Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
meantone
a system of tuning in which the two JI
"whole tones"
(with ratios of 9:8 and 10:9)
are conflated into one "mean tone" which lies between
the two, the objective being to produce or approximate
JI "major 3rd"s
(with a ratio of 5:4) but to eliminate
the syntonic comma and its associated
problem of commatic drift by
having only one size of whole-tone.
The "5th"
is flatter (by a fraction of a syntonic
comma) than the just 3:2, and this is normally how the tuning is
described. Thus the names "Quarter-comma meantone", "2/9-comma meantone", etc.
[from Joe Monzo, JustMusic: A New Harmony]
[I had originally stated that the '5th' is flattened slightly in
order to produce a closed system where a "cycle of 5ths" eventually
returns to the origin. Paul Erlich corrected me, saying that ETs
create a closed system, but most meantones do not.]
(see also my paper explaining the derivation of
W. S. B. Woolhouse's optimal meantone.)
[from John Chalmers:]
...the definition of
meantone could be tightened up a bit as it implies that all
meantone-like tunings have 5/4 major thirds. I would describe meantone
[that is, meantone proper] as the temperament whose fifth is equal to the fourth root of 5 and is
thus 1/4 of a syntonic comma flatter than 3/2.
Meantone-like temperaments are those cyclic systems which have fifths
flatter than 3/2 by some fraction (rational or
irrational) of the
syntonic comma and which form their major thirds by four ascending
fifths reduced by two octaves. Such tunings divide the ditone (which is
here equivalent to the major third) into two equal "mean tones"
Well-known examples include the 1/3-comma (just 6/5 and 5/3), the
1/5-comma (just 15/8, 16/15), and John Harrison's tuning whose major
third equals 2(1/pi) and which has been revived and promoted in an
extended form by Charles Lucy.
Certain meantone-like temperaments are audibly equivalent to equal
temperaments of the octave. For example, the 1/4-comma system
corresponds to 31-tet and the 1/3 comma to 19-tet. Arbitrarily close
equivalences may be found by using continued fractions or Brun's
Algorithm.
The upper limit of meantone-like fifths is the fifth of 12-tone ET (700
cents) and the upper limit of meantone-like major thirds is the major
third of 12-tet
(400 cents).
(While Pythagorean tuning could be considered
as a meantone-like system
in which the tempering fraction is zero, temperaments whose fifths are
larger than 700 cents are best thought of as positive systems, which
make their major thirds by a chain of 8 descending fifths (or ascending
fourths). These major thirds are formally diminished fourths (e.g.,
C-Fb). Other relations occur as the fifth become still sharper (9 fifths
up, augmented second, C-D#).)
[from John Chalmers, personal communication]
[from Paul Erlich:]
From: Paul H. Erlich
Sent: Monday, December 20, 1999 7:24 PM
To: 'tuning@onelist.com'
Subject: Summary of optimal meantone tunings
Here is a summary of the meantone tunings (giving the size of fifth in
cents, the fraction of the syntonic comma by which the fifth is reduced,
and the first known advocate or reference to a TD posting by me from Brett
Barbaro's e-mail) that are optimal under various error criteria for the
three "classical" consonant interval classes: the one 3-limit interval,
the p4/p5; and the two 5-limit intervals, the m3/M6 and the M3/m6.
* For some cautionary annotations by Margo Schulter
concerning Aron and Vicentino in this table, see her posting in
Onelist
Tuning Digest # 532, message 12.
(to download a zip file of the entire Dictionary,
click here)
I welcome
feedback about this webpage:
| Max. error |Sum-squared error|Sum-absolute error|
---------+------------+-----------------+------------------+
Inverse | 697.3465 | 696.5354 | 696.5784 |
Limit | 3/14-comma | 63/250-comma | 1/4-comma |
Weighted | Riccati |TD 162.10 5/5/99 | Aron |
---------+------------+-----------------+------------------+
Equal | 696.5784 | 696.1648 | 696.5784 |
Weighted | 1/4-comma | 7/26-comma | 1/4-comma |
| Aron | Woolhouse | Aron |
---------+------------+-----------------+------------------+
Limit | 695.9810 | 696.0187 | 696.5784 |
Weighted | 5/18-comma | 175/634-comma | 1/4-comma |
| ~Smith | Erlich (TTTTTT) | Aron |
---------+------------+-----------------+------------------+
m3/M6 & | 695.8103 | 695.9332 | 696.5784 |
M3/m6 | 2/7-comma | 7/25-comma | 1/4-comma |
only | Zarlino | This may be new | Aron |
---------+------------+-----------------+------------------+
================================================================
From: "Paul H. Erlich"
corrections, improvements, good links.
Let me know if you don't understand something.