Tuning and temperament (5): Meantone and the syntonic comma

The Pythagorean comma may be considered as the difference in two ways of tuning 84 semitones: either as 12 fifths, or 7 octaves. The just intonation of each is 1.5^{12} and 2^7 respectively, and so

    \[ P = \frac{1.5^{12}}{2^7} \]

which has a cents value of

    \[ 1200\log_2(P)\approx 23.46. \]

The syntonic comma may be considered as the difference in two ways of tuning 28 semitones: either as four perfect fifths, or two octaves and a major third, for which the just values are 1.5^4 and 2\times 2\times 5/4=5 respectively. This has a value in cents of

    \[ 1200\log_2(1.5^4/5)\approx 21.51. \]

Meantone tunings are based on dealing with the syntonic comma; for example, by setting all fifths so that four fifths will be equal to two octaves and a major third; thus each fifth will have a value of 5^{1/4}\approx  1.4954, or about 696.58 cents. Note that 12 of these are a bit less than 7 octaves:

    \[ (5^{1/4})^{12}=5^3=125. \]

This means that in order to fill up our octaves, one fifth is going to have to be very sharp: this will be the “wolf” fifth. Starting at C, we want the wolf fifth to be as far away from C as possible, which in the cycle of fifth is F#:

Rendered by QuickLaTeX.com

So our basic meantone tuning will have 11 fifths tuned to 5^{1/4}, and the final fifth (F\sharp-C\sharp) tuned to

    \[ \frac{2^7}{5^{11/4}}\approx 737.63\mbox{cents}. \]

This will be a very sharp fifth. To see the intervals, use the intervals program of the previous post.

[In ]:  fifth = rat2cents(5.0**0.25)
[In ]:  wolf = 8400 - 11*fifth
[In ]:  mt = [fifth]*12; mt[6]=wolf
[in ]:  print intervals(mt)['D']

[[ -5.38  -0.     5.38  -5.38]
 [-46.44  -0.   -35.68  -5.38]
 [ -5.38  -0.     5.38  -5.38]
 [-46.44  -0.     5.38  -5.38]
 [ -5.38  41.06   5.38  -5.38]
 [ -5.38  -0.     5.38  -5.38]
 [ -5.38  41.06   5.38  35.68]
 [ -5.38  -0.     5.38  -5.38]
 [-46.44  -0.     5.38  -5.38]
 [ -5.38  41.06   5.38  -5.38]
 [ -5.38  -0.     5.38  -5.38]
 [ -5.38  41.06   5.38  -5.38]]

We see immediately that the fifths are a bit flat; the fourths correspondingly sharp, with the exception of the wolf which is way off. Note that many of the major thirds are pure – this is a facet of meantone. We might compare it with equal temperament (ET):

[In ]: print intervals([700]*12)['D']

[[-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]
 [-15.64  13.69   1.96  -1.96]]

The fifths and fourths are closer to pure in ET, but all the thirds are considerably out.

Meantone is so called because the size of the first two tones is equal. If we list the scale:

[In ]: print intervals(mt)['S']

[0.0, 117.11, 193.16, 310.26, 386.31, 503.42, 579.47, 696.58, 813.69, 889.74, 1006.84, 1082.89, 1200.0]

we see that the first tone C-D is 193.16 cents, and the next tone, D-E is 386.31-193.16=193.15 cents. This is not the case for just intonation, where the tones are 10/9 and 9/8, noting that

    \[ \frac{10}{9}\times\frac{9}{8}=\frac{5}{4} \]

which is a major third.

Meantone tuning has one of the longest durations in history, being in use from the early 1500’s into the 1700’s. Some modern scholars believe that much of the music of Mozart and Haydn can be played to great effect in meantone. The meantone above is known as “quarter-comma meantone”, since every fifth (aside from the wolf) is tuned down one quarter of a syntonic comma. We can see this by noting that meantone fifth (in cents) is 300\log_2{5}, and the syntonic comma is

    \[ 1200\log_2(81/80)= 1200(4\log_2(3)-4-\log_2(5)). \]

Reducing a pure fifth by one quarter of this produces

    \[ 1200\log_2(3/2)-1200\log_2(3)+1200-300\log_2(5)=-300\log_2(5). \]

Alternatively, note that the distance between 5^{1/4} and 1.5 in cents, is

    \begin{align*} 1200\log_2(3/2)-1200\log_2(5^{1/4}&=1200\log_2(3/2)-300\log_2(5)\\ &=300\log_2((3/2)^4)-300\log_2(5)\\ &=300\log_2(81/80) \end{align*}

which is a quarter of the syntonic comma.

This quarter comma meantone was first described by the musician and theorist Pietro Aron (1480 – 1545) in his treatise Toscanello in musica, published in 1525. According to the software engineer and musician Graham Breed, quarter-comma meantone is optimal in the sense of minimizing the root-mean square value of the differences of the thirds with their just values. I haven’t checked this myself, yet!

Other meantones

Aron’s quarter comma meantone is just one of many. The “1/6 comma meantone” tunes each fifth down by one sixth of the syntonic comma:

    \[ 1200\log_2(3/2)-\frac{1}{6}1200\log_2(81/80)=1200\log_2\left(\frac{3}{2}\frac{80^{1/6}}{81^{1/6}}\right). \]

If we take out the fraction on the right:

    \[ \frac{3}{2}\frac{80^{1/6}}{81^{1/6}}=\left(\frac{3^6}{2^6}\frac{80}{81}\right)^{\1/6} \]

and this fraction reduces down to

    \[ \left(\frac{45}{4}\right)^{1/6}. \]

This is the value for the fifth, and is approximately 1.4969, or 698.37 cents. It is closer to the true fifth than 5^{1/4}. For a quick test:

[In ]:  fifth = rat2cents(11.25**(1.0/6))
[In ]:  wolf = 8400 - 11*fifth
[In ]:  mt6 = [fifth]*12; mt6[6]=wolf
[in ]:  print intervals(mt6)['D']

[[-10.75   7.17   3.58  -3.58]
 [-10.75  26.72   3.58  15.97]
 [-10.75   7.17   3.58  -3.58]
 [-30.31   7.17   3.58  -3.58]
 [-10.75  26.72   3.58  -3.58]
 [-10.75   7.17   3.58  -3.58]
 [-10.75  26.72   3.58  -3.58]
 [-10.75   7.17   3.58  -3.58]
 [-30.31   7.17 -15.97  -3.58]
 [-10.75   7.17   3.58  -3.58]
 [-30.31   7.17   3.58  -3.58]
 [-10.75  26.72   3.58  -3.58]]

The fourth and fifths are better; the wolf is much reduced, and although none of the thirds are pure, they are better in general that with equal temperament.

Another meantone which was extensively discussed by early theorists was 2/7-comma meantone, where every fifth is tempered by 2/7th of a syntonic comma. This has the effect of approximating a fifth by

    \[ \left(\frac{50}{3}\right)^{1/7}\approx 1.4947. \]

Then there’s 1/3 comma meantone, which tempers every fifth by one-third of a syntonic comma, resulting in fifths equal to

    \[ \left(\frac{10}{3}\right)^{1/3}\approx 1.4938. \]

And of course one-fifth comma meantone, with fifths

    \[ \left(\frac{15}{2}\right)^{1/5}\approx 1.4963. \]

And so on.

More recently, Charles Lucy has promoted a tuning originally developed by the horologist John Harrison. Harrison was the subject of Dava Sobel‘s entertaining biography “Longitude”. Harrison’s idea was to approximate the fifth (in cents) by

    \[ 600+\frac{300}{\pi}\approx 695.49297 \]

which leads to a fifth frequency of approximately

    \[ 1.494412 \]

Tunings based on this value are now known as “Lucy tunings”.

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