Tempermental

I used to think I was reasonably musical. I reckoned I could tune a guitar, understand music theory, improvise on the piano, and generally appreciate music better than the average person. Not having yet met the average person I cannot be sure that I was wrong but I have met several non-average people and have come to the conclusion that there are many who are a good deal more musical than I am. Which peeves me because I thought I had a corner. Hence my Song of Albion posts. Which I might yet continue anyway.

As an example - I love the piano - both playing it and listening to it being played. Of course I can tell if a piano is wildly out of tune but I am unaware of the nuances of out-of-tune-ness that some of my friends claim they can detect. But at least I am interested in out-of-tune-ness.

I invented my musical slide rule whilst at school and was intrigued by the idea of enharmonic notes which it manages to discriminate between. Which led to discovering just intonation and the Pythagorean comma.

Pythagorean Comma

Which goes like this:  The seven notes of the major scale are kind of "God-given" in that they result naturally from the vibrations in pipes or strings. The third harmonic of a string or open pipe is a "just" perfect fifth above the second harmonic, the ratio of the frequencies being exactly 3:2. The just perfect fifth is the purest sounding interval after the octave (ratio 2:1) because the ratio is so simple. So two notes are consonant if the ratio of their frequencies is simple. If you start at the bottom A of the piano keyboard and ascend by fifths 12 times you will get to the top A, seven octaves up. If these had been just fifths you arrive at slightly higher than A, this discrepancy being the "comma".

just perfect fifth = frequency ratio 3:2 = 1.5
1.5 raised to power of 12 = 129.746337890625 = 1.0136 x 128 (2 raised to the power 7, thus the top A)
equal temperament semitone = 2 raised to power of 1/12 = 1.0595

thus the Pythagorean comma is about 1/5th of an equal temperament semitone.

The purpose of equal temperament tuning is to divide the octave in 12 exactly equal semitones and thus to avoid the comma.  At the expense of intervals no longer being "just" and therefore not being truly consonant. But it means that music can modulate into remote keys - whereas with just intonation remote keys are out of tune. So equal temperament has heralded the complex harmonies that are commonplace in today's music.

Thus all keys should sound the same thus dispelling the myth that flat keys sound soft and sharp keys sound crisper. Or is it a myth? In fact the piano is not tuned to equal temperament. Because of the stiffness of the piano "strings" the higher harmonics are out of tune - this is called "inharmonicity". To compensate so that higher and lower notes played together don't sound discordant the piano is actually "stretched" or deliberately tuned away from equal temperament.

Railsback curve

Which begs the question - which is the more correct? Just intonation seems to be closer to what I describes as "God given". But then, frankly, who cares? The keyboard (which almost implies equal temperament) is here to stay. And who wants to go back to Gregorian chant? And most of us don't understand the distinction and of those that do, most cannot tell the difference.

And then most of the keys on a piano strike three strings or "unisons" because they are in theory tuned in exact unison see here. Except that this site says "a piano tuner's "secret" is that not all three strings have to be perfectly in tune, a slight difference (which you'll get anyway in a couple of months with regular playing) actually beefs up the tonal quality of the note (chorus effect)". This site claims the reason is to increase the loudness to compensate for the larger bass strings, but I think this is weak reasoning. The force imparted by a given amount of thump on a piano key will be divided three ways when striking three strings thus defeating this argument. Wikipedia and other sites explain that, because the three strings are coupled by the bridge, having three strings lengthens the decay rate although giving an initial sharp decay provided they are tuned in unison - this sounds more believable to me. A side effect of coupling is that it will affect the frequency of vibration thus giving another potential departure from true equal temperament.

This site claims that the three "unisons" may intentionally be de-tuned by an amount that is culture dependent - "The effect is to change what stage of a played note the tuning resolves for maximum appeal. In the American style, the attack, or initial sound is emphasized for optimum 'in-tuneness' tonality. In the Japanese style, the sustain, or the sound of the note after the initial strike fades is where the highest degree of 'in-tuneness' resolves. In the European style, influencing the overall color of the note is the goal."

Which views are rather confusing and not always consistent.

I have access to several pianos including a grand. I cannot comment on their relative degrees of in-tune-ness - at the moment they all sound OK to me - but the one I hands-down prefer is an upright: for its crispness of tone, its responsiveness to and lightness of touch, and its great dynamic range.