The Pythagorean Tetractys, or how multiplying by 2 & 3 gets you all the notes you need, with stops at the Pythagorean Comma and Equal Temperament along the way. Like my previous post, a deep rabbit hole, but — to me at least — a very rewarding one.
(Originally posted as part of #100daysofpractice 2023)
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Fascinating! The Pythagorean sequence sounds like “stretch tuning.” Do human tuners and/or electronic keyboards incorporate to some degree the enharmonic discrepancy you’ve demonstrated?
Some do, to various degrees
That was quite a magical video thank you! This ought to be taught in music schools.
Hi Dan,
Great vid on the Tectractys. I wasn’t aware of this. I did know about Chinese theory of construction of scales by building in fifths. Which also produces the Pythagorean comma. Your explanation was much clearer.
I played in Lamont Youngs Theater of the Eternal Music Big Band in the 90’s, The piece was based on 60 HZ and used Lamont’s tuning system based on prime numbers. It was an eye opening experience which has shaped my musical exploration.
I want to mention a demonstration I attended some years ago. It was 2 keyboards a piano and a harpsichord. The Harpsichord was tuned to Pythagorean,(if I remember correctly), and the piano to equal temperament. The Pythagorean overtones roared in an extremely vibrant manner while the equal temperament sounded lifeless and dull. It was an eye opener for me about tuning and our standard system. When the Pythagorean tuned instrument got about 3 keys from the tuned key it was chaos. I think we have sacrificed a lot to be able to modulate to different keys, but then we wouldn’t have bebop! or so much other great music. I play trombone so intonation is very different than most instruments other than fret-less strings. This gives me an excuse to be out of tune, I can claim I’m in other tuning’s, (lol). Thanks for your work, and I look forward to seeing more and checking out your music.
Peter
There’s no question that equal temperament is a big tradeoff.
E# and B#. Extremely cool. Thanks for the journey!
Thank you Fred!
Can I get the program please?