Perfect Tuning is a Mathematical Impossibility

[Mugre]

someone talked about the 3 trumpet effect. Over on TalkPiano (not really, it Piano World) we've been chatting about how pure the unisons under one hammer should be. Many pianists prefer the "bloom" of one of the strings being just slightly off--not enough to actually wobble but not quite perfect either.

In my life as a recorder player I sometimes go to "early music concerts". This means Baroque and earlier. Many of these are done on "early" instruments, the forerunners of today's bassoon, trumpets, saxophones etc playing with strings that can play in Just Intonation. These instruments do not play in ET but in one of the many (I'm told over a hundred) systems of tuning before ET took hold. Therefore they confine themselves to the friendly keys of each group of instruments. The sound is glorious. So coherent and harmonious and lacking the twangy quality of the narrow fifths and other ET compromises. It also helps that A is as low as 392hz.

There's a book for anyone who wants to dive deep on this called "How Equal Temperament Ruined Harmony." By Ross W. Duffin. You can read the intro and part of the first chapter in a preview on Google Books.

Personally I don't believe we should be chasing systems or tying to define perfect. We should be chasing expression. That way I can like Bessie Smith AND Debussy.

 

[LBS-bass]

The precise mathematical value of the tuning of my bass going through my Boss TU-3 tuning pedal on stage in a bar about to play rock and country to at least 100 adoring fans is......

(Takes deep breath)



.....plenty close enough.

Ah! Close enough for you, but... this comes into play in the real world all the time. It's the reason so many guitar players are endlessly trying to get their B strings in tune. Most don't realize that it's actually impossible so they'll tune with the tuner, notice that something sounds weird and it's most noticeable between the B and high E strings, so then comes the endless noodling with the tuning pegs to try and "fix" what can never be fixed. Drives me crazy when I see players doing this.

 

[micguy]

The OP is right, in that perfect tuning is impossible (sorta), but he's citing the wrong reasons. The fact that the 12th root of 2 is a irrational number is not the issue. The fact that the 12th root of 2, raised to various powers, doesn't quite give you the right integer ratios for certain harmonies, is. An octave is a ratio of 2 to 1 - that ratio sounds good to our ears, and the 12th root of 2 raised to the 12th power gives you a 2 to 1 ratio - that's fine. A musical fourth, a musical fifth - those sound best if they're precisely 4:3 or 3:2 ratios in frequency, which, in the equal tempered scale, they're close, but not exact - the 12th root of 2 raised to the 5th or 7th powers are not quite right. If you don't do equal tempering, you set one key for the right ratios, and everything else sounds worse.

So, what we have it terms of tuning is an approximation that makes all keys sound close, but not perfectly in tune. Perfection can be achieve (theoretically) with fretless instruments, but practically......that's a whole other issue.

One other thing that gets in the way here, is that instruments with large ranges (like a piano) are not tuned to perfect octaves. Our ears don't quite hear 2 to 1 as a perfect octave, so a piano tuner stretches octaves - higher notes are a bit sharp. Lower ones are a bit flat compared with perfect 2 to 1 ratios.

Even that is an approximation, as your perception of pitch depends on how loud something is. Loud Soprano singers often sing flat - to the audience. To them, they're on pitch, but their ears are hearing things really loud, and high pitches played loud sound higher than they really are, so they're compensating by singing a bit flat so, to them, it sounds on pitch. In the audience, not so much. In ear monitors, if you run them sensibly, can help with this - lower volume at the singer's ears means she'll sing more on pitch.

 

[jmlee]

You guys gave me a giggle this morning. [Context: My first degree was in honours mathematics.] It's not everyplace you'd find an argument about whether a ratio is irrational.

 

[IamGroot]

Never having passed a math class I'm more interested in The Implications (if there are any) of the split between math and nature. Einstein had to add a fudge factor to make his theory work but practically, as a trumpet player I'm always needing to use my ears and pull myself flat as I go higher or else I'll wind up too sharp (and why 2 or 3 trumpets work like piano strings, all slightly imperfect making a fat sound) Automatically (in Bb or any tuned brass) A and E in the staff are sharp, and C# and D below then staff are sharper, so that any non beginner horn has a trigger to lengthen the 3rd valve slide and pro horns may trigger the 1st and 3rd valve slides for intonation. That's 2/3 of the valves have slides on them just to stay in tune. Welcome to crazy world.

The fudge factor that Einstein had in his original General Theory of Relativity, the cosmolgical adjustment factor, was a source of great embarrassment for him. In the last half year, there has been a verinteresting paper released which shows Einstei may have been right after all.

 

[IamGroot]

You guys gave me a giggle this morning. [Context: My first degree was in honours mathematics.] It's not everyplace you'd find an argument about whether a ratio is irrational.

So, did you get the joke?
BTW, my partners in crime include a few retired math professors and physicist. I like to stir crap up like "is it possible that transcendental numbers do NOT sit on the continuum.

 

[JRA]

Aside from tempered tunings on a piano, perfect tuning is impossible on a fretted (or fretless) because you must take into account that as a string is pushed down to a fret, it is stretching sharp.

'fretlessness' can compensate for this --- fretted cannot = you're stuck with where the fret stops the string.

Perfection can be achieve (theoretically) with fretless instruments, but practically......that's a whole other issue.

...............

 

[IamGroot]

someone talked about the 3 trumpet effect. Over on TalkPiano (not really, it Piano World) we've been chatting about how pure the unisons under one hammer should be. Many pianists prefer the "bloom" of one of the strings being just slightly off--not enough to actually wobble but not quite perfect either.

In my life as a recorder player I sometimes go to "early music concerts". This means Baroque and earlier. Many of these are done on "early" instruments, the forerunners of today's bassoon, trumpets, saxophones etc playing with strings that can play in Just Intonation. These instruments do not play in ET but in one of the many (I'm told over a hundred) systems of tuning before ET took hold. Therefore they confine themselves to the friendly keys of each group of instruments. The sound is glorious. So coherent and harmonious and lacking the twangy quality of the narrow fifths and other ET compromises. It also helps that A is as low as 392hz.

There's a book for anyone who wants to dive deep on this called "How Equal Temperament Ruined Harmony." By Ross W. Duffin. You can read the intro and part of the first chapter in a preview on Google Books.

Personally I don't believe we should be chasing systems or tying to define perfect. We should be chasing expression. That way I can like Bessie Smith AND Debussy.

I brought up non equal tempered systems to a piano player and she was incredulous. Stll bent out of shape over it.

 

[JPDub]

NB there's no law of nature that says you have to measure vibrations in Hertz.
Hertz = vibrations / (time it takes Earth to rotate on its axis / (24 x 60 x 60))
Use a different unit and A440 may or may not be rational.

 

[honeyiscool]

So what is the precise value?

There's nothing particularly precise about how we normally measure a second. Officially, a second is "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom," but every single tuner pedal uses a crystal instead of an atomic clock so even 440 Hz, also known as 440 cycles per second, is never precise when measured by a digital tuner, either.

I get that you're trying to be funny, but in my mind, the meta joke is that I find it hilarious that people can find numbers in front of the unit like 440 to be so important to get correct when the reality is that the unit itself is hard enough to measure correctly.

 

[LBS-bass]

'fretlessness' can compensate for this --- fretted cannot = you're stuck with where the fret stops the string.
View attachment 3277091

...............

Fretlessness can't compensate at the nut, which is where tuning the instrument itself takes place. But yes, as the instrument is played you can compensate with a fretless to the best of your ear's ability.

 

[micguy]

The fudge factor that Einstein had in his original General Theory of Relativity, the cosmolgical adjustment factor, was a source of great embarrassment for him. In the last half year, there has been a verinteresting paper released which shows Einstei may have been right after all.

Dark energy (The energy of free space, the thing that makes the cosmological constant of Einstein have meaning) has been discussed since about 1999. And it turns out that it's most of what exists in the universe - there's more dark energy than there is dark matter or regular matter.

 

[Mugre]

I brought up non equal tempered systems to a piano player and she was incredulous. Stll bent out of shape over it.

"It is the mark of an educated mind to be able to entertain a thought without having to accept it."
Aristotle

 

[LBS-bass]

I brought up non equal tempered systems to a piano player and she was incredulous. Stll bent out of shape over it.

That's so funny. This thing about tempered tuning is one of the first things I learned about when I started out in music and it's particularly relevant to piano players. Anyone who has ever tried to tune their own piano will quickly come to realize that there's something wrong, unless they have ears of stone.

 

[JRA]

But yes, as the instrument is played you can compensate with a fretless to the best of your ear's ability.

yes.

 

[Tommyc]

This is why Bach pushed for a new temperament , equal temperament. The old system of mean tone temperament rendered keys using many accidentals to sound out of tune so free modulation to remote keys from a given tonic was generally avoided, at least in fixed pitched instruments like keyboards and lutes. In 1722 he wrote the first Book of the Well-Tempered Clavier to demonstrate that free modulation could be done in all 12 keys utilizing the new tuning system. Tempering IS supposed to be a mathematical compromise. It's about the ears, NOT the numbers.

 

[redwingxix]

This is why Bach pushed for a new temperment , equal temperment. The old system of mean tone temperment rendered keys using many accidentals to sound out of tune so free modulation to remote keys from a given tonic were generally avoided, at least in fixed pitched instruments like keyboards and lutes. In 1722 he wrote the first Book of the Well-Tempered Clavier to demonstrate that free modualtion could be done in all 12 keys utilizing the new tuning system. Tempering IS supposed to be a mathmatical compromise. It's about the ears, NOT the numbers.

That was the basic focus of that book I was talking about and what a battle it was for the new standards to become adopted.

Music is heard by people, not calculators and abacuses (abaci?)

 

[redwingxix]

This is why Bach pushed for a new temperment , equal temperment. The old system of mean tone temperment rendered keys using many accidentals to sound out of tune so free modulation to remote keys from a given tonic were generally avoided, at least in fixed pitched instruments like keyboards and lutes. In 1722 he wrote the first Book of the Well-Tempered Clavier to demonstrate that free modualtion could be done in all 12 keys utilizing the new tuning system. Tempering IS supposed to be a mathmatical compromise. It's about the ears, NOT the numbers.

That was the basic focus of that book I was talking about and what a battle it was for the new standards to become adopted.

Music is heard by people, not calculators and abacuses (abaci?)

 

[lfmn16]

But yes, as the instrument is played you can compensate with a fretless to the best of your ear's ability.

If you took each note even the best fretless player in the world played and analyzed them I doubt they'd be in exact tune. Some times good enough is actually good enough.

 

[wizerd]

I'd love to invent a tuning system involving i (square root of -1).

I know a singer who uses that system, ... I I I me me me..