How many tone rows are there?

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billiamo
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How many tone rows are there?

Postby billiamo » Wed Feb 18, 2009 10:54 pm UTC

This one's for the mathemticians and the musicians both!

How many arrangements of the twelve pitches in the equal temperament system are? Obviously, you could just say that it's 12!, but I want to follow the ideas of the serialist and twelve tone composers, and thus disregard the transpositions of a given tone row, as well as their retrogrades and inversion. So this leaves us with 11!/4 - something like 9.9 million.

But this won't be complete. For example, take our initial tone row to be an ascending chromatic scale from A to G sharp. An ascending chromatic scale beginning on on B flat and going to A will not only be the inversion and retrograde of the initial tone row, but also one of its transpositions.
There are a couple of other problems like this, and I don't have the maths knowledge to work it out more precisely. Any suggestions or solutions?

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lira_riu
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Re: How many tone rows are there?

Postby lira_riu » Thu Feb 19, 2009 4:23 am UTC

I don't have my textbook on me, but I think there's a listing of which hexachords (and therefore tone rows) are only partially invertible or transposable. I might look it up when I get back home after my interview this weekend.

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Masily box
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Re: How many tone rows are there?

Postby Masily box » Fri Feb 20, 2009 6:51 am UTC

The answer is that there are 9,985,920 distinct "row classes," though how you get there is a little complex.

Luckily, that's what music theory journals are for:

Julian Hook, "Why Are There Twenty-Nine Tetrachords? A Tutorial on Combinatorics and Enumeration in Music Theory" in Music Theory Online, Vol. 13 No. 4, December 2007:

http://societymusictheory.org/mto/issue ... .hook.html

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ChocloManx
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Re: How many tone rows are there?

Postby ChocloManx » Fri Feb 20, 2009 3:53 pm UTC

how funny that it all sounds the same, then.

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Re: How many tone rows are there?

Postby Marbas » Fri Feb 20, 2009 7:30 pm UTC

how funny that it all sounds the same, then.

burnnnn


Says the guy who listens to prog-rock.

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Antimatter Spork
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Re: How many tone rows are there?

Postby Antimatter Spork » Sat Feb 21, 2009 9:32 pm UTC

ChocloManx wrote:how funny that it all sounds the same, then.

burnnnn

No it doesn't. It sounds as different as is technically possible!
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Re: How many tone rows are there?

Postby ChocloManx » Sat Feb 21, 2009 11:45 pm UTC

I was just kidding by the way, I quite like some dodecaphonic musics.
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Masily box
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Re: How many tone rows are there?

Postby Masily box » Sun Feb 22, 2009 5:42 am UTC

Antimatter Spork wrote:No it doesn't. It sounds as different as is technically possible!



Not sure what you mean by this. That it sounds incredibly different from common practice music?

billiamo
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Re: How many tone rows are there?

Postby billiamo » Mon Feb 23, 2009 11:20 am UTC

I think that he meant it has the most possible variation within any style, as you're working with all possible pitches in every one of their possible 9,985,920 permutations, arranged in various rhythmic and registral patterns.

And how does prog rock even remotely all sound the same?

Thanks for the article, Maisly Box, I'm reading it right now.

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Masily box
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Re: How many tone rows are there?

Postby Masily box » Mon Feb 23, 2009 4:54 pm UTC

I see what you mean. But actually, the requirement that you cycle through an entire row before reusing a pitch class places really strict limits on what kinds of sounds are possible. Statistically, in a piece of tonal music, you'll get lots more of certain notes, whereas in 12-tone music, things can be pretty homogeneous. It's one of the ironies of music that the more differentiated you try to make a piece sound, the more likely it is to sound "all the same". (At least, if you take the path of the serialists, but the other major path that I see is that of the stylistic polyglot, which threatens to do the same thing.)

As for prog rock (maybe what I'm saying is obvious, in which case please ignore), it's an experience I've had often. When first introduced to 16th century masses, musical theater, Frank Sinatra, Indian music... my first reaction is always "but this all sounds the same!" As far as I've reasoned, it's because a novice isn't able to distinguish aspects of the genre from details of the individual piece. Every genre has its own range of constrained variation, and it's only after you figure out that range that you can understand what makes a particular song unique and interesting.

You're welcome for the article... always happy to spread the music theory love :D

billiamo
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Re: How many tone rows are there?

Postby billiamo » Tue Mar 03, 2009 1:29 pm UTC

I never said I agreed with him, though! What you say is true, you need to get a knowledge of a certain style to appreciate its differences.

And then today I had my first class dealing with Fortean analysis and set theory and stuff. Turns out there's only 208 chords? Pretty cool.

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Masily box
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Re: How many tone rows are there?

Postby Masily box » Tue Mar 03, 2009 6:45 pm UTC

billiamo wrote:Turns out there's only 208 chords?



Only to the extent that C major and F minor are the same chord. :D

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Antimatter Spork
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Re: How many tone rows are there?

Postby Antimatter Spork » Sat Mar 28, 2009 2:56 pm UTC

Masily box wrote:
billiamo wrote:Turns out there's only 208 chords?

Only to the extent that C major and F minor are the same chord. :D

Well, in an equal-tempered environment, all major chords are identical in terms of sonority, so kind of, I guess. Though I suppose if you look at interval content, but the difference in sonority between major and minor chords should be enough to argue that the order of the intervals in a chord does matter but whatever I'm not going to get into a huge set theory argument with myself here.

Also 12-tone music is almost never written in what people perceive as the "strict" way, where you actually follow all the rules. Composers usually repeat notes, sustain notes, include repeating motives, and so forth. Some people have actually written tonal music that is also 12-tone (Berg actually did this kind of a lot), and other people (like Hindemith) have done tonal analysis of the atonal 12-tone works of people like Schoenberg. Also, you don't even need to use 12-tones to write atonal music. In his late period, Stravinsky wrote serial music using shorter rows (often 5 or 6 notes). The music is still definitely serial and atonal, but it isn't 12-tone.

And any kind of music that a listener is not familiar with is going to "sound the same" until they can become familiar with the surface conventions enough to hear the actual differences between a couple of given pieces (or else all that they'll be able to pay attention to will be the surface qualities like "this is played on electric guitar" or "it uses the western major-minor tonal system" or "the harmonies sound very dissonant compared to what I'm used to hearing")
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Masily box
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Re: How many tone rows are there?

Postby Masily box » Sat Mar 28, 2009 9:26 pm UTC

Antimatter Spork wrote:but whatever I'm not going to get into a huge set theory argument with myself here.



Ah, but those are among the most fun kinds of arguments to have. :D

FWIW, I think set theory is fantastic, but it's worth keeping close to mind what it's assumptions about equivalence are. (And remembering that equivalence != identity.)

And, of course, set theory can be useful in largely tonal contexts, too, like Prokofiev and even Wagner.

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Antimatter Spork
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Re: How many tone rows are there?

Postby Antimatter Spork » Sat Mar 28, 2009 9:40 pm UTC

That's because both Wagner and Prokofiev were completely nuts as far as pitch goes.

And yes, set theory rules.
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