Due to the stiffness of real strings, the overtones do not occur at even multiples of the fundamental frequency.
This means that when tuning a piano, you need to compromise between tuning so that the fundamentals are in tune with distant notes but having the overtones sound discordant with nearby notes, or having sweet overtone matching with nearby notes, causing notes to be very far off when harmonizing at longer distances across the keyboard.
With my upright, I've had more success tuning more based on the fundamental than on the overtones, because it has enough inharmonicity that tuning to the overtones causes right-hand-to-left-hand harmonies to sound noticeably off. But this makes single-hand chords sound messier.
A piano with looooooooooooooooong bass strings can have skinnier strings that are flexible and behave closer to ideal, so you don't have to compromise as much when tuning.
Excellent explanation. We can get a little quantitative if we remember that the fundamental frequency of a vibrating string is sqrt(T/μ)/(2L), where T is the tension, μ is the linear density, and L is the length. So if we make the string twice as long, (T/μ) has to be increased by a factor of four; if we keep the tension the same we can use a string with 1/4 the linear density, which I guess would decrease the inharmonicity significantly.
The square root magnifies the effect of increasing the length, so you get a lot from making the piano a bit longer. The project is a wonderful idea.
Normally, the lowest few octaves on a piano all sound a little weird; the individual notes are a little hard to tell apart and multiple low notes played together often blend into a rumbly mess. This is because the strings for those notes are shorter than they should be for their frequencies -- they're artificially made to vibrate at a lower frequency by making the strings heavier.
This effect seems to be significantly reduced on the Alexander piano.
I hear solely through a hearing aid and always thought the low frequencies were just being butchered by my hearing aid. Glad to learn it's not just me!
The sound of inharmonicity comes from dispersion -- different frequencies travel through the string at different velocities. The lower the frequency, the less the wave notices the stiffness of the string, so to speak, and the stiffer the string the quicker a wave will travel through it. If you've ever tapped on a wire fence or played with a slinky, you'll be familiar with the "pew" sound. A pure impulse consists of all frequencies at once, but when it's traveled through the fence, bounced off a post, and come back, you hear the high frequencies first, which is why the "pew" descends in frequency.
Anyway, I've found that you can notice this on the attacks of bass notes on pianos. Only the very lowest notes of the Alexander piano seem to really have it audible, and even then it's much more slight.
I thought the cause of inharmonicity was that higher frequencies cause steeper bending of the string at the attachment points, which makes the effective length of the string shorter. Sound on Sound's article on synthesizing guitar sounds (which have the same problem) has a good diagram explaining it (see figure 12):
Thanks for sharing that article. My understanding is that this bridge effect is actually what is happening across the entire length of the string; it's just that the boundary condition makes the effects of stiffness clearer.
Stiffness causes there to basically be a radius of curvature in the string when you apply a force. The boundary condition of a guitar string is that the displacement and first derivative of displacement of the string are zero at both ends. So, this radius of curvature will be visible there. But, even when plucking a string, rather than having a sharp peak at the plectrum, it will necessarily be similarly smoothed out. (Though, through time in a frequency-dependent way.)
In the wave equation, stiffness involves a factor with a coefficient proportional to Young's modulus. Based on the stress/strain graphs I could find, Young's modulus of a guitar string increases with tension, increasing inharmonicity. Of course, the pitch of the string also increases with tension, so there's a lot going on.
(I have to admit that the zero-first-derivative boundary condition having no additional effect is coming from my intuitions about linearity of solutions to the wave equation, but maybe it still has some interesting effect. I think the overall effect of stiffness would dominate this one, however.)
The harmonic series is a really neat thing to learn about -- one of those "art is math" kind of moments.
This video does a good job of explaining it in a relatively short and simple way, but goes in-depth enough that even people with an understanding of music might learn something:
Inharmonicity is when the overtones (frequency multiples) are not whole number multiples of the fundamental frequency. It sounds bad. Discordant, like playing two notes directly next to each other on the keyboard.
When designing a string or percussion instrument (or any resonator, in general, I suppose) one of the challenges is ensuring that you aren't creating undesired harmonics. With something like a piano with hundreds or thousands of strings that might be induced to resonate undesirably if even slightly out, or whose own movement might alter other parts and affect them, it becomes a significant challenge.
When played a note will emit overtones -- the same note an at higher octaves. So a low C will also have middle C, high C, etc at different volumes depending on the instrument. This is playing low notes without low volume overtones. In short, the low notes are more purely low.
That's not quite right. The overtones are the harmonic series, not just octaves (1:2, 1:4, 1:8, etc..) but an octave and a fifth (1:3), two octaves and a third, (1:5), etc..
In a piano, the bass and treble strings have a lot of inharmonicity. They don't behave like ideal strings for various reasons, so the harmonics aren't exact multiples of the fundamental frequency. Piano tuners deal with this be stretching the octave, so that the piano sounds in tune even though it kind of isn't.
Having a super long bass string instead of a short double-wound bass string probably behaves more like an ideal string (with harmonics that are closer to whole-number multiples of the fundamental), so it should sound like it has a more definite pitch and less like a gong.
Interestingly, we don't really perceive the fundamental frequencies of the low notes in a piano much at all in the first place. If you filter them out, the notes will sound about the same -- we take most of our perceptual cues from the higher harmonics, and our brain just inserts the implied bass fundamental. In the same way, playing a major chord sort of implies a bass note a couple octaves below the root of the chord -- if all the harmonics are there, then we expect it to be there.
“playing a major chord sort of implies a bass note a couple octaves below the root of the chord”
Wow! Of course! Lightbulb moment. This isn’t something that occurred to me independently but makes so much sense I wonder how it hadn’t, and it explains a lot of what chords are really all about.
Yeah, chords are interesting. Major triads are basically a 4:5:6 ratio, which is the 4th, 5th, and 6th harmonic of some low bass note. (Different inversions get you 2:3:5 or 3:4:5 or whatever.) A dominant 7th chord gets you one more step in the harmonic series: 4:5:6:7. That 7th harmonic is way out of tune in 12-tone equal temperament, though.
Minor chords are a 10:12:15 ratio, which kind of looks like a bunch of arbitrary chosen numbers, but we could rewrite that as 60/6 : 60/5 : 60/4 and then it sort of make sense that minor would be a sort of mathematical reciprocal of the major. Minor chords can be interpreted either as the 10th, 12th, and 15th harmonics of some bass note or as the first three prime subharmonics that are integer divisions of the fundamental note. Subharmonics don't really occur in nature, though.
>That 7th harmonic is way out of tune in 12-tone equal temperament
You can sing in any tuning though. If you want to hear just-intonation 4:5:6:7 chords, listen to barbershop music, where this chord is a defining part of the style. See:
