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Can we take up the comma by stretching octaves?
#2933756 01/14/20 02:11 PM
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You know, the comma of Pythagoras, the amount that 12 fifths are wider than 7 octaves. It's a quarter of a semitone, from what I read, which is 25 cents, measured at the top of the treble I suppose. We already stretch the extreme octaves by 20 cents. Why isn't it possible to tune pure fifths all the way and have the slightly wider octaves?
As I understand it, the Pythagorean tuning puts the entire comma into one fifth, which is the intolerable "wolf". The various mean-tone systems divide the comma among four to eight fifths; equal temperament divides it among all twelve. Am I right?

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Re: Can we take up the comma by stretching octaves?
edferris #2933792 01/14/20 03:38 PM
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Yes you can but with tradeoffs. With a pure 5th tuning throughout the whole compass the octaves and most other intervals become intolerably wide. All octaves have beats and 3rds scream. The term "Cordier" tuning has been given to a pure 5th tuning on a keyboard.


Chris Leslie
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http://www.chrisleslie.com.au
Re: Can we take up the comma by stretching octaves?
edferris #2933793 01/14/20 03:39 PM
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It's not "the comma". The real kicker is that each "circle" of intervals that comes back to an octave -- whether thirds, fourths, fifths -- or even minor seconds or major sixths -- has a different "comma". The comma from thirds is about 39 cents, for example. Other commas have different values.


So the challenge is to "stretch" or "compress" the various intervals to fit into the octave to achieve some level of tolerability. This process is called "tempering". The dominant method for that nowadays is to tune an 'equal' temperament and has been considered preferable to re-tuning the piano every time you play in a different key. (There are other temperaments that permit playing in other keys, as well. As I said, "equal" is the overwhelmingly most commonly used -- although esthetically not necessarily better than some of the "well" temperaments.)


Keith Akins, RPT
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Re: Can we take up the comma by stretching octaves?
edferris #2933821 01/14/20 04:38 PM
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Here's a link to discussion of impure-octave tuning:
http://www.pykett.org.uk/impureoctaves.htm

The author is tuning an organ, but seems to think that the perfect-fifth (Cordier) tuning is workable. The major thirds are sharp, but not much more than in Equal Temperament. He points out that the ear hears beat frequencies, not cents, so a one-cent difference at the high end of a two-foot rank can be intolerable. Narrowing the octaves makes thirds, fourths, and fifths more consonant than in Equal Temperament.

Devising temperament schemes is a traditional way to go crazy, second only to making improved piano actions.

Re: Can we take up the comma by stretching octaves?
edferris #2933835 01/14/20 05:14 PM
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There's a whole book written on it called "The New Tuning". Written about 30 years ago I think. I have it. I don't like it. Pure 12th ET is pretty close to it but not quite.

Pwg


Peter W. Grey, RPT
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