1/6-Pythagorean-comma mean-tone?? — Temperament Part I

(Originally posted in March, 2010)

If you just thought this title didn’t make sense, no, it does not make sense, at least to you and me.  Perhaps some of you can help me understand this stuff better — I don’t have a mathematical mind and I can only go so far as to understand the temperament issues, but I don’t think I’m too stupid to understand the basics.  Some of those claims out there doesn’t quite make sense to me….

Now, those of you who are thinking you’ll be in over your head talking temperament, you’ve got to stay with me.   Let’s clarify what the mean-tone temperament is.  Well, if we are to polarize the temperament systems with their characters, we have melodic systems on one side and harmonic systems on the other.  The Pythagorean system is basically a stack of pure (just intonation) perfect 5ths, and because of the sharp mediants and leading tones it has, it is a great melodic temperament for monophonic lines.  if you play something unaccompanied on a violin-family instrument with perfect 5th tuning, it is likely that you are playing in a temperament that’s pretty close to the Pythagorean.  But because of pretty wide major 3rds that you find in this system, this isn’t a great temperament harmonically — basic triads in this system would sound pretty ugly.  If you can obtain pure (just intonation) major 3rds in this temperament, chords would sound much nicer!

So, how are we going to incorporate narrower and pure 3rds into the stack of perfect 5ths?  Let’s figure out what the difference between pure perfect 5th stack from C to E and pure major 3rd interval C to E is.

Just intonation is natural physics.  The frequencies of the diatonic notes in just tuning are related by ratios of small whole numbers.

Frequency ratio 2:1 = octave; 3:2 = perfect 5th; 4:3 = perfect 4th;

5:4 = major 3rd; 6:5 = minor 3rd….  and so on.

If we are to get the interval of a pure perfect 5th in cents (1 octave = 1200 cents, in equal temperament one semitone is 100 cents; perfect 5th is 700 cents):

log (3:2) x 1200 ÷ log (2) = 701.955 cents  (don’t ask me about logarithm!)

In order for us to get to an E just stacking up perfect 5ths from the C in the cycle of 5ths, we need four 5ths.

C — G, G — D, D — A, A — E

701.955 + 701.955 + 701.955 + 701.955 = 2807.82 cents

Pythagorean E is 2807.82 cents away from the bottom of the stack C.  The interval between the bottom C and this E is a major 17th.  Why don’t we flatten this interval and make it a simple major 3rd?

2807.82 – 2400 (two octaves) = 407.82 cents

Now, let’s get the pure (just intonation) major 3rd in cents:

log (5:4) x 1200 ÷ log (2) = 386.3137….

To compare this with the Pythagorean C-E interval:

Pythagorean major 3rd [407.82] – Just intonation major 3rd [386.31] = 21.51 cents

Tada!  This is the syntonic comma.  As you remember, in the cycle of 5ths, we needed to stack up four perfect 5ths to get to the E.  So, if you make each of those 5ths narrower by a 1/4 syntonic comma, you get the harmonically happy major 3rd from C to E.  And if you keep narrowing the rest of 5ths by 1/4 commas, that’s the mean-tone temperament.  That’s why, in the specific sense, mean-tone IS the 1/4-comma mean-tone, and when people just say “mean-tone,” it means the 1/4-comma mean-tone specifically.

Now you understand why the title of this entry doesn’t make sense.  The mean-tone tuning is a syntonic-comma-based system, not a Pythagorean-comma-based system.  But some piano workshop website has the instruction for tuning in 1/6-Pythagorean-comma mean-tone IN ADDITION TO 1/6-comma modified mean-tone and 1/6-Pythagorean-comma (Vallotti & Young) temperaments…

What is the broader definition of the term ‘mean-tone?’ here anyway…?