Perfect Tuning is a Mathematical Impossibility

[crazyBassClown]

Just get a fretless, and play every note out of tune.

A surprising number of people manage this on fretted instruments as well.
Gotta love the discordant wailings of a newbie guitarist playing a chord with half the notes bent out of tune by poor technique.

 

[Bipslapper]

Yes, when the seventeen drunk girls on summer break on Cape Cod are dancing to a silly pop tune our (shameful) cover band is playing in a sold out ocean-side club in July notice that my low "B" string is not perfectly in tune when I hit the low C# on "Shut up and Dance" things go to hell. I concur....

 

[Ekulati]

And a physical impossibility too.

 

[red_rhino]

I love when musician talk math. Like it matters.

 

[MVE]

Almost 8 pages... and so much misunderstanding.

It's pretty easy.
A "perfect" fifth doesn't beat.
A perfect, non beating fifth's frequency is 3:2 ratio of the root.
A perfect octave is 2:1.
Start doing the math for the circle of 5ths from any arbitrary root tone (I will use A4 at 440 for no particular reason)
440/2 x 3 = E660
660/2 x 3 = B990
990/2 x 3 = G#1485 ect
If you keep going all the way around the circle back to A, you will end up at 57,088.3887

But if you go up by 2:1 octaves and just keep doubling 440 by octaves you eventually get to A11 at 56,320.

I know, I know only a dog or maybe some insects can hear that, but that's not the point. The point is assending by perfect 5ths doesn't quite line up with perfect octaves.

So there you go.

We could tune a keyboard to have "perfect" non beating 5ths but the octaves would be off at either end and you could only play it in one key. So, we have pushed the tuning of the 5th to be a little out of tune; it is "tempered" to allow the octaves to meet up.

This is not "stretch". Stretch is something else that is done in response to the overtones that are never quite in tune with the fundamental.

So, to make a long story short, that $150,000 Steinway D is NOT IN TUNE!

 

[Ekulati]

Almost 8 pages... and so much misunderstanding.

It's pretty easy.
A "perfect" fifth doesn't beat.
A perfect, non beating fifth's frequency is 3:2 ratio of the root.
A perfect octave is 2:1.
Start doing the math for the circle of 5ths from any arbitrary root tone (I will use A4 at 440 for no particular reason)
440/2 x 3 = E660
660/2 x 3 = B990
990/2 x 3 = G#1485 ect
If you keep going all the way around the circle back to A, you will end up at 57,088.3887

But if you go up by 2:1 octaves and just keep doubling 440 by octaves you eventually get to A11 at 56,320.

I know, I know only a dog or maybe some insects can hear that, but that's not the point. The point is assending by perfect 5ths doesn't quite line up with perfect octaves.

So there you go.

We could tune a keyboard to have "perfect" non beating 5ths but the octaves would be off at either end and you could only play it in one key. So, we have pushed the tuning of the 5th to be a little out of tune; it is "tempered" to allow the octaves to meet up.

This is not "stretch". Stretch is something else that is done in response to the overtones that are never quite in tune with the fundamental.

So, to make a long story short, that $150,000 Steinway D is NOT IN TUNE!

Yup, the old Pythagorean comma. But more importantly, what color corresponds with that out of tune high A?

 

[Mr_Moo]

Almost 8 pages... and so much misunderstanding.

It's pretty easy.
A "perfect" fifth doesn't beat.
A perfect, non beating fifth's frequency is 3:2 ratio of the root.
A perfect octave is 2:1.
Start doing the math for the circle of 5ths from any arbitrary root tone (I will use A4 at 440 for no particular reason)
440/2 x 3 = E660
660/2 x 3 = B990
990/2 x 3 = G#1485 ect
If you keep going all the way around the circle back to A, you will end up at 57,088.3887

But if you go up by 2:1 octaves and just keep doubling 440 by octaves you eventually get to A11 at 56,320.

I know, I know only a dog or maybe some insects can hear that, but that's not the point. The point is assending by perfect 5ths doesn't quite line up with perfect octaves.

So there you go.

We could tune a keyboard to have "perfect" non beating 5ths but the octaves would be off at either end and you could only play it in one key. So, we have pushed the tuning of the 5th to be a little out of tune; it is "tempered" to allow the octaves to meet up.

This is not "stretch". Stretch is something else that is done in response to the overtones that are never quite in tune with the fundamental.

So, to make a long story short, that $150,000 Steinway D is NOT IN TUNE!

That's right, but ICBS (it can be shown) that Groot's Conjecture is a physicist's approximation of a math joke. It follows, by the UC Lemma, that applying the Jordan form and treating the bass guitar as a perfect sphere, the joke is homeomorphic to a previous joke and the solution punchline solution can be shown to exist. The details are left as an exercise for the reader.

 

[fdeck]

Tuning is subjective. It's whatever you want it to be, that sounds good for the music that you're playing. Wind instruments have no well-defined "temperament" to speak of. Players push the notes around as needed by how they control their breath and embochure. String instruments are tuned in perfect intervals, and players adjust the notes slightly to make the intervals sound better. Early temperaments were known to be a compromise, but a musician had to be able to tune their own instrument quickly, because keyboard instruments went out of tune. Equal temperament required a skilled technician and an instrument that was stable enough to be worth tuning in that way. Electronic instruments are whatever you want. I remember a Korg analog synth that had a row of 12 knobs so you could choose your own temperament.

My son's cello teacher has a viola da gamba tuner with one knob for the tuning note, and another knob for choosing from a variety of temperaments. I'm in the process of building a new gamba tuner using a microprocessor, since the maker of the teacher's tuner has gone out of business. So I've got a spreadsheet full of different temperaments.

 

[chuck3]

I only sorta understand all this stuff but I like reading about it. All very interesting to me. Besides upright and electric bass my other main instrument is mandolin and a mando is never perfectly in tune, IMV. Different mandos by different makers (Collings vs. Gibson for example) are out of tune in different ways.

 

[honeyiscool]

That may be the modern calculation, but it originates as a division of the length of a day. A 60th of a 60th of a 24th of a day. Invented by people working in base 6.
Base 6 is cool. You can count up to 35 using only your fingers.

Thanks for explaining what a traditional second is. I didn't know.

 

[MVE]

Tuning is subjective. It's whatever you want it to be, that sounds good for the music that you're playing. Wind instruments have no well-defined "temperament" to speak of. Players push the notes around as needed by how they control their breath and embochure. String instruments are tuned in perfect intervals, and players adjust the notes slightly to make the intervals sound better. Early temperaments were known to be a compromise, but a musician had to be able to tune their own instrument quickly, because keyboard instruments went out of tune. Equal temperament required a skilled technician and an instrument that was stable enough to be worth tuning in that way. Electronic instruments are whatever you want. I remember a Korg analog synth that had a row of 12 knobs so you could choose your own temperament.

My son's cello teacher has a viola da gamba tuner with one knob for the tuning note, and another knob for choosing from a variety of temperaments. I'm in the process of building a new gamba tuner using a microprocessor, since the maker of the teacher's tuner has gone out of business. So I've got a spreadsheet full of different temperaments.

Yes, but if we think of the concept of harmony and being in tune as a function of sound waves' peaks and valleys lining up or not lining up, then it becomes quite objective and defined.
Of course, things like overtones, reflections, our own perceptions and maybe even the shape of our ears make this concept practically subjective as it relates to music.
But as a function of wave physics, it is very objective and definite.

 

[GoesThump]

I'm by no means well-versed (illiterate) in music theory, but a couple years a go I read a book called Temperament by Stuart Issacoff that was a history of how western tuning standards arose over the last several centuries. I highly recommend it to anyone even remotely curious about this stuff. There was some math but even glossing over that it was a really good read.

View attachment 3277001

This is a really good book - highly recommended.

I often used it to backstop arguments about tuning when playing fretless. It was a desperate attempt to hide the fact that I suck.

GT

 

[fdeck]

Yes, but if we think of the concept of harmony and being in tune as a function of sound waves' peaks and valleys lining up or not lining up, then it becomes quite objective and defined.
Of course, things like overtones, reflections, our own perceptions and maybe even the shape of our ears make this concept practically subjective as it relates to music.
But as a function of wave physics, it is very objective and definite.

Definitely. The 12 tone scale is the simplest consonant scale. So I suspect it's no accident why it's so widespread in traditional music. There are exceptions of course. But I suspect that once the 12 tone system is accepted, there's a lot of leeway in terms of how it's realized on instruments.

 

[thewildest]

So A is the only note with a precisely defined value. All other notes are an approximation to whatever degree of accuracy is desired.

Thank you for a thread that sparks deep intellectual exchange, and mixes humour and healthy discussion; I agree with some comments that the issue is not generating true temperament, but how we do "approximations" to our 12 note system.

Just for kicks, if I ever desing a perfect temperament instrument, I will try it first with a Chorus, as I won't stand its perfection anyway.

 

[JPDub]

Thanks for explaining what a traditional second is. I didn't know.

The point is that as a chunk of time, it's a pretty arbitrary measure. Defined that way, it's also constantly changing as the rotation of the Earth gradually slows.
440Hz is only an integer because of how a Hz has been defined.

 

[MVE]

Thank you for a thread that sparks deep intellectual exchange, and mixes humour and healthy discussion; I agree with some comments that the issue is not generating true temperament, but how we do "approximations" to our 12 note system.

Just for kicks, if I ever desing a perfect temperament instrument, I will try it first with a Chorus, as I won't stand its perfection anyway.

What do you mean by "perfect temperament?"
Temperament is, by definition, the act of making it less perfect.
There are plenty of examples of harmonically perfect (string quartets, choirs, acapella, etc...)
Temperament is just something we do to make instruments with specifically defined notes (fretted guitars, pianos) work across a range of keys.
Frankly, to me, it's all a bunch of semantics and scholarly pursuits anyway as I, personally, can barely tell the difference between a M3 or m3, much less the difference between a harmonically perfect C# or Db.

 

[tubedude]

Yes indeed!

Also, as our fingers transmit warmth to the strings and changing their temperature, the strings stretch. Constantly going out of tune as we constantly play.

Ya know, I don't think a perfectly in tune instrument is going to sound right anyway. That's what makes the violin section sound so beautiful. Everybody is a little bit out of tune.[/QUOTE

That's the definition of a minor 2nd, two fretless players playing in unison.

 

[Wisebass]

You can count up to 35 using only your fingers.

Believe me or not. I can count to 35 without using my fingers!

Not that I will ever need that. 1, 2, 3, 4 is good enough for Rock& Roll!


greetings

Wise(b)ass

 

[tubedude]

There is a theory that vibrato evolved in order to emulate the impossible perfect pitch

You'll hear this with fretless players and guitar slide players. Even Jeff Beck on the Ronnie Scott's album, where he uses the slide up near the pickups where microns make big pitch changes, he vibratoes his way to getting the pitch correct.

 

[tubedude]

*Pulls out liberal arts degree*

Being "in tune" is, like, a social construction.

Yeah, well that's just like your opinion man.