[CAUT] Meantone (Which?)

[Fred Sturm]

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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