonic side went from a twelfth above the root pitch to a double-octave above
that root pitch, a pitch change of a fourth. And by golly, the pitch of the
longer side is a perfect forth above the root pitch.

This relationship occurs only if you stick with the "sweet spots" - the
shorter side sticking with 1/2, 1/3, 1/4, 1/5, and so forth of the open string
length.

As for the other topic - other fingering patterns to pick out on your
jumbush's fretless fingerboard - you might want to try subharmonic pitch
relationships. You can get them by placing imaginary frets at exactly equal
distances apart all the way up the neck. That rather than the usual
ever-decreasing distance as you move your way up the neck toward the bridge.
These subharmonic frequencies are 1/2, 1/3, 1/4, 1/5, 1/6, and so forth of a
very high fret position.

But there's a very important constraint here: You must split the open string
length into a whole-number of equal-sized pieces! As I vaguely recall from Ivor
Darreg's jumbush, the open string length is about 13 inches long. You can
therefore split that up into 13 uniform 1-inch long imaginary fret positions.
But if your open string length is 13 and a half inches long, DON'T use a 1-inch
sized imaginary fret spacing!, leaving a 1/2-inch piece at the end (especially
not at the bridge end!). Lengthen this uniform distance between your imaginary
frets slightly so that you've segmented that 13.5-inch distance into an even 13
pieces, or shorten it slightly into an even 14 pieces.


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