Rev asks

“Unfortunately, even with all those pitches, the thirds are never in tune. Equal temperament be like that. Violins can play thirds in tune—and even if they don’t, they can just fudge it up with vibrato so nobody knows the difference.”

So why aren’t pianos tuned with the thirds in tune?

What is a “third” anyway?

——

Human musical experience is governed very much by frequency ratios.

Consider the E D C that opens Mary Had a Little Lamb. The interval between E and D is sensed very much as the interval from D to C. Taking C to be middle C, the frequencies are about 329.6 Hz 293.7 Hz 261.6 Hz. They form a geometric succession with r = 2^(-1/6).

What if I multiply them all by r? Then we’d have about 370 Hz 329.6 Hz 293.7 Hz. F# E D.
This would feel like the same tune. If I play the whole of MHALL to you, with any such frequency shift, you’d immediately know it was the same tune.

Increasing by an octave doubles the frequency. Middle C, also called C4, is 261.6 Hz. C5 is 523.2 Hz etc.
There are 12 semitones to the octave, and hence going up a semitone is the same as increasing freq by 2^(1/12).

An instrument tuned with equal temperament can be equally well used to play tunes in any key.

Hanging in complement with this is the fact that simple fraction harmonies also sound … satisfying? They don’t have a bunch of frequent beats.

The simplest harmony might be between two notes an octave apart, ie having 2:1 ratio.

A perfect fifth is an interval with a 3:2 ratio. Plenty of instruments can be played with arbitrary pitch, including fretless stringed instruments, so you can reproduce this interval exactly (or as close as you can get with your level of skill).

However on a, shall we say, discretely intervalled instrument such as a piano, equal temperament tuning is the norm

It happens that 2^(7/12) is 1.49830707688, which is close enough to a perfect fifth for government work.

It happens that several powers of 2^(1/12) are pretty close to simple ratios.

Perfect third, also called major third, is a 5:4. This is prrretty close to 2^(4/12), which would be the difference between a C and an E on a piano.

A just fourth is 4:3, which is close to 2^(5/12), the interval between a C and an F on a piano.

This Rodgers organ can be played in equal temperament, pythagorean or mean tone.

Here are some demonstrations of all three tunings:

https://www.youtube.com/playlist?list=PL-0jRwu33pSid5VUnw_3SLy_jwawmZGLJ

And talking of having no frets:

Achilles and the Tortoise happen upon each other
in the park one day while strolling.

Tortoise: Good day, Mr. A.
Achilles: Why, same to you.
Tortoise: So nice to run into you.
Achilles: That echoes my thoughts.
Tortoise: And it’s a perfect day for a walk. I think I’ll be walking home soon.
Achilles: Oh, really? I guess there’s nothing better for you than walking
Tortoise: Incidentally, you’re looking in very fine fettle these days, I must say.
Achilles: Thank you very much.
Tortoise: Not at all. Here, care for one of my cigars?
Achilles: Oh, you are such a philistine. In this area, the Dutch contributions are of markedly inferior taste, don’t you think?
Tortoise: I disagree, in this case. But speaking of taste, I finally saw that Crab Canon by your favorite artist, M. C. Escher, in a gallery the other day, and I fully appreciate the beauty and ingenuity with which he made one single theme mesh with itself going both backwards and forwards. But I am afraid I will always feel Bach is superior to Escher.
Achilles: I don’t know. But one thing for certain is that I don’t worry about arguments of taste. De gustibus non est disputandum.
Tortoise: Tell me, what’s it like to be your age? Is it true that one has no worries at all?
Achilles: To be precise, one has no frets.
Tortoise: Oh, well, it’s all the same to me.
Achilles: Fiddle. It makes a big difference, you know.
Tortoise: Say, don’t you play the guitar?
Achilles: That’s my good friend. He often plays, the fool. But I myself wouldn’t touch a guitar with a ten-foot pole!

(Suddenly, the Crab, appearing from out of nowhere, wanders up excitedly, pointing to a rather prominent black eye.)

Crab: Hallo! Hulloo! What’s up? What’s new? You see this bump, this lump? Given to me by a grump. Ho! And on such a fine day. You see, I was just idly loafing at the park when up lumbers this giant fellow from warsaw. A colossal bear of a man playing a lute. He was three meters tall, if I’m a day. I mosey on up to the chap, reach skyward and manage to tap him on the knee, saying, “Pardon me, sir, but you are Pole-luting our park with your mazurkas.” But wow! he had no sense of humor-not a bit, not a wit-and POW!-he lets loose and belts me one, smack in the eye! Were it in my nature, I would crab up a storm, but in the time-honored tradition of my species, I backed off. After all, when we walk forwards, we move backwards. It’s in our genes, you know, turning round and round. That reminds me-I’ve always wondered, “Which came first-the Crab, or the Gene?” That is to say, “Which came last the Gene, or the Crab?” I’m always turning things round and round, you know. It’s in our genes, after all. When we walk backwards, we move forwards. Ah me, oh my! I must lope along on my merry way-so off I go on such a fine day. Sing “ho!” for the life of a Crab! TATA! iOle!

(And he disappears as suddenly as he arrived.)

Tortoise: That’s my good friend. He often plays the fool. But I myself wouldn’t touch a ten-foot Pole with a guitar!
Achilles: Say, don’t you play the guitar? Tortoise: Fiddle. It makes a big difference
Achilles: Oh, well, it’s all the same to me.
Tortoise: To be precise, one has no frets.
Achilles: Tell me, what’s it like to be your age? Is it true that one has no worries at all?
Tortoise: I don’t know. But one thing for certain is that I don’t worry about arguments of taste. Disputandum non est de gustibus.

Achilles: I disagree, in this case. But speaking of taste, I finally heard that Crab Canon by your favorite composer, J. S. Bach, in a concert other day, and I fully appreciate the beauty and ingenuity with which he made one single theme mesh with itself going both backwards and forwards. But I’m afraid I will always feel Escher is superior to Bach
Tortoise: Oh, you are such a philistine. In this area, the Dutch contributions are of markedly inferior taste, don’t you think?
Achilles: Not at all. Here, care for one of my cigars?
Tortoise: Thank you very much.
Achilles: Incidentally, you’re looking in very fine fettle these days, I must say.
Tortoise: Oh, really? I guess there’s nothing better for you than walking.
Achilles: And it’s a perfect day for a walk. I think I’ll be walking home soon.
Tortoise: That echoes my thoughts.
Achilles: So nice to run into you.
Tortoise: Why, same to you.
Achilles: Good day, Mr. T

Bubblecar said:

This Rodgers organ can be played in equal temperament, pythagorean or mean tone.

Here are some demonstrations of all three tunings:

https://www.youtube.com/playlist?list=PL-0jRwu33pSid5VUnw_3SLy_jwawmZGLJ

If PWM isn’t in here doing a rogers organ joke then the world has gone mad.

The Rev Dodgson said:

Achilles: Good day, Mr. T

How very interesting

dv said:

Bubblecar said:

This Rodgers organ can be played in equal temperament, pythagorean or mean tone.

Here are some demonstrations of all three tunings:

https://www.youtube.com/playlist?list=PL-0jRwu33pSid5VUnw_3SLy_jwawmZGLJ

If PWM isn’t in here doing a rogers organ joke then the world has gone mad.

Dylan Thomas Under Milk Wood:

Third Woman: And how’s Organ Morgan, Mrs. Morgan?
First Woman: you look dead beat
Second Woman: it’s organ organ all the time with him
Third Woman: up every night until midnight playing the organ.
Mrs. Organ Morgan: Oh, I’m a martyr to music.

dv said:

The Rev Dodgson said:

Achilles: Good day, Mr. T

How very interesting

From Godel Escher Bach.

In case that wasn’t obvious.