Why It's Impossible to Tune a Piano
5 years agoThis also gives a nice explanation for why it is natural to have 12 notes in a chromatic scale, and not some other number: powers of the twelfth root of two have a tendency to be surprisingly close to simple rational numbers. For instance it's fifth power is strangely close to 4/3, it's seventh power is close to 3/2, it's 4th power is close to 4/5, etc. Powers of, say, the 11th root of 2 would not play so nicely.
Great video!
Actually, my next video will also have a little something to do with this :)
+3Blue1Brown you answered! Awesome haha I found your videos about 3 days ago, so it was a cool coincidence to find your comment on this video. I am a math fan, and your different way of explaining complicated problems is like a fresh breeze in a rather straight forward field. Keep up the good work! I'll keep recommending your channel to others like us who appreciate the beauty of mathematics. Cheers!
I have made music where the notes are all based on the eleventh root of two, and the harmonies are really odd I must say. None of them are harmonise in as simple of a ratio as in 12ET.
Now try the fifty third root of two, or f = 1.013164143...
f^31 = 1.49994... ~= 3/2
f^22 = 1.3333858... ~= 4/3
f^17 = 1.24898... ~= 5/4
f^14 = 1.200929.... ~= 6/5
f^9 = 1.124911... ~= 9/8
Of course, the issue there is that 53 is prime, so you would have to
use a digital device that could re-tune itself each time you changed
keys. Otherwise the spacing would be inconsistent.
3Blue1Brown yeah so do powers of the 53rd root of 2. Don't pretend like 12tet isn't arbitrary.
3Blue1Brown and it's not a strange tendency, 7/12 is one of the convergents to log2(3/2)
Teddy Dunn the problem with 53-tet is that there are 53 notes
@Cooper Gates "Of course, the issue there is that 53 is prime, so you would have to
use a digital device that could re-tune itself each time you changed
keys. Otherwise the spacing would be inconsistent."
There's no problem with 53 being prime. It is a functional Equal-step Tuning as well, named 53ET (53 Equal-step Tuning) or 53edo (53 equal divisions of the octave). The problem is that it is not as practical as it would require many more keys on the piano to cover the same pitch range.
@groszak1 I suppose you could skip certain intervals depending on what keys and chords are most used, so there'd be gaps of 2^(2/53) or 2^(3/53) or more, and something like 24 keys for each octave.
I assume thats also why, since ancient times up to the French revolution, we used twelver systems. Not decimal systems.
Grant! I'm a few years late, but, happy to see you here 😆
I thought 12 was chosen because 12 is a highly composite number. Macro- and microtonality, though, are not uncommon outside Western music. Audible to humans are intervals as small as 5 cents (which is one 20th of a semitone), but even a 15-cent interval is probably difficult to discern for most people. The neutral third, which lies midway between the major third and the minor third, is one of the most common intervals produced by human infants. I think that there are good arguments to be made in favor of certain intervals being thought of as more "universal" or "fundamental" than others; on the chromatic scale, these would be the traditional "perfect consonants", in descending order: the unison, the octave, the fifth and the fourth.
@ewqdsacxz765 12 is a reasonable number of increments in an octave. 53ET, on the other hand, while it can obtain much higher accuracy, would require irregular subdivisions depending on the intervals and keys desired. 1 step, 2, steps, 4 steps, or some number of powers of 2^(1/53).
@Cooper Gates 53ET would be fine for making microtonal music in, though, no? With the added benefit of approximating the 3:2 and 4:3 pitch ratios even more accurately than is allowed for in 12ET.
@ewqdsacxz765 You should be able to even make a physical piano using 53ET, except you wouldn't want to include all 53 intervals in each octave, which would produce, for instance, 318 keys for 6 octaves, so you'd have limited options to modulate a particular set of chords, because keys would be missing.
@Cooper Gates Right, unless you limit yourself to only playing within the range of a single octave, which, to be fair, could be pretty limiting. I guess microtonal music is a lot easier to compose on a computer than on a more traditional instrument.
@ewqdsacxz765 Another thing I've wondered is why octaves are based on powers of 2. Because it's the first prime? I almost wish someone had a piano based on powers of 3, but I haven't checked which fractional powers of 3 can best approximate the most common harmonics yet.
@Cooper Gates Well, acoustically, an octave is simply a doubling of a frequency. I doubt it has anything to do with primality; just a matter of "2" being the smallest plural element of the set of natural numbers (provided that "plural" is defined as such that possess a cardinality greater than 1). If you attempt to do the same with a smaller integer, multiplying a numerator by 1 fails to produce any novel value. Multiplication of acoustic frequency by zero would imply a perfect vacuum, if I'm not mistaken, and that's a quantum-mechanical impossibility. If you try to use negative integers, it would be just as unusable as multiplication of frequencies by zero, since there is no such thing as subzero acoustic frequencies, either (unless you redefine them to mean something weird, like antiparticle acoustics).
Audition (or auditory perception) of the octave, however, is as much of a psychological phenomenon as it is physical/acoustic. Humans and chimpanzees are both believed to experience octave equivalence (perceptually), whereas birds are believed to not. "Sound" is an ambiguous word, since it can refer to either acoustic waves (as studied in physics) or the perception thereof (as studied in psychology and musicology). "If a tree falls in a forest and no one is around to hear it, does it make a sound?" Depends, on how you define "sound."
Men's and women's voices are typically an octave or so apart, and that's handy for choir singing. Here's one last trivium for fun: music and language have probably co-evolved, in the biological sense. This applies to both pitch and rhythm. The pitch range of musical instruments typically overlaps closely with the pitch range of human voices, as used for both song and speech. It's safe to assume that vocals were the first musical instrument ever used, and that other non-percussive instruments imitate the human voice to some degree. As for rhythm, the range of tempo in music roughly corresponds to the range of heart beat rates, and people further adjust their heart beats to music tempo by rhythmic motion (i.e. dance).
This also gives a nice explanation for why it is natural to have 12 notes in a chromatic scale, and not some other number: powers of the twelfth root of two have a tendency to be surprisingly close to simple rational numbers. For instance it's fifth power is strangely close to 4/3, it's seventh power is close to 3/2, it's 4th power is close to 4/5, etc. Powers of, say, the 11th root of 2 would not play so nicely.
Great video!
Actually, my next video will also have a little something to do with this :)
+3Blue1Brown you answered! Awesome haha I found your videos about 3 days ago, so it was a cool coincidence to find your comment on this video. I am a math fan, and your different way of explaining complicated problems is like a fresh breeze in a rather straight forward field. Keep up the good work! I'll keep recommending your channel to others like us who appreciate the beauty of mathematics. Cheers!
I have made music where the notes are all based on the eleventh root of two, and the harmonies are really odd I must say. None of them are harmonise in as simple of a ratio as in 12ET.
Now try the fifty third root of two, or f = 1.013164143...
f^31 = 1.49994... ~= 3/2
f^22 = 1.3333858... ~= 4/3
f^17 = 1.24898... ~= 5/4
f^14 = 1.200929.... ~= 6/5
f^9 = 1.124911... ~= 9/8
Of course, the issue there is that 53 is prime, so you would have to
use a digital device that could re-tune itself each time you changed
keys. Otherwise the spacing would be inconsistent.
3Blue1Brown yeah so do powers of the 53rd root of 2. Don't pretend like 12tet isn't arbitrary.
3Blue1Brown and it's not a strange tendency, 7/12 is one of the convergents to log2(3/2)
Teddy Dunn the problem with 53-tet is that there are 53 notes
@Cooper Gates "Of course, the issue there is that 53 is prime, so you would have to
use a digital device that could re-tune itself each time you changed
keys. Otherwise the spacing would be inconsistent."
There's no problem with 53 being prime. It is a functional Equal-step Tuning as well, named 53ET (53 Equal-step Tuning) or 53edo (53 equal divisions of the octave). The problem is that it is not as practical as it would require many more keys on the piano to cover the same pitch range.
@groszak1 I suppose you could skip certain intervals depending on what keys and chords are most used, so there'd be gaps of 2^(2/53) or 2^(3/53) or more, and something like 24 keys for each octave.
I assume thats also why, since ancient times up to the French revolution, we used twelver systems. Not decimal systems.
Grant! I'm a few years late, but, happy to see you here 😆
I thought 12 was chosen because 12 is a highly composite number. Macro- and microtonality, though, are not uncommon outside Western music. Audible to humans are intervals as small as 5 cents (which is one 20th of a semitone), but even a 15-cent interval is probably difficult to discern for most people. The neutral third, which lies midway between the major third and the minor third, is one of the most common intervals produced by human infants. I think that there are good arguments to be made in favor of certain intervals being thought of as more "universal" or "fundamental" than others; on the chromatic scale, these would be the traditional "perfect consonants", in descending order: the unison, the octave, the fifth and the fourth.
@ewqdsacxz765 12 is a reasonable number of increments in an octave. 53ET, on the other hand, while it can obtain much higher accuracy, would require irregular subdivisions depending on the intervals and keys desired. 1 step, 2, steps, 4 steps, or some number of powers of 2^(1/53).
@Cooper Gates 53ET would be fine for making microtonal music in, though, no? With the added benefit of approximating the 3:2 and 4:3 pitch ratios even more accurately than is allowed for in 12ET.
@ewqdsacxz765 You should be able to even make a physical piano using 53ET, except you wouldn't want to include all 53 intervals in each octave, which would produce, for instance, 318 keys for 6 octaves, so you'd have limited options to modulate a particular set of chords, because keys would be missing.
@Cooper Gates Right, unless you limit yourself to only playing within the range of a single octave, which, to be fair, could be pretty limiting. I guess microtonal music is a lot easier to compose on a computer than on a more traditional instrument.
@ewqdsacxz765 Another thing I've wondered is why octaves are based on powers of 2. Because it's the first prime? I almost wish someone had a piano based on powers of 3, but I haven't checked which fractional powers of 3 can best approximate the most common harmonics yet.
@Cooper Gates Well, acoustically, an octave is simply a doubling of a frequency. I doubt it has anything to do with primality; just a matter of "2" being the smallest plural element of the set of natural numbers (provided that "plural" is defined as such that possess a cardinality greater than 1). If you attempt to do the same with a smaller integer, multiplying a numerator by 1 fails to produce any novel value. Multiplication of acoustic frequency by zero would imply a perfect vacuum, if I'm not mistaken, and that's a quantum-mechanical impossibility. If you try to use negative integers, it would be just as unusable as multiplication of frequencies by zero, since there is no such thing as subzero acoustic frequencies, either (unless you redefine them to mean something weird, like antiparticle acoustics).
Audition (or auditory perception) of the octave, however, is as much of a psychological phenomenon as it is physical/acoustic. Humans and chimpanzees are both believed to experience octave equivalence (perceptually), whereas birds are believed to not. "Sound" is an ambiguous word, since it can refer to either acoustic waves (as studied in physics) or the perception thereof (as studied in psychology and musicology). "If a tree falls in a forest and no one is around to hear it, does it make a sound?" Depends, on how you define "sound."
Men's and women's voices are typically an octave or so apart, and that's handy for choir singing. Here's one last trivium for fun: music and language have probably co-evolved, in the biological sense. This applies to both pitch and rhythm. The pitch range of musical instruments typically overlaps closely with the pitch range of human voices, as used for both song and speech. It's safe to assume that vocals were the first musical instrument ever used, and that other non-percussive instruments imitate the human voice to some degree. As for rhythm, the range of tempo in music roughly corresponds to the range of heart beat rates, and people further adjust their heart beats to music tempo by rhythmic motion (i.e. dance).
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