[CAUT] Meantone (Which?)

[Jim Busby]

Thanks Fred!

 

As always from you - a "textbook" answer worthy of posting on the wall, with a human/approachable touch.

 

Highest regards,

Jim Busby

 

________________________________

From: caut-bounces at ptg.org [mailto:caut-bounces at ptg.org] On Behalf Of Fred Sturm
Sent: Tuesday, May 29, 2007 11:12 AM
To: College and University Technicians
Subject: Re: [CAUT] Meantone (Which?)

 

On May 14, 2007, at 1:27 PM, Jim Busby wrote:

	

	 

	A harpsichord here at BYU is supposed to be kept in meantone, but I'm not sure which one. There are 10 or so listed on my Verituner. 1/5 comma, ¼ comma, etc.

	 

	By the way, is there a quick explanation of "comma".  (Hey, I dunno, I just tune 'em...)

	 

	Thanks.

	Jim Busby BYU

Hi Jim,

            I've been off on vacation a couple weeks, so I'm pretty late on this. You've had some good responses, but I'd like to add a bit. 

            First, I'd go with 1/4 comma. Why? For pedagogical reasons (ie, to help the students). I like your prof's notion of giving them a mixed palette to work on and listen to. 1/4 comma gives dead on, beatless M3s, which they may never ever have another opportunity to hear (especially counting the students who are not harpsichordists). You might also find the opportunity to teach some students to tune this. It is probably the easiest tuning to learn aurally.

 

            I'm going to give a little slower explanation of comma, with a bit more detail. Not just to fill space in cyber-world, but to try to demystify the subject, and make it a bit more straightforward.

            Essentially there are two commas, syntonic and Pythagorean. Syntonic is the difference between four beatless 5ths and a beatless major third. If you tune four beatless fifths from, say, F (FC, CG, GD, DA), you end up with a M3 (FA). It has a pretty fast beat rate, significantly faster than in ET. You can tune the A down so that it will form a beatless M3 with F. The difference between those two pitches for A is the syntonic comma. It is about 22 cents. If you want to tune your four fifths so that you end up with a beatless M3, you can distribute the comma evenly between the fifths, 1/4 each. Which means you make each fifth 22/4 cents narrow (= 5.5 cents). (If you want a "less extreme" tuning, you can do less tempering, as in 1/6 comma. That means fifths that are 22/6 cents narrow - about 3.5 cents - and ends up with M3s that beat slowly, about 2 - 4 bps in the temperament area).

            The Pythagorean comma is the difference between 12 beatless fifths and an octave. If you tune a series of 12 beatless fifths, you end up 24 cents wide of a beatless octave. So a Pythagorean comma is 24 cents. You can distribute those 24 cents evenly between all the fifths, by tuning 1/12 comma fifths: 24/12 cents narrow (= 2 cents narrow).

            The two commas are very close in size to one another, so for most practical purposes they can be used interchangeably. The basic principal is that you have to have the 12 fifths add up to a total of 24 cents narrow to produce clean octaves. In 1/4 syntonic comma meantone, you tune 11 fifths narrow by 5.5 cents each (for a total of 55 cents narrowing), and so you end up with a very wide fifth (55 minus 24 = about 31 cents wide), the "wolf," to complete the octave. For WTs, parts of the comma are distributed unevenly among the various fifths.

            I've used the word "beatless" above. Theoreticians would use "just." For most practical purposes, the terms are interchangeable, but if you want to be precise, there will be differences arising from inharmonicity. For harpsichords, this is generally insignificant.

            I strongly recommend reading A Guide to Musical Temperament by Thomas Donahue (The Scarecrow Press, 2005) for a good, basic account of all this, plus a lot more details if you are interested. I think every tuner should have at least this much basic information. It is really not nearly as complicated as it often seems (mostly because of unfamiliar jargon), and it's what we are working with every day.

Regards,

Fred Sturm

University of New Mexico

 

 

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