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.

    The elimination of the syntonic comma means that the essential feature of meantone and meantone-like systems is the equation of the "major 3rd" with four "5ths".

    [Note that while ETs create a closed system, most meantones do not.]


    While the whole point of meantones, as explained above, is to collapse the 2-dimensional "8ve"-equivalent 5-limit lattice into a 1-dimensional linear temperament which emulates (in notation, but not in tuning) the linear Pythagorean system, it so happens that certain meantones can provide very good approximations to higher odd/prime limits.

    Below is a graph showing the deviation of the basic intervals in meantone from their JI equivalents in the 11-limit. The horizontal axis represents the fraction-of-a-comma amount by which the meantone generator (i.e., the "5th") is tempered (narrowed) from the Pythagorean "pure 5th" of ratio 3:2. The vertical axis shows the deviation in cents from the JI intervals, of 1 generator from 3:2 (blue line), 4 generators ("8ve"-equivalent) from 5:4 (pink), 10 generators from 7:4 (yellow), and 18 generators from 11:8 (cyan). Some common meantones are drawn vertically as illustrations. (In this graph an alternative mapping of 11:8, to -13 generators, is not shown. All of these graphs choose one particular mapping of 11-limit ratios, as shown at the bottom of the "meantone error" applet which can be viewed by clicking on the small graph below.)


    Below is a graph showing the average absolute deviation for each set of ratios of prime-factors.

    Below is a closeup of the central section from the above graph.

    If one follows Partch's observation that the sensitivity of a ratio to mistuning is directly proportional to its prime- (or odd-) limit, it can be seen from the above graph that the area between 0.229- and 0.217-comma meantone is pretty close to ideal for a meantone representing 11-limit harmony using the mapping I chose.


    		Click on the graphic at left to open a new window with a mouse-over applet which shows the error of various specific meantones from JI.


    [from Joe Monzo, JustMusic: A New Harmony]

    (see also my paper explaining the derivation of W. S. B. Woolhouse's optimal meantone, and Graham Breed's excellent Meantone Temperaments webpage.)

    ...............................

    [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 Tuning List posting of Sun Feb 13, 2000 11:24pm, which appeared in Onelist Tuning Digest # 532, message 12.

Note also that while Aron was the first theorist to give a thorough description of 1/4-comma meantone, Arnold Schlick was the first to desribe to describe it at all, in 1511 [Schlick, Arnold. Spiegel der Orgelmacher und Organisten. Speyer, 1511. Monatshefte für Musikgeschichte, Heidelberg, 1869. Reprint 1959. English translation by E.B. Barber, Frits Knuf, Buren, 1980.]. See this German webpage.


Click on the image at left to open a window with animated graphs comparing various meantone chains.


Meantone cycles which are based on tempering by a fraction of a syntonic comma, which is a 5-limit interval, can have some of their pitches graphed on a lattice diagram. (I explained the mathematics of one example of this -- 1/6-comma meantone equating to a cycle of 64:45's -- in Yahoo Tuning List message #30125, Date: Tue Nov 13, 2001 6:55 am, Subject: Re: New Telemann page - audio files).

Below is a 5-limit lattice illustrating partial cycles of the most common meantone tunings; the cell containing the reference pitch "C" is colored black because all the cycles contain it, and thus all the axes which represent the different cycles pass thru and are centered on that point on the lattice.

Applet: representation of meantone tunings as JI-interval cycles

©2001 by Joe Monzo

(inspired by this lattice applet by Ming Sun Ho)

Below is an applet illustrating several examples of the cycles on the above diagram.

MouseOver the names of the various fraction-of-a-comma meantone cycles to see a graph of the notes on a 5-limit lattice diagram. Each blue square on the graph represents a "cell" or ratio-point on the lattice, with powers of 3 along the horizontal axis and powers of 5 along the vertical axis. The plots of these cells gives each meantone cycle a unique angle and density.

The ratios of generator g in n/d-comma meantone are plotted as 3x * 5y, where x = (d-(4*n))*g and y = n*g.

These meantone cycles can be approximated arbitrarily closely by various EDO cycles, for example, 1/3-comma equates to 19-EDO (generators -9 to +9), 1/4-comma to 31-EDO (generators -15 to +15), 1/5-comma to 43-EDO (generators -21 to +21), 1/6-comma to 55-EDO (generators -27 to +27), 1/11-comma to 12-EDO (generators -6 to +5 or -5 to +6), etc. I have not taken this into account on these graphs, and simply arbitrarily extend each cycle from -22 to +22 generators.

Meantones as JI-interval chains


©2001 by Joe Monzo

1/2 1/3 2/7 5/18 7/26 1/4 2/9 3/14 1/5 2/11 3/17 1/6 1/11


I expanded on the above idea to create a set of diagrams which I think are really useful: Lattice diagrams comparing rational implications of various meantone chains.

Below are some crude ASCII lattice diagrams which accurately portray the nature of meantone intervallic relationships. These are flat projections of what are intended to be cylindrical diagrams. The dotted line represents a circle whereby pitches wrap around the cylinder to occupy the same point.

Here is the meantone diatonic C-major scale: 



                      ,,-F
               ,,-C-''  / \
        ,,-G-''  / \   /   \
  ..D-''../.\.../...\./..,,-D..
     \   /   \ /  ,,-A-''
      \ /  ,,-E-''
       B-''

Here is a typical 12-tone meantone system: 



                              
                  ,,-Eb
           ,,-Bb''   / \
     ..F-''.../.\.../...\....,,-F..
         \   /   \ /  ,,-C-''  / \
          \ /  ,,-G-''  / \   /   \
         ..D-''../.\.../...\./..,,-D..
            \   /   \ /  ,,-A-''  / \
             \ /  ,,-E-''  / \   /   \
            ..B-''../.\.../...\./...,,-B..
               \   /   \ /  ,,-F#-''
                \ /  ,,-C#-''
               ..G#-''

And meantones were also used with pitch-sets larger than 12 notes, thus: 



         ,,-Fb
   ..Cb''.../.\...........,,-Cb..
       \   /   \   ,,-Gb''   / \
        \ / ,,-Db''   / \   /   \ 
      ..Ab''.../.\.../...\./..,,-Ab..
          \   /   \ /  ,,-Eb''  / \
           \ /  ,,-Bb''  / \   /   \
         ..F-''.../.\.../...\./..,,-F..
             \   /   \ /  ,,-C-''  / \
              \ /  ,,-G-''  / \   /   \
             ..D-''../.\.../...\./..,,-D..
                \   /   \ /  ,,-A-''  / \
                 \ /  ,,-E-''  / \   /   \
                ..B-''../.\.../...\./...,,-B..
                   \   /   \ /  ,,-F#-'' / \
                    \ /  ,,-C#-'' / \   /   \
                   ..G#-''./.\.../...\./..,,-G#..
                      \   /   \ /  ,,-D#''  / \
                       \ /  ,,-A#-'' / \   /   \
                      ..E#''../.\.../...\./..,,-E#..
                         \   /   \ /  ,,-B#''  / \
                          \ /  ,,-Fx''  / \   /   \
                         ..Cx''../.\.../...\./..,,-Cx..
                            \   /   \ /  ,,-Gx''
                             \ /  ,,-Dx''
                            ..Ax''

The proper way to lattice a meantone, or any linear temperament, is as a cylinder which runs perpendicular to the vector of the interval being tempered out. Here is a Monzo lattice of 1/4-comma meantone, with a typcial 12-tone Eb...G# scale showing along the meantone chain on the face of the cylinder.

The cylinder for all meantones will always have the same angle to the lines on my JI lattice, because they all temper out the same interval, the syntonic comma. On my lattices, each meantone chain itself will spiral around the cylinder differently depending on the fraction of a comma by which the "5th" is tempered.

Comparison of 19edo and 31edo

Two of the earliest, commonest, and most convenient forms of meantone tuning are 19edo and 31edo. Below is a table and graph comparing their "cycle of 5ths".

Updated:

2003.08.01 -- added final section comparing 19edo and 31edo chains
2003.03.13 -- added graphs and text showing absolute average error from JI
2003.03.03 -- added "meantone deviation" graph
2003.02.28 -- corrected date of 2/7-comma description to 1558
2003.02.26 -- added the "chronology of meantone advocacy" graph
2002.09.22 -- added the bit about how 4 "5ths" = "major 3rd"
2002.01.27
2001.12.9
2001.11.13

(to download a zip file of the entire Dictionary, click here)

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