13. Inharmonicity and Octave Stretching
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- เผยแพร่เมื่อ 3 ส.ค. 2024
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In theory, all overtones are just whole number multiples of the fundamental frequency. However, for this to be the case strings have to be perfectly flexible, which they are not. They are made out of stiff metal which doesn’t quite bend perfectly. So in reality, there may be overtones which are not whole number multiples of the fundamental. This is called: inharmonicity. The shorter, thicker and looser a string is, the more inharmonicity it exhibits - precisely because shorter and thicker strings cannot bend as easily. And unfortunately, the lower strings on an upright piano are short, think and loose - which means they are very inharmonious. This inharmonicity contributes to the richness of the sound of a piano, but it also creates some dissonance when you play a low note on an upright piano. And it’s partly the reason why recreating the sound of an acoustic piano on a digital piano is quite difficult. It’s not as simple as just creating an fundamental frequency and adding some overtone.
Now, inharmonicity caused by the stiffness of the strings also has another interesting feature. The inharmonic overtones tend to be a little bit too high. The 16th overtone is about a semitone higher than it should technically be. And the 50th overtone is about a Perfect 5th higher than it should technically be. This is one of the reason that you cannot tune a piano.
If you tune two notes on the piano to a perfect octave, that is, so the fundamental frequencies are a ratio of 2/1, then because of this inharmonicity, the notes will sound out of tune. Which is an interesting thing to think about. If a piano is perfectly in tune, it is actually out of tune.
I hope you are ok with all this correction. Another: Your example of A3 with 2nd partial = 444 Hz puts the 2nd partial as 16 cents sharp. (1 Hz = 4 cents at 440 Hz) That is way too much stretch. No piano string has this amount. I know you are using these numbers just to represent but it's good to estimate with numbers that are typical.
Correction: Bass strings on an upright are not loose. In fact, they are tighter than the other strings.
Exactly. They're quite stiff on upright pianos as they're much shorter and thicker.
Your example at 2:59 is what we call a 2:1 octave. It is not stretched enough. Standard stretch is between a 4:2 and a 6:3.
Excellent ... thanks for sharing this great information.
This was short and pretty clear!
Wow, absolutely changed my mind. Gotta go to try recreating this effect with additive synthesizer (Harmor)
Well, I were able to make bass sound really natural, but rest got more like a slightly detuned guitar. I can explain what I did if anyone is interested
@@user-xl4tb2mb6h I’d love to read what you did!
@@joshjams1978 I'd love to too :)
I'm thinking this might explain why my short-scale bass guitar sounds a little "off" even though my tuner tells me it's in tune.
It is. Happens to me on guitar with the low string especially when moving up the fretboard. I tune 5-8 cent flat on that string
I think this explains why the notes on the low E string of my short-scale bass sound out of tune even though the bass/guitar tuner says it’s perfectly in tune. Question: how might one resolve this issue on a short-scale bass in terms of tuning - what is the reference point for getting the E string stretched flat sufficiently so that it’s in tune with the rest of the instrument and if a bass/guitar tuner won’t work - what else can be used to achieve this? I though maybe about adjusting 440hz down on my tuner but by how much? Thanks
So wait, you're saying the stretched tuning is due to physical reasons. I've heard other people saying it's due to the nature of human hearing.
Correct, it’s due to physical properties. Human hearing has nothing to do with why we stretch piano tuning
It's both. The combination of physics of strings and psychoacoustics of humans is what motivates this. Otherwise why bother octave stretching (to eliminate effects such as beating) if there's no human perception to motivate it, right? Thus, human hearing has everything to do with it. The whole point of building, tuning and playing pianos is to stimulate humans with the sounds they produce. It's not like we play pianos to do vacuum cleaning.
Why would an overtone be more noticeable than the fundamental? Seems odd that tuning the fundamentals to the "right" note is heard as out of tune because of the overtones, but tuning the overtones doesn't make the differing fundamentals sound out of tune.
I believe the point is that dissonances are more noticeable when they refer to notes that should be at the same exact frequence, like a fundamental frequence of a note, and the first harmonic of the lower octave.
I wonder if the same thing happens on guitar and bass. Sometimes an instrument will sound nicer if I tune it by ear rather than with a tuner.
With guitar it comes down to the frets. The guitar is technically flawed (although I love the sound of those flaws). Straight frets across the neck don't allow for perfect intonation. The string doesn't divide at exactly the right point to produce a perfect interval. So tuning via a tuner is making a compromise between all frets so it sounds as in tune as possible across the whole neck. But you can tune to a specific chord to get it "perfect" if you want, but then all the other chords will sound out.
th-cam.com/video/D8EjCTb88oA/w-d-xo.html Check this out
@@lanehillard1596 Yeah, I know that traditional frets don't allow for perfect intonation. But even still, tuning it by ear often produces better results to my ear than using a tuner. Haven't done an apples to apples comparison along the entire neck, but a big enough portion sounds good.
@@dougthemoleman That can also be because you have learned to play out of tune, not a bad thing as I agree that when tuning by ear the guitar will sound better. The catch is it only sounds better to me, to others it might sound okay, or others who play the guitar and have been accustomed to their own tuning it will sound worse.
or you can now buy a fretless guitar like a violin
I have a question: If the middle A has 220 Hz, and the A one octave above has 440 Hz, then all the harmonic overtones, including fundamental, of the latter, would also be the harmonic overtones of the former, so theoretically if on a perfect piano that only produces harmonic overtones and has no inharmonicity, if you strike both together, won't your ear only hear one pitch, which is the former, and not both?
So if that's true, what makes our ear able to hear both on an actual piano? Is it the existence of inharmonicity, or something else?
Just a guess, remember overtones have much less volume, so the two fundamentals would be louder? But inharmonicity probably also plays a part in discerning octaves.
Sometimes it indeed is hard to discern if an upper octave is played in Chopin's common upper technique. It just sounds like the piano all in the sudden sounds more bright.
Jazzz, I think it's a yes. That harmonic building is highly utilized in pipe organ playing. Once a key is pressed, several pipes speak together, interlocked in harmonic structure, to make a single loud, bright and rich sound.
This video th-cam.com/video/_DyvmwLyWWk/w-d-xo.html at 07.26 show how that magic works.
not all harmonic overtones of the former are harmonic overtones of the latter. 220hz + 220 equals 440, which + 220 equals 660. 660 is not part of the harmonic overtones of 440hz. go back to the missing fundamental video.
So, is being exposed to the real pianos (with stretched tuning) the culprit for this auditory tendencies?
The tendency being sharper we want to hear as pitch goes higher and flatter we want to hear as pitch goes lower.
Or does this tendency have nothing to do with our exposure to the pianos with stretched tuning? In which case, it would be the human brain tendency.
Let me know. I’m curious.
worst case when the customer thinks you are wrong but you are not , some bass lowest strings can only be tuned using intervals
Do you think a piano should be tuned differently if it will be played predominantely old school classical, vs modern or jazz?
There are definitely old school pieces that benefit from period appropriate tunings. You can find examples all over TH-cam!
It would be great to see/hear some actual, real world examples of this. A semitone too high at the 16th overtone? Is that an exaggeration? That kind of discrepancy makes your overtone theory of consonant and dissonant ratios more ideological than actual.
Bear in mind that very high harmonics tend also to be very quiet as a component of the whole note. I think you would hear a small discrepancy between the fundamental and a low, loud overtone much more readily than between the fundamental and a high, quiet overtone. Louder, lower harmonics will also tend to mask higher, quieter harmonics.
All adjusted systems of tuning aim at reducing inharmonicities to an acceptable level - they can't eliminate them entirely. Fretted string instruments are particularly bad for this. Fortunately, human hearing isn't perfect, either, particularly at the extremes.
As for the degree of out-of-tuneness: I'm not sure where the OP has sourced his information. All the charts of harmonics that I've seen go up to the sixteenth harmonic (fifteenth overtone), which is an octave of the fundamental and stated to be in tune.
On the other hand, the 11th harmonic (10th overtone) is given as 49 cents flat, relative to the 'true' pitch, which is nearly a quarter of a tone. The 13th harmonic (12th overtone) as almost as bad in the other direction, at 40 cents sharp of 'true'. (A cent is 1/100th of an equally tempered semitone.)
So there may be no exaggeration for the 16th overtone (17th harmonic), though I too find the idea of a full semitone (100 cents) sharp unlikely. You can see an example chart here: annelanzilotti.com/blog/2016/2/11/on-harmonics
Further to this, I've managed to find a chart that goes up to the 20th harmonic. This shows the the 17th harmonic (16th overtone) as only 5 cents sharp. andrewmeronek.com/music-tools/blue-notes-and-the-harmonic-series/
I think your example of harmonic charts is more something that's fundamental to the tuning system (e.g. an equal temperament fifth is not exactly a perfect fifth, though they're pretty close). And those will only go up to +/- 50 cents since things will always be "rounded" to the nearest note. What the video is talking about is differences that are due to the strings being "physical strings" rather than "ideal strings".
Anyway, for that part of the video I think he's still talking about the upright piano (given the graphic in the corner), for which the low notes is one of the places where the inharmonicity will be exaggerated the most. I imagine most other instruments will have a much smaller degree of inharmonicity, though I wouldn't be able to give you specific numbers on anything.
I'm inclined to agree with you - especially as it's well known that, as the OP says, short-stringed pianos suffer more from inharmonicity - but it perhaps needs to be made clearer in the video that 'inharmonic overtones' and 'harmonic overtones' are not the same thing, and that removing inharmonicities caused by the physical properties of real strings may still result in an instrument sounding out of tune because the harmonic overtones themselves are 'out' to varying degrees.
As for specific numbers - overtones are usually related to a fundamental, but when the OP states that the 16th (inharmonic) overtone is a semitone sharp, he doesn't say which fundamental pitch, or range of pitches, he is talking about. I assume he means some note in the lowest bass octave.
Grand pianos have single strings in the bass and 3 strings for each key in the treble, which are intentionally detuned a few cents with each other to add richness to the sound. Like uprights the octaves are also stretched, and more so as you go higher.
50th overtone = P5??? You can't say that because it's not true. Every piano has a different amount of inharmonicity and every string on every piano has a different inharmonicity. It's not a standard amount. So you can't say a certain partial is a specific amount sharp.
But what does this have to do with actually tuning a piano by beat rates? Inharmicty is not a word. Piano wires are actually not stiff. Steel stretches. You're working against the stretch of a piano wire when tuning not the stiffness of the wire.
It's not that strings are or aren't stiff. Stiffness is a quality of all tensioned strings. It's essentially a measure of how much resistance there is working against the string oscillating (that is, its horizontal/lateral motion) in a mathematically ideal way. The more massive the string, the more tension it is placed under, and/or the shorter its vibrating length, the more stiff it is. It's this stiffness that displaces overtones from their mathematically ideal position, i.e., inharmonicity. For nearly all string instruments, this string inharmonicity is so small that it can effectively be ignored. But because the modern piano has such stiff strings and such a wide range (8 octaves), it becomes noticeable.
I think some information is wrong in this, of course you can tune a piano. I'm questioning what you're saying.
i aggree with you . pianos can be tuned perfectly , this guy is 100% wrong, well almost wrong, beause of how the ear and brain works pitches that are very low or very high can sound slightly off to the ear/brain, therore the [only] very top notes on a piano are tuned slightly sharp and the [only] bass notes tuned slightly flat. and the ear/brain recognises this as perfect tuning. but of course the vast middle note sction of the piano is tuned to exact equell temperd scale. this guy needs to do some studying. main fault to his argument if a note is tuned to a pitch its errelervent how the string vibrates as it will vibrate at tuned pitch regardless of its overtones are off or whatever he is talking about which doesnt make any sense to me. whats your thoughts on this gareth. cheers.
But can you tune a fish?
Well, the information in this video in incomplete. If you learn just intonation and temperaments, you know what I mean.
@@wakerickman You missed something, the stretch is actually caused by the physical properties of the strings.
@@angelicamartacahyaningtyas9083 hhmm the point i was making was that the video was seeming to infer that all the strings where not all tuned to concert pitch and theofore every single noe slightly out of tune with say a synthersize kerboard where all its notes where tuned correctly to the equel tempared scale, the video infered that a piano is not tuned to the equal tempered scale and therfore not be in tune with all standerd synthersizer keyboards which have no stretching even if the synth was 88 notes for synthersizer sounds even on that 88 note keyboard would still all be tuned to the equal tempered scacle. its ownly a comman thing for piano perticuarly orchestrial piano that a stretch tunining would be used ie the very low notes slightly flat and the very high notes are tuned sharp. this is my understanding of how a piano is tuned by convention . and not sure what you mean by the stretch is caused by the physical propertise of the strings as seems by the way as i belive the strings are just tuned normaly and the physical properties of the strings are just automaticly taken into account as the strings are tuned. often with a digital chromatic tuner nowadays. cheers
"You cannot tune a piano" Huh? I think you need to clarify that. Doesn't make sense. I tune pianos all the time. It's my profession. 😂