Counting Past 12...
A while back... last century, Arnold Schoenburg decided that we had run out of possibilities for music based on our 12 tone equal tempered scale. So he invented 12 tone serialism in an attempt to creat an atonal music and move beyond the limitations of harmony.
His system required some very rigid rules to make sure tonality didn't accidentally creep into the music.
There's an easier way. I'm not sure why Schoenburg didn't think of it... all he had to do was change to 13, 14, 15 tones instead of 12. Equally tempered (ET) scales that include a perfect fifth are fairly rare. Technically, no ET scale contains perfect fifths because (3^x)/(2^y)=2^(z/n) only if x,y, and z all equal 0 for integer values of x,y,z regardless of the positive integer value of n. (Don't worry about the math if you didn't get it.)
But some ET scales have an interval close enough to a perfect fifth that you can't really hear the difference. The 12 tone scale is one of these.
So why's that important? Well, the perfect fifth actually exists as an audible physical reality. There are other intervals implied by the sequence of overtones - which is a feature of physics... but we really can't hear them in nature. As a result, scales which ignore all these intervals can sound tonal... but lacking a perfect fifth makes a tonal sound pretty much impossible.
So, if Schoenburg had only used a 13 tone scale, which has no good approximation of a perfect fifth... anything he played would have sounded atonal...
Of course, getting a piano tuned to a 13 tone scale might have been difficult.
On a slighty different tack...
I just finished reading a science fiction book, The World is Round by Tony Rothman. (I highly recommend this book).
How does this tie in? Well, the book is set in a culture that uses a 31 note scale in their music. Rothman doesn't say whether the scale is ET, but it might be. An ET scale of 31 notes has a pretty good fifth: 1.496 vs 1.500 (perfect fifth). Actually when you get a large number of notes, close to perfect fifths start getting common. The 31 note scale also has 1.529 - you could probably hear the difference but it would sound like an out-of-tune fifth.
The music in the book got me thinking about trying some scale other than 12 tone... I don't really want to mess with 31 tones, but maybe something more than 12.
But, I want to stay tonal at least at first... It turns out that 17 tones has a pretty good fifth: 1.503 (For reference, the 12 tone scale fifth is 1.498).
So now I just need an instrument to play it on...
That's what I love about cigar box guitars... for $10 in parts and a few hours of work... I have a perfectly good instrument tuned to a 17 tone ET scale.
It even has a cool looking head stock:
His system required some very rigid rules to make sure tonality didn't accidentally creep into the music.
There's an easier way. I'm not sure why Schoenburg didn't think of it... all he had to do was change to 13, 14, 15 tones instead of 12. Equally tempered (ET) scales that include a perfect fifth are fairly rare. Technically, no ET scale contains perfect fifths because (3^x)/(2^y)=2^(z/n) only if x,y, and z all equal 0 for integer values of x,y,z regardless of the positive integer value of n. (Don't worry about the math if you didn't get it.)
But some ET scales have an interval close enough to a perfect fifth that you can't really hear the difference. The 12 tone scale is one of these.
So why's that important? Well, the perfect fifth actually exists as an audible physical reality. There are other intervals implied by the sequence of overtones - which is a feature of physics... but we really can't hear them in nature. As a result, scales which ignore all these intervals can sound tonal... but lacking a perfect fifth makes a tonal sound pretty much impossible.
So, if Schoenburg had only used a 13 tone scale, which has no good approximation of a perfect fifth... anything he played would have sounded atonal...
Of course, getting a piano tuned to a 13 tone scale might have been difficult.
On a slighty different tack...
I just finished reading a science fiction book, The World is Round by Tony Rothman. (I highly recommend this book).
How does this tie in? Well, the book is set in a culture that uses a 31 note scale in their music. Rothman doesn't say whether the scale is ET, but it might be. An ET scale of 31 notes has a pretty good fifth: 1.496 vs 1.500 (perfect fifth). Actually when you get a large number of notes, close to perfect fifths start getting common. The 31 note scale also has 1.529 - you could probably hear the difference but it would sound like an out-of-tune fifth.
The music in the book got me thinking about trying some scale other than 12 tone... I don't really want to mess with 31 tones, but maybe something more than 12.
But, I want to stay tonal at least at first... It turns out that 17 tones has a pretty good fifth: 1.503 (For reference, the 12 tone scale fifth is 1.498).
So now I just need an instrument to play it on...
That's what I love about cigar box guitars... for $10 in parts and a few hours of work... I have a perfectly good instrument tuned to a 17 tone ET scale.
It even has a cool looking head stock:
Well this post has gotten pretty long. Next time: 17 tone music theory and maybe a sample mp3.
posted by Doctor Oakroot at
2:05 PM
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