Goofed instructions to piano tuner yield instructive demonstration

I asked my piano tuner to give me a mathematically accurate quarter comma
mean tone E5 above A 440 to record for demonstration purposes before tuning it
to a just fifth E 660, as I wanted to demonstrate the contrast between a just
fifth and a tempered fifth. My thought was that the demonstration would
substantiate the idea that tempering the fifth 5.4 cents flat would leave its
character essentially intact.

He was working with his frequency meter and I said: "The E is 5.4 cents
flat". He set his meter to minus 5.4 cents and was working his tuning hammer
this way and that as strobe lights on his meter would rotate first clockwise
and then counterclockwise. Finally he got the E pretty well at the -5.4 cent
point and played the A 440 - tempered E 660 fifth. As I heard it - with its
disappointingly off sound - I felt a wave of discomfort. I hadn't thought a
quarter comma mean tone fifth sounded that bad. My piano tuner, a twelve
equal temperament fan, said: "Whew!". Then I began speculating as to why, in
this situation, the tempered fifth sounded so bad. Perhaps it was because not
all three unisons were being used. So he tuned the other E 660 unisons and
the fifth continued to sound "off". I'd heard this "off" sound quite a lot
with my quarter comma mean tone tuned piano, and I'd felt that when strings
went slightly out-of-tune in such a direction as to make a tempered fifth even
flatter, it didn't take much to make it sound "off". But if the tuning was
accurate, the 5.4 cent flat fifths sounded to me like acceptable fifths as
long as I didn't listen too critically to them.

I thought: "Well, I guess they sound worse than I'd thought. If that's the
way they sound, that's the way they sound, and that's what I'll demonstrate to
the music students." Then the thought hit me - "The frequency meter reads
minus 5.4 cents, but that's 5.4 cents less than the EQUAL TEMPERAMENT
frequency for the E5, which is ALREADY tempered close to 2.0 cents flat! I
felt a sense of relief and asked him to set the E to minus 3.4 cents, which he
did. He played the less tempered fifth again and it sounded "right" for a
quarter comma fifth to me - just barely OK. By that I mean that when I hear
it, it has that feel of a musical, significant fifth with some kind of
emotional impact. One knows it's "tempered", but it doesn't seem to have lost
too much of its character. The 7.4 cent flat fifth, on the other hand,
sounded immediately recognizable to my ears - and my piano tuner's - to be
"off".

What this inadvertent lesson brought home to me was something I'd already
come to believe - that pitch differences as small as 2 cents on an instrument
such as the piano can make a significant difference to the character of a
musical interval's sound.

When he tuned the E5 to a pure fifth above A 440, setting his frequency meter
to plus 1.9 cents, the pure fifth on the piano really sounded good. As he was
getting close, I found myself wanting to hear the E sharper and just a bit
sharper, etc. Then it sounded really good. I didn't place that fifth next to
an equal tempered fifth, but will do so when I prepare the demonstrations in
my computer. I'm pretty sure, though, that the just fifth really does sound
different than a fifth which is only two cents flat.

He also tuned the C sharp 5 to both an equal tempered third above the A4 and
to a Pythagorean third +7.8 cents on his meter while the E was a just fifth
above A4. On the piano, the Pythagorean third sounded stranger and more
inharmonious than an equal tempered third. The just triad on A4 which I
played and recorded (other piano strings damped) sounded really good - sweet
and high and bright. The Pythagorean triad on A4 sounded - to me - really
strange - not terrible, but quite strange.

What I'm learning with the piano is that in some cases, quite small changes
in pitch can appreciably affect a chord's character.

Note: I've had the piano in quarter comma mean tone temperament for about six
months and have been playing it a lot, practicing and improving my piano
technique. Yesterday, after setting the demonstration fifths and thirds to be
recorded then, my piano tuner put the piano into sixth comma mean tone
temperament. It will be in that tuning for a week while I make comparison
recordings of pieces I've already recorded in 1/4 comma mean tone temperament.
Then next week he'll tune the piano to equal temperament so that I can make
comparison recordings of these numbers in equal temperament. Soon after that,
he'll return the piano to 1/4 comma mean tone temperament, which I find very
enjoyable to play.

Dave Hill, La Mesa, CA