Alternative Temperaments

[Richard Moody]

Forwarded by R Brekne from Paul Erlich

>I started with piano but now
> play more guitar, and I have a 22-tone one that I play once in a while.
> I would love to see a 22-tone piano constructed -- it would be a great
> help for the advancement of music.

> Other meantones "derived" throughout history include 5/18-comma meantone
> (Robert Smith), 3/14-comma meantone (?), 2/7-comma meantone (the first,
> Zarlino) etc.
> 
> -Paul




	The way to figure the x/y comma meantones is to first make sure which comma is
being referred to.  For meatone it must be the comma of Didymus. (syntonic) This
comma is the excess of four pure fifths (reduced two octaves) over a pure third.  
The original object of Meantone was to acheive pure thirds by tempering four fifths.
 This we assume was done for the first keyboard instruments and may have been
proposed as far back as `1382. Since the comma is divided among four Fifths it was
called the "1/4 comma Meantone". Why the "mean" is in Meantone is another topic. In
practice it is not difficult to tune a pure sounding third.The difficulty is in
tuning a circle of  four fifths (equally narrow) and ending up with a pure third.
Also in practice it was found that if the thirds were a little wide the fifths
didn't have to be tempered so much. Since this is a forum of piano tuners, the
experience of tuning four fifths narrow to produce one pure third is worth a
thousand or two words. But such is the  nature of temperaments and scale
construction.  One can go on and on and on.  There are thousands of combinations to
try and hundreds of schemes to propose. 
	As a tuner and facing the task of "laying the bearings" of 1/4 comma meantone,
knowing the fifths must be reduced by "1/4 comma" doesn't tell you much about  how
they should sound, much less the beat rate.   2/7 comma (Zarlano) doesn't tell you
much of anything.  It gives a Third of LESS than pure, but what is the objective?? 
	The mathematical reality of reducing a fifth by 1/4 comma is not a simple
fractional divison. You are seeking to knock off a small amount so that when the
fifth is multpilied (played on top of each other) four times it will will give a
pure third two octaves above the starting note.   A number times itself four times
is that number to the fourth power.  To reduce a number so that four times itself
will give a specified amount less than its fourth power means you have to reduce it
by its fourth root. Thus the Fifth in Meantone must be diminished by the fourth root
of the comma.  That comma is expressed as 81/80.   Today computation of
this and beat rates is rather straight forward with a calculator, but  from the 14th
century to the 19th the theory of tuning and the practice of tuning probably had
little in common. 
	One development that has given a "common denominator"  to the theory and practice
of temperaments and scales is the math term "cents"   100 cents to a semitone, 1200
cents to the octave.   The comma of Didymus, (or syntonic) used in Meantone is
21.506 cents.  With cents you can get results by simple division, and make mental
comparisons. For example a pure third is 386 cents, an ET Third is 400.     Thus in
"1/4 Meantone" pure Fifths are narrowed by 21.5/4 or  5.375 cents.  2/7 comma would
be 21.5/(2/7) or 6.14.   Hmm since the fifths are narrowed more, this should give a
third narrower than pure......... In ET Fifths are narrowed by 2 cents. 
	If the octave has 1200 cents, then the 12 note octave should have 100 cents per
note.  The 19 note octave (equally divided) then should have 1200/19, or 63.16 cents
per note. The 22 equal note octave would be 54.5454 cents.    But sometimes the "12
note octave" is called a thirteen note octave with 12 intervals.   Do we know how
the 19 or 22 note octave is set up?
  
	" Salinas's 1/3 comma mean-tone (1577) with its pure minor 3rds at the expense of
major 3rds....the intervals are virtually identical with an equal division of the
octave into 19 parts"   New Groves, "Temperament"  Mark Lindley  


Richard Moody 


  
> From: Richard Brekne <richardb@c2i.net>
> To: PTG <pianotech@ptg.org>
> Subject: Alternative Temperaments
> Date: Wednesday, August 25, 1999 2:30 PM
> 
> Hi list.
> 
> To those of you who are interested in this kind of thing, one Paul
> Erlich has recently contacted me and asked me to put him in contact with
> some of you.
> 
>> Thank you for including a link to my paper Tuning, Tonality, and
>> Twenty-Two-Tote Temperament! How did you find out about it? What do you
>> think of it? I should let you know that an html version of the paper
> (currently with some problems, hopefully to be revised soon) is
>> available,
>> and both versions are linked to from
>> http://www-math.cudenver.edu/~jstarret/Erlich.html.
>> 
>> Keep up the good work!
> 
>> -Paul
> 
>