Equal vs. Meantone Temperaments

 Equal vs. Meantone Temperaments 

For many years I have puzzled over the best way to explain the dilemmas posed by Equal Temperament. I discovered some of them for myself in 1986 when I put together the Spinet kit I bought from John Storrs of Chichester (https://www.friendsofsquarepianos.co.uk/spinets-harpsichords-and-clavichords-for-sale/john-storrs-spinet/). These problems barely show on a piano and, in any case, I think one gets used to equal temperament. There is something about the twang of a spinet/harpsichord that shows up the shimmering 'beats' of the discordant thirds. 

By equal temperament I refer to the practice of (a) defining an octave as an exact doubling of the frequency of a note, and of (b) dividing the octave into twelve equal semitones.

Equal Tempered Semitone (ets) = 12√2 =1.059463094359

Multiplier	If fundamental  =440 Hz	True ratios (Pythagorean) intervals frequencies	Error as ratio (or 1/ratio)
1.0000	A440	Fundamental
1.0000*ets=> 1.0595	A♯, B♭	Semitone
1*ets^2=> 1.1225	B	Pythagorean tone 9/8 = 1.125)
1*ets^3=> 1.1892	C	Minor third = 6/5 = 1.20000	0.991(1.0091)
1*ets^4=> 1.2599	C♯, D♭	Major third = 5/4 = 1.25000	1.0079
1*ets^5=> 1.3348	D	Fourth = 4/3 = 1.33333	1.0011
1*ets^6=> 1.4142	D♯, E♭
1*ets^7=> 1.4983	E	Fifth = 3/2 = 1.500000	0.9989 (1.0011)
1*ets^8=> 1.5874	F
1*ets^9=> 1.6818	F♯, G♭
1*ets^10=> 1.7818	G
1*ets^11=> 1.8877	G♯, A♭
1*ets^12=> 2.0000	A	Octave = 2/1 = 2.00000

From the table we can see that the 'harmony' of the equal tempered fifth and fourth are fairly close to the true harmonies of the Pythagorean scale. Nevertheless, the equal tempered fifth is too small by a small amount, and the equal tempered fourth is too big by a similar and therefore compensating amount. 

However, the equal tempered major third is too big while the equal tempered minor third is too small; and these errors are 8 to 9 times greater than the errors of the fourths and fifths. Once again these error tend to cancel out, but not completely. Thus, a major plus a minor third equals a fifth, where the error is 8 times smaller than that of the major third.

An interesting question arrises as to how the instrument tuner in 1600 or 1700, or even 1950 tuned without the aid of an electronic tuner. You might naively start (as I did ) by tuning A4 using a tuning fork at A4=440 Hz.  Then, by tuning by ear up a fifth and down a fourth, you get to B. Repeating the process you get to C♯ etc, till you get back to A (jumping an octave when you need or want.)  But you crash into the shattering fact that it is a different A; it is 1.36% too sharp. The gap between the two is called a comma. It you narrow the  fifth slightly, from a true fifth to make it an equal tempered fifth, you can spread the comma round the octave. That takes skill, listening for, and counting, the 'beats' per second that appear and speed up as you depart from perfect harmony. The same thing happens if you simply go up a fifth over and over 12 times, for (3/2)12 =129.746 while 7 octaves (27 =128); 129.746/128=1.01364; you get exactly the same discrepancy or comma.

I learned an easier method for tuning to the so-called "quarter comma" meantone temperament, which gives very sweet-sounding harmonies for rennaissance and baroque pieces in 'unadventurous' keys (e.g.  from 2 flats to 2 sharps).  Remember, octaves are always perfect. Tune your A4 to the A440Hz tuning fork (or signal), then A3 (220 Hz) to that.  To A3, tune a perfect third to F3. Go back to A3 , and from it tune D3 almost a pure fifth below but leave it slightly sharp; about 2 'beats' per second. From that tune up a fourth again nearly perfect to get a G3, again a tad sharp; and from that down to C3 again leaving it slightly sharp.  The instructions are to "spread the comma between these 3 intervals evenly: A3 ➞D3, D3, ➞G3, G3 ➞C3 . Check it is tolerable with the chord C3 + F3 + A3. All the other intervals are tuned as pure major thirds (5/4), or octaves (2/1):  C➞E, D➞F♯, D➞B♭,  B♭➞B♭, E➞G♯, G♮➞E♭, A➞C♯,➞C♯, G➞B♮➞B♮.  (See Figure below.)

Having set 'the temperament', use pure octaves to complete the instrument. Electronic tuners are quicker and easier, and are available free for most smart mobile phones. I use a Table of deviations from equal temperament (in cents) given me by a friend 40 years ago. After considerable searching I found a very comprehensive table online at (http://www.instrument-tuner.com/TemperamentTables.html)