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Author Topic: Just Intonation Composition  (Read 52461 times)
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« Reply #20 on: Jan 24, 2008, 02:26AM »

ooops. ron was the right name yeah :)
i've been to a composition class with him once. his a cool guy

and heh no it's not me. it's a norwegian singer called siri gjære :p
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« Reply #21 on: Jan 24, 2008, 03:46AM »

siri gjære

Wow; I don't think I could pronounce that if I tried.
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« Reply #22 on: Jan 24, 2008, 03:49AM »

Ron Nagorcka, perhaps?

http://www.ronnagorcka.id.au/

Hmmm . . .

At first glance, he seems to take to some of John Cage's ideas. Or perhaps native Japanese music. Reflecting nature and stuff.
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« Reply #23 on: Jan 24, 2008, 04:01AM »

By the way, here are some useful websites I've found concerning just intonation:

http://www.kylegann.com/microtonality.html

The crash course on Mr. Gann's page is excellent.  Wikipedia has some good stuff, too:

http://en.wikipedia.org/wiki/Microtonal_music
http://en.wikipedia.org/wiki/Just_intonation
http://en.wikipedia.org/wiki/Tonality_diamond
http://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale

The Microtonal Music article has a huge list of composers at the end, and for many tells what general category of theory they used, whether it was just intonation or some form of equal temperament or something else.
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« Reply #24 on: Jan 24, 2008, 04:19AM »

. . . so how do you notate all of this?

I'm late joining this conversation, but the same question came up when I was putting together my little just intonation scale and arpeggio practice book, Breakfast.  There I show adjustments in just one place, in a major scale and in a minor scale on page 13, expressed as cent adjustments, like -14, relative to the equal tempered scale. Elsewhere the performer trusts his ear and the accompaniment.

Fine barbershop quartets - for example the Buffalo Bills in the television production of The Music Man a few years ago - certainly don't need notational guidance to sing in just intonation. And surely that is how they sing. So, as Dave T. said up top, trust the performers.

On the computer your music notation program may have a pitch bend function which lets you adjust each note. That function may not print the adjustments on paper, but you can see them on the screen. The program takes care of the documentation and notation for you.

For a thorough discussion of the topic I highly recommend W.A. Mathieu's Harmonic Experience.  At the link, select table of contents to see the broad scope of this 576 page book.  In an appendix, Glossary of Singable Tones in Just Intonation, Mathieu identifies sixty intervals within the octave by ratios to the tonic!

Good luck, Andrew.

David

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« Reply #25 on: Jan 24, 2008, 07:43AM »

For a thorough discussion of the topic I highly recommend W.A. Mathieu's Harmonic Experience.  At the link, select table of contents to see the broad scope of this 576 page book.  In an appendix, Glossary of Singable Tones in Just Intonation, Mathieu identifies sixty intervals within the octave by ratios to the tonic!

Good luck, Andrew.

David

Well, well. That book looks like exactly what I'm looking for.

Thanks! Good!
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« Reply #26 on: Feb 01, 2008, 10:46AM »

Here's a slight update:

I received some just-tuning composed music, and some of it is rather awful, and some of it is absolutely awesome. I'll cover the awesome:

The Bells of New Albion by Terry Riley. Riley is known as one of the important minimalists, but he really doesn't deserve the title. This piece is a 2 hour long solo piano work, but it involves a whole lot of improvisation as well as the tune structures which do involve repetitive elements (but not strict like in early Steve Reich or Phillip Glass works) - I really like it a lot. You don't need to listen to the entire two hours at once; he divides the piece into many discernable chunks, and you can listen to each chunk (5-20 minutes) at your leasure. That just-tuned piano is really something else!

And the other masterpiece I found is a set of string quartets written by Ben Johnston and performed by the Kepler Quartet. Man oh man, is this a great album, and man oh man will this album challenge your ears if you're not ready for it! Unlike a piano, in which no matter how creative you get can only really use twelve different pitches, stringed instruments are a lot more flexible in their tuning, and Johnston really exploits that. Not only is the music harmonically and rhythmically challenging, he writes great melodies and gets a huge palette of tone color out of the strings, and the Kepler Quartet really put forth a passionate performance of this stuff; I'd say it's probably the best recorded string quartet performance I've ever heard.
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« Reply #27 on: Feb 08, 2008, 06:44AM »

How do you suppose he notated that string quartet music? 
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« Reply #28 on: Feb 08, 2008, 09:38AM »

I did get an explanation from Kyle Gann of the notation used in the links below:

Quote
FAC, CEG, and GBD are purely tuned 4:5:6 triads

# = 25/24
b = 24/25
+ = 81/80, adjusting for the syntonic comma
- = 80/81
7 = 35/36
upside-down 7 = 36/35
^ (arrow pointing up) = 33/32
v (arrow pointing down) = 32/33
13 = 65/64
upside-down 13 = 64/65

It's not really possible to do justice to this in typing. The arrows are really arrows, and the sevens are sometimes combined with sharps and flats as a little diagonal line hanging from the top or rising from the bottom. I've been looking around the web for a sample of one of Ben's scores, and all I've found is this one, which doesn't use many of the accidentals:

http://www.smith-publications.com/catalog/samples.pl?id=29

Actually, here's another, which has a little more going on:

http://www.smith-publications.com/catalog/samples.pl?id=11

What people object to are the use of pluses and minuses, which sometimes seem counterintuitive; for instance, C-G, E-B, and F-C are perfect fifths, but D-A is not - the A needs a + or the D a -. There's a competing notation called HEWM using the same symbols but starting with the Pythagorean scale instead of pure triads, which I don't like. And in any case, while the notation makes the relationships perfectly precise, it still doesn't provide much easily digested information for the performer, who has to pretty much memorize where the pitches are anyway.
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« Reply #29 on: Feb 27, 2008, 12:19AM »

Just a useless random addition to this whole idea:

I was thinking about how 3/2 is the perfect fifth ratio, which puts a pitch exactly between the octaves, but our ears tell us that the equal tempered tritone is actually right in between, based on how many steps there are to get to the tritone.

So I was thinking this, then I ended up doing some math.

For starters, the equal tempered tritone is simple: starting pitch * square root of 2.

And I was thinking that it's funny how the various just tuned tritone intervals seem to evenly spread around this equal tempered pitch. For example, 7/5 is 17 cents flat, and 10/7 is 17 cents sharp. And then I thought some more, and in fact there are a whole bunch of just tuned pitches that do the same thing, and cents away from the tritone also correspond to cents away from the octave root. For example, that 7/5 tritone, being 17 cents flatter than the equal tempered tritone, is also 583 cents away from the lower octave root, and the 10/7 tritone is 583 cents away from the upper octave. Major thirds are the same way: a major third is 14 cents flatter than the equal tempered third and 386 cents above the lower octave and the minor sixth is 14 cents sharp, 386 cents below the upper octave.

So, I was thinking this, and I was thinking that it might be cool if I could find out why this pattern was happening. I did a bit of basic math remembering (and a little research to fill in the gaps), and I came up with this:

log2(x/y)=1-log2(2y/x)

This is log base 2, and x and y are the numerator and denominator of the pitch ratio in question. It's actually pretty simple; simply multiply both sides by 1200 and each side represents the number of cents toward the octave. And seeing how this example requires the log base to be 2, it becomes easier to see why the pitches also "revolve" around the square root of 2. Oh, and 2y/x is how you get the inverse interval: a major third, being 5/4 has as it's inverse a minor sixth: 4*2/5 or 8/5. Same with the above-mentioned tritone: 7/5 and 5*2/7=10/7

Another way of thinking about this is that our ears hear intervals via multiplication of frequencies, not by addition of frequencies. Multiply the square root of 2 (the E.Q. tritone) by itself, and you get 2, which DOES give you double the starting pitch. Multiple 3/2 (the perfect fifth) by itself and you get 9/4, which does NOT double your starting pitch. Duh.

Hooray for basic algebra!!! Idea! Idea! Idea! Good! Clever Clever Amazed Eeek!
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« Reply #30 on: Feb 27, 2008, 07:56AM »

Another way of thinking about this is that our ears hear intervals via multiplication of frequencies, not by addition of frequencies.

Doesn't this result just pop right out of the fact that intervals are ratios?

I think what you've done, interestingly enough, is start from the calculations for equal temperament and derive the octave, rather than the other way around.  You've also run into some of the wonderful properties of exponents and logarithms.  It's great stuff.

This kind of exercise is why I'd like to teach math someday.  I think these results are pure beauty, and I want to help other people see it.
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« Reply #31 on: Feb 27, 2008, 09:14PM »

Doesn't this result just pop right out of the fact that intervals are ratios?

Yeah; I suppose that is a pretty good clue.


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« Reply #32 on: Mar 02, 2008, 11:20AM »

A piece of software you can use to help experimentation (create a MIDI with performance or notation software) is Scala:
http://www.xs4all.nl/~huygensf/scala/

(I mentioned it before, but I'll explain how I used it to experiment with Just Intonation.)

The feature I liked about it was that it Can retune existing MIDI files. You can convert a standard MIDI file to be in any tuning via pitch bend commands or a MIDI Tuning Standard tuning specification.

This feature makes it a useful tools for sort of quickly previewing simple music using Just Intonation. Basically, you have to export each tonal center as a different MIDI, create 12+ scale files (one for each key), apply appropriate scale files to the appropriate sections, and then find a way to playback the sections consecutively. For complex music, or note perfect music, you would still have to find a way to manually adjust for the different types of intervals (e.g. grave minor seventh versus minor seventh), but at least a bulk of the work could become automated when you're producing simple previews.

- - -

If you're looking for beatless music, Just Intonation won't go that far in all situations. (For example, the ensemble trick is to play a just dominant chord, but to weaken the volume of the dominant seventh to mask the beats produced by the seventh in a just setting.)

To make reading easier, most composers make use of the five-line notation system and create a way to notate deviations to that system. Of course, in the realm of easy-to-play, one could use regular notation with a prepared piano or one could write music playable by computer using any exotic notation that can eventually be read by the computer.

- - -

The math used to calculate this stuff is pretty simple. A spreadsheet I lost did this, but it calculated for any given intonation standard (e.g. 440) and base frequency a list of the cents from root for ET, cents from root for Just (by converting the frequencies to the relative [per]cent notation), and found the exact adjustments through subtraction. This is still all abstraction and is a few steps removed from being able to calculate how to adjust any set of frequencies to be as beatless as possible (or the math--probably simple math considering sound revolves around multiples--needed to describe beatless combinations of frequencies).
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« Reply #33 on: Mar 03, 2008, 12:44AM »

If you're looking for beatless music, Just Intonation won't go that far in all situations.

Actually, no. I'm looking for more direct control over consonance vs. dissonance.

That program does look interesting. I'll have to check it out the next time I have a chance.
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« Reply #34 on: Mar 30, 2008, 07:53AM »

Just a slight update on my literature search.

As some of you may know, a lot of folk music (not all) is tuned to just intonation. Perhaps the most commonly heard of those is bagpipe music.

My roommate is a big fan of early European music, and he showed me a really interesting CD that uses just intonation:

"Myths from Medieval Iceland," by a group called "Edda Sequentia." Sung in Icelandic, with some traditional instruments; the most identifiable to me being a kind of viol. Some of the voice leadings are very unique; it caught my ear. I thought some of the voice crossing resulting in major second intervals was particularly striking, given that those seconds were still tuned to the original just tuned scale, meaning that each second interval was not necessarily the same.
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« Reply #35 on: Mar 31, 2008, 11:58PM »

As some of you may know, a lot of folk music (not all) is tuned to just intonation. Perhaps the most commonly heard of those is bagpipe music.


Just idly wondering about something.

Inharmonicity.

For example, the piano is not actually tuned to equal temperament.  Because of inharmonicity (the stiffness of steel strings causes the overtones to not quite line up) the piano sounds bad if the fundamental of each pitch is tuned to ET, because then the overtones clash.  So the piano is stretched, moving the fundamentals away from ET, so that the overtones are close enough that the piano sounds like it is ET even though it is not quite. 

The same thing must apply to just intervals (not sure just temperament itself is possible.  But intervals are.) 

But not all instruments have this problem.  An organ wouldn't, for example.  The pipes are driven by constant input, therefore the overtones line up.  Bowed strings would be the same.  Plucked strings would be different. 

So, we have folk music with bagpipes or hurdy gurdy, no inharmonicity.  But with guitar or hammered dulcimer, large inharmonicity.  Accordions with steel reeds, even more. 

I'm not asserting any point here, just bringing up something I started wondering about. 
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« Reply #36 on: Apr 01, 2008, 12:51AM »

 Don't know

I'm not quite sure you're correct about piano tuning not quite being equal tempered. Obviously, a piano tuned by ear alone will not be. But symthesized pianos definitely are. And pianos tuned with the aid of a tuner should be pretty darn close. And I'm not sure I agree mathematically the reason for adjusting away from equal temperament, either. By definition, nothing besides an octave is in tune in equal temperament, so you're not necessarily making things more "out of tune." Overtones of non-just tuned strings should clash regardless of whether it's equal tempered or not.

Are you talking about a 19th century tuning which was really a refinement of the well temperament that Bach's famous piece was written for? In that temperament, notes aren't quite evenly spaced; in the link below, look for Thomas Young's system:

http://www.kylegann.com/histune.html
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« Reply #37 on: Apr 01, 2008, 03:40AM »

Don't know

Obviously, a piano tuned by ear alone will not be.

Pianos tuned by ear are tuned to have a specific number of beats sounding for various intervals.

But you listen for these beats in the upper partials, not in the fundamentals.

Pianos tuned electronically (ETDs, Electronic Tuning Devices) actually mimic the ear.  They have software that detects different partials, measures the inharmonicity, and calculates the stretch - which is how far off you have to tune the fundamental.  That is, how far off from ET.  You can set this software to detect whatever partial you want.  TuneLab has a free download if you want to play with it. 
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« Reply #38 on: Apr 03, 2008, 11:37PM »

Just to add something I learned yesterday:

Even octaves are not tuned ET on the piano.  Yes, that was a surprise to me. 

Octaves are tuned to a compromise between 4:2 and 6:3.  That terminology was new to me.  An octave has a nominal frequency ration of 2:1, obviously, for the fundamental.  But the tuner doesn't listen to the fundamental.  He can make the 4th partial of the lower octave correspond to the 2cnd partial of the upper octave, OR he can make the 6th partial of the lower octave correspond to the 3rd partial of the upper octave, but not both.  But the articles I read say that the best results come from tuning the octave somewhere between those choices. 
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« Reply #39 on: Apr 04, 2008, 02:23AM »

Hmmmm . . .

Something tells me that I won't actually understand what's going on until I start doing wave analyses with the sound.

But, upon thinking about what you've told me so far, I am reminded that inharmonicity may make the octave partials not exactly correspond to the whole number ratios. Wierd, given that octaves are pretty much THE simplest interval besides a unison.

Do you know if this is a specific cent or microcent difference between the pitches to tune a proper octave on the piano due to this effect?

This reminds me that on the trombone, the partials aren't exact integer multiples of each other either, primarily because the trombone is not a perfectly cylindrical instrument.
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