Microtonal Music

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Adam H
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Microtonal Music

Postby Adam H » Mon Nov 03, 2014 11:14 pm UTC

This is a very specific topic that not very many people in the general population are interested in, but maybe I'll get a nibble?

(I'm not sure Microtonal is the right term for what I'm talking about. Wikipedia suggests "xenharmonics" and I would have just named it "non-standard tunings".)

Here's the wikipedia page if you want to dip your toes in. Basically, the common western 12-tone scale is not the only one that produces consonant harmonies. In fact, the equal tempered 12-tone scale - which we're trained to accept - is pretty bad at it!

My feeble attempt at explaining why tuning systems are more complex than most people think:
Spoiler:
If you play a note on any instrument, there tend to be audible "overtones" on top of the base frequency. If the base frequency is 100hz, the harmonic overtones are 200hz, 300hz, 400hz, 500hz, etc. These overtones have varying loudness which gives an instrument its unique sound. E.g. clarinets have notably quiet even-numbered overtones (2x, 4x, 6x, etc).

An octave above any particular note is calculated by multiplying the base frequency by two. So if the base note is 100hz, the octaves above that are 200hz, 400hz, 800hz, 1600hz, etc. Now you can see that there are overtones between the octaves: 300hz, 500hz, 600hz, 700hz, 900hz, etc. That means that when you play a note, you are playing overtones that are neither the base note nor octaves of the base note. The overtones are new notes that all sound "consonant" when played with the base note.

OK, let's define 100hz as a G note, since it's pretty close. Then 200hz and 400hz are both G's as well. The second overtone, 300hz, is close to a D(perfect fifth), and the fourth overtone, 500 hz, is close to a B (major third). I don't think it's a coincidence that those first two non-G overtones form the very pleasing major G chord... And unsurprisingly, the next few overtones also correspond with other notes in the 12-tone scale. 700hz is an F (minor seventh), 900hz is an A (major second), etc, etc, etc.

Except here's the problem. If you define a semitone by dividing an octave into 12 equal intervals - like we commonly do in western music; it's called equal-tempering - you can't get any of the non-octave overtone frequencies by going up and down by semitones. If your base frequency is 100hz, the closest you can get to 300hz by using semitones is 299.7hz. Which, granted, is pretty darn close. But the closest you can get to 500hz is 504.0hz, which is not really close at all!

The end result of all this is that when you play G and B together, you're playing 100hz, 504hz, and all of the overtones, including 500hz. Therefore it sounds like you are playing 500hz and 504.0hz together, which sounds out of tune even though it's precisely in tune for how we've defined the tuning system. If you are a good performer you will tend to bend the pitches slightly to compensate, but only if you are using an instrument that can pitch-bend, like your voice or a wind instrument. But on a piano, for example, bending the pitch is not going to happen.

So why do we use the equal tempered 12 tone scale? It's actually pretty arbitrary IMO. It approximates the overtone frequencies better than other numbers of equally spaced tones, up until the 53 tone scale which is nearly spot on. But it's hard to tell the difference between the semitones in the 53 tone scale, not to mention inconvenient.

Instead of defining a semitone by dividing the octave equally, we could define every note by using the frequencies in the harmonic overtone series (called "just tuning"). But then when we move to a new base note (e.g. playing a song in the key of A instead of the key of G), every single note needs to change frequencies slightly in order be in tune.


On top of all that crap, there are harmonic overtones that we don't even attempt to hit with the 12-tone scale. Relatively consonant harmonies that you may have never even heard before. ;)


My favorite nonstandard tuning is the Bohlen-Pierce scale. It sounds totally alien at first, but after I get reoriented it clicks into place and can become hypnotic. And the theory behind it fascinates me. Plus, it's super convenient how the equal tempered version is very close to the just version.

Unfortunately, it appears as if I've run out of Bohlen Pierce music to listen to! Youtube and google seem to be tapped out; I'm not sure if that's because google has tailored my searches or because there's very little music out there. I keep running into Elaine Walker ("Stick Men" is pretty good), and Richard Boulanger's "Solemn Song for Evening" was recommended in the book I'm currently reading. There are a few excellent clarinet solos on youtube. There are a couple versions of Canon on youtube (which IMO are neither creative nor pleasant to listen to). There's a dubstep Bohlen-Pierce song on youtube (Sevish - Mako Haze) that is amazing. And my favorite has been some of the stuff off of Charles Carpenter's record "Splat". You can listen to a few songs here.

Has anyone dabbled in composing with microtonal systems? I'm trying, but the only method I've come up with is to have Audacity (awesome free program) generate sine waves which I import into my main recording software (Cubase). It would be nice to use my midi controller but I don't know how to set the keys to play audio files. Pretty sure it can do it. Maybe I'll sit down with the manual this week and figure it out.
-Adam

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Re: Microtonal Music

Postby azule » Tue Nov 04, 2014 1:50 am UTC

Interesting topic. It would probably work better if I could listen to those samples, but good thing I understood your technical explanation.

So did I read it right that 53 is the magical number of divisions?

This sounds neat. Would some chords still be possible that have a darker, dissonant feel?
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Re: Microtonal Music

Postby ahammel » Tue Nov 04, 2014 5:58 am UTC

[url=/https://en.wikipedia.org/wiki/Spectral_music]Spectral music[/url] makes extensive use of microtones and is awesome. I like Tristan Muriel's Gondwana and Seven Lakes Drive. I think he also wrote an electric guitar solo, if that's your kind of a thing.
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Re: Microtonal Music

Postby Adam H » Tue Nov 04, 2014 4:37 pm UTC

azule wrote:Interesting topic. It would probably work better if I could listen to those samples, but good thing I understood your technical explanation.

So did I read it right that 53 is the magical number of divisions?

This sounds neat. Would some chords still be possible that have a darker, dissonant feel?
There's actually no magical number of equal divisions that allow you to create mathematically "perfect" intervals. To divide an octave into equal intervals you have to use irrational numbers because frequency is exponential. Basically, you are trying to get nice even fractions like 3/2 and 5/4 from the equation f=2^(a/b), where b is the number of divisions and a is the scale step. Example: for b=12 (12 tone scale) and a=4 (4 semitones up from the base note), f equals 1.2599 which is "close enough" to 5/4 (the interval of a major third). Compare that with the 53 tone equal tempered scale: for b=53 and a=17, f=1.2490 ~ 5/4. Similarly the other intervals that we care most about (perfect fifth, minor seventh, major second, etc) are closer to the "ideal" in the 53 tone scale, but they aren't perfect. And there are other values for b that work well, including 19 and 31.

As for dissonance... I don't know if this is a thing that is true and music theorists would agree with, but I would say there are two types of dissonance (for how the layperson defines dissonance). One type is when something sounds out of tune and you hear beats from the wavelengths going in and out of phase. The other is that sense that a harmony just sounds... weird. Playing a tritone (e.g. C and F#) on a piano has both of those types of dissonance. But there's an interval not found on the piano that is very close to it (7/5 * base frequency) that sounds dissonant in a good way, IMO. When there are no beats you are left with a very pure sounding interval that still sounds dark and alien.

So IMO microtonal music can do dissonance really well, because when you take away the bad kind of dissonance, you can dwell on the good kind of dissonance. That's just my opinion though, I don't know if anyone who really understands this stuff agrees with me.

Another artist to take a look at is Dolores Catherino. She uses this 106 tone (53*2) keyboard that I can't even fathom. I think this is the video I saw (youtube is blocked at my work so I can't check).

I'm excited to check out spectral music, thanks for the tip. :)
-Adam

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Re: Microtonal Music

Postby Envelope Generator » Tue Nov 04, 2014 5:55 pm UTC

I remember experimenting a little with microtunings using ZynAddSubFX which supported arbitrary tunings. One cool thing I found was a scale where a "semitone" was something like 44.5 cents and a "whole" tone was around 89 cents. Then you could span an octave with a repeating pattern of tone-tone-semitone (or tone-tone-tone-semitone, can't remember), producing a symmetric scale. Since this "whole" tone wasn't hopelessly out of tune to my ear w/r/t a 12-TET semitone, I could produce quite normal sounding music and then chuck in 45-cent semitones that would throw everything off in a way that was hard to pin down.

In general, though, notwithstanding the good run that quartertones had for nightmare soundscapes in late 20th century classical music, the problem with the western learned microtonality is that it attracts few top tier musicians. It's always been a domain of eccentrics.
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Re: Microtonal Music

Postby DR6 » Tue Nov 04, 2014 7:05 pm UTC

Adam H wrote:
Here's the wikipedia page if you want to dip your toes in. Basically, the common western 12-tone scale is not the only one that produces consonant harmonies. In fact, the equal tempered 12-tone scale - which we're trained to accept - is pretty bad at it!

What do you mean? Equal tempered 12-tone scale is very good at accomodating consonant harmonies, because due to its periodicity intervals don't produce dissonancies when accumulating. Sure, it produces a ratio of 1,4983 rather when it should be producing 1.5, but who cares? (Definitely not human ears)

That doesn't mean other scales can't be worth it: of course their have their own advantages, but saying that equal tempered 12-tone is "pretty bad" is unfair.

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Re: Microtonal Music

Postby Adam H » Tue Nov 04, 2014 9:20 pm UTC

DR6 wrote:
Adam H wrote:
Here's the wikipedia page if you want to dip your toes in. Basically, the common western 12-tone scale is not the only one that produces consonant harmonies. In fact, the equal tempered 12-tone scale - which we're trained to accept - is pretty bad at it!

What do you mean? Equal tempered 12-tone scale is very good at accomodating consonant harmonies, because due to its periodicity intervals don't produce dissonancies when accumulating. Sure, it produces a ratio of 1,4983 rather when it should be producing 1.5, but who cares? (Definitely not human ears)

That doesn't mean other scales can't be worth it: of course their have their own advantages, but saying that equal tempered 12-tone is "pretty bad" is unfair.

Meh, I feel comfortable saying that thirds and sixths in the equal tempered 12-tone scale are "pretty bad". They're off by 10-15 cents. YMMV.
-Adam

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Re: Microtonal Music

Postby shieldforyoureyes » Wed Nov 05, 2014 6:07 am UTC

I had a 1/4-step fretted guitar custom built back in the early 90s. It was certainly interesting, but I never was really productive with it.

On the rare occasion that I do anything musical these days, it's some sort of algorithmically-generated ambient/drone/harsh-noise for movie soundtrack, and I tend to break my octaves into divisions other than 12. Usually primes.

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Re: Microtonal Music

Postby tarascon » Tue Jun 09, 2015 2:39 pm UTC

On another site I'd started a thread called "Smashing the Envelope" which focused on--for lack of a better word as well as attempting to be all-inclusive--avant-garde music since pre-WWI. This included Minimalism, Serialism, 12-Tone, Noise, etc. A few folks liked the topic, most didn't and the thread died a slow, lingering death. Fading away like the echo of a minor chord.

My pick for this first post: Morton Feldman.


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