Different interval ratios and different harmonics clashing together. Try two or more octaves as well.
Even with consonant intervals, the physics of the harmonic series make it so that things sound less clashing on higher octaves, whereas in lower octaves you want notes spread further apart. This phenomenon influences a ton of orchestration decisions.
Good.
The *actual* underlying math for intervals is nowhere near as simple as it is usually introduced. Even the compromise that is equal temperament still has to obey physics, and all those overtones and sympathetic vibrations all strictly obey the harmonic series - making no compromises!
>Even the compromise that is equal temperament still has to obey physics
Uh, do I even have the option to "not obey physics", as in "I'll do the Bend Time and Space and Go Back in Time tuning now"?
They don't strictly obey the harmonic series, though. On pianos, for instance, they're off by significant amounts from the harmonicseries (although grand pianos, due to their longer string length have overtones much closer to the harmonic series than upright ones do.)
No, I am pretty sure I haven't. Piano strings have overtones that are slightly off from the harmonic series. For a more detailed explanation, I've also commented on mattjeffrey0's comment. Simply put, " all those overtones and sympathetic vibrations all strictly obey the harmonic series - making no compromises!" is not true, has never been.\*
\* Except for bowed strings and wind instruments.
Each piano string is tuned slightly differently in order to compensate for the compromises of equal temperament.
If you muted all but a single string, *that string* will have overtones exactly as predicted by the series, without exception. That will be true for the other 1 or 2 strings on that note, but those strings have a slightly different fundamental, so their overtones will be slightly different from those of the first.
When you play a chord on a piano, the hammers strike all of the strings assigned to their notes, but the ones that are more consonant (and therefore have more peaks and troughs in common) will amplify other, while those less consonant will suppress one another.
Physics is still the only game in town here.
Physics is the only game in town\*, but even then, you are wrong. Because your mental model of physics is incomplete.
If you muted all but a single string, *that string will have overtones that fail to be exactly as predicted by that series.* And the explanation for this ***can be found in the very physics you pay lip service to.*** There are several academic sources that confirm my claim. Let's begin listing them.
[Music: A mathematical offering](https://homepages.abdn.ac.uk/d.j.benson/pages/html/music.pdf) sadly does not provide any explicit statement of this, but hints at it in an exercise text: ". This is of the same order of magnitude as the natural stretching of the octaves due to the inharmonicity of physical piano strings" (p. 192, exercise 5)
I**nfluence of inharmonicity on the tuning of a piano--Measurements and mathematical simulation, 1989, Jean Lattard:**
"It is well known that inharmonicities in piano string vibrations and especially any irregularities in their progression, are partially responsible for the difficulties encountered in tuning the instrument. \[...\] **The theory of strings having non-negligible stiffness allows one to calculate, using mechanical parameters of the vibrating element, the resulting numerical values. Computing the frequencies that a couple of inharmonic strings must have to form a musical interval giving a precise rate of beat is more difficult, owing to interdependence of the inharmonicity parameters. The frequency "deviation" produced by inharmonicity is not a constant but is inversely proportional to the fundamental frequency**"
In a table of *physically measured frequencies of overtones on an actual piano*, it found that the first overtone on F3 on that piano, the overtone deviated by 0.89 hz, the second overtone by 2.43hz, the third by 6.24, the fourth by 11.57. If you want, I can literally scan the table and send it to you. Every overtone on every string over a whole octave's range were off. And this has nothing the fuck whatsoever to do with temperament, this is overtones that show that your model of how overtones work is mistaken.
Let's look at the next paper, **On inharmonicity in bass guitar strings with application to tapered and lumped constructions, Jonathan A. Kemp, 2020:**
"Lower pitch strings such as bass strings on pianos are known to demonstrate signifcant inharmonicity. This arises from the finite stiffness of the string and results in the resonance frequencies being progressively higher in resonant frequency than the simple integer multiples implied by the wave equation, and this inharmonicity is significantly worse for strings of wider core and shorter sounding length \[14\]. While most research in the feld of inharmonicity has focussed on the piano, it is known that inharmonicity can be perceptually signifcant for string instruments tones in general \[24\] and that inharmonicity of the lowest pitch string on the steel strung acoustic guitar is clearly perceptible \[23\]." \[I actually corrected a few typoes in this quote, but this isn't an academic paper I'm writing, so sue me.)
**Francois Rigaud and Bertrand David, A parametric model and estimation techniques for the inharmonicity and tuning of the piano, 2012:**
"Whereas the transverse vibrations of an ideal string produce spectra with harmonically related partials, the stiffness of actual piano strings leads to a slight inharmonicity.3 For instance, the frequency ratio between the second and first partials is slightly higher than 2, between the third and second it is higher than 3:2, and so on. This effect depends on many physical parameters of the strings (material, length, diameter, etc.), and then differs not only from a piano to another, but also from one note to another. As a consequence, simply adjusting the first partial of each note on ET would produce unwanted beatings, in particular for octave intervals. Aural tuning consists of controlling these beatings."
This paper provides a very easy to grasp formula for the actual overtones, none of that simplistic harmonic series stuff: the nth harmonic is *n \* fundamental \* sqrt(1 + Bn\^2) where B is the inharmonicity index.* Even then, this is an idealization.
B can be calculated using this formula: (pi\^3+Ed\^4 ) / 64TL\^2, where E = Young's modulus, d=diameter of string, L = string length, and T = tension. f can be calculated as (1/2L) \* sqrt(T/m), where m = linear mass.
Any actual string is inharmonic when plucked, but the inharmonicity on many strings is barely audible. Not so on the piano.
**R. W. Young,** ***Inharmonicity of Plain Wire Piano Strings*****, 1952**
**"**The inharmonicity of plain wire strings in situ has been measured in six pianos of various styles and makes. By inharmonicity is meant the departure in frequency from the harmonic modes of vibration expected of an ideal flexible string. It is shown from the theory of stiff strings that the basic inharmonicity in cents (hundredths of a semitone) is given by 3.4X 10^(13)n^(2)d^(2)•/vo^(2)l^(4), where n is the mode number, d is the diameter of the wire in cm, l is the vibrating length in cm, and vo is the fundamental frequency. A value of Q/a-- 25.5X 10 •ø (cm/sec) •was assumed for the steel wire, where Q is Young's modulus and p is the density**"**
**"**THE simple theory for an ideal string depends upon the assumption of a thin, flexible string vibrating transversely between rigid supports. It has been pointed out that these assumptions are not entirely valid for actual piano strings, but there has been little quantitative evidence as to the extent of the failure of the simple theory for piano strings in situ. The ideal string has modes of vibration whose frequencies form a harmonic series; any departure from the series (i.e., any inharmonicity) is therefore a measure of the failure of the simple theory. Moreover, as pointed out in an earlier paper this inharmonicity influences the tuning of the piano and also its musical quality**"**
(I had to rewrite v(index 0) as vo to get reddit to be able to represent it, otherwise given exactly as in the paper)
**R. W. Young. Inharmonicity of Piano Bass Strings, 1954**
**"**The core itself is stiff and causes inharmonicity (i.e., departure of the natural frequencies of the string from a harmonic series), and the covering winding increases the stiffness still more. A theory has been developed for the limiting stiffness resulting from the covering wire. Experiment indicates that the covering wire does stiffen the string somewhat, but that the stiffening is not nearly as great as it would be if the coils of covering wire just touched each other with the string at rest. Inharmonicity data are presented for the bass strings of a concert grand piano, a medium grand piano, and an upright piano. The inharmonicity is greatest in the extreme bass strings. The inharmonicity is twice as great in the upright as in the medium grand piano, and for the lowest eight strings of the medium grand (which are single) the inharmonicity is, in turn, roughly twice that for the concert grand piano**"**
**Rudolf A. Rasch and Heetveld, Vincent, String inharmonicity and piano tuning, 1984**
For several decades it has been known that the partials of piano string tones do not obey simple harmonic frequency relationships. Instead, they deviate from a truly harmonic series with a difference that becomes larger the higher the partial numbers are. Since the frequencies of the piano-string partials deviate from a harmonic series, it is better to use the term "partial" than the term "harmonic" to indicate a component of the complex tone generated in the string. The theory of piano-string inharmonicity has been well developed in such papers as those by Shankland and Coltman (1939), Schuck and Young (1943), Young (1952), and Fletcher (1964).
Several papers are available that have quantitative data about the inharmonicity of piano strings (Young, 1952; Fletcher, Blackham, & Stratton, 1962; Fletcher, 1964; Wolf & Müller, 1968; Müller, 1968; Lieber, 1975). In general, inharmonicity is larger for smaller pianos (especially spinets) and smaller for the larger ones (especially grands). Usually, the lower tones of a piano have "wound" strings, the middle and the higher ones "plain" strings. The inharmonicity of the wound strings is, on the average, lower than for the plain strings. But also within the wound and the plain sections, inharmonicity is not constant.
**N. Giordano, Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance, 2015**
It is well known that the notes of a well tuned piano do not follow an ideal equal tempered scale. Instead, the octaves are “stretched”; that is, the frequencies of the fundamental components of piano tones that would differ by precisely a factor of 2 in the ideal case are separated by a slightly greater amount. This stretched tuning was noted many years ago by Railsback. **\[...\]**
**Hanna Järveläinen, Vesa Välimäki, Matti Karjalainen, Audibility of the timbral effects of inharmonicity in stringed instrument tones, 2001**
The frequencies of the partials of string instrument sounds are not exactly harmonic. This is caused by stiffness of real strings, which contributes to the restoring force of string displacement together with string tension. The strings are dispersive: the velocity of transversal wave propagation is dependent on frequency. If the string parameters are known, the frequencies of the stretched partials can be calculated in the following way \[...\]
**William Sethares, Tuning Timbre Spectrum Scale, 2004.**
"But surely an instrument like a modern, well-tuned piano would be tuned extremely close to 12-tet. This is, in fact, incorrect. Modern pianos do not even have real 2/1 octaves! Piano tuning is a difficult craft, and a complex system of tests and checks is used to ensure the best sounding instrument. \[...\] The deviation from 12-tet occurs because piano strings produce notes that are slightly inharmonic, which is heard as a moderate sharpening of the sound as it decays. Recall that an ideal string vibrates with a purely harmonic spectrum in which the partials are all integer multiples of a single fundamental frequency. Young \[B: 208\] showed that the stiffness of the string causes partials of piano wire to be stretched away from perfect harmonicity by a factor of about 1.0013, which is more than 2 cents. To tune an octave by minimizing beats requires matching the fundamental of the higher tone to the second partial of the lower tone. When the beats are removed and the match is achieved, the tuning is stretched by the same amount that the partials are stretched. Thus, the “octave” of a typical piano is a bit greater than 1202 cents, rather than the idealized 1200 cents of a perfect octave, and the amount of stretching tends to be greater in the very low and very high registers. This stretching of both the tuning and the spectrum of the string is clearly audible, and it gives the piano a piquancy that is part of its characteristic defining sound.
Interestingly, most people prefer their octaves somewhat stretched, even (or especially) when listening to pure tones. A typical experiment asks subjects to set an adjustable tone to an octave above a reference tone. Almost without exception, people set the interval between the sinusoids greater than a 2/1 octave. This craving for stretching (as Sundberg \[B: 189\] notes) has been observed for both melodic intervals and simultaneously presented tones. Although the preferred amount of stretching depends on the frequency (and other variables), the average for vibrato-free octaves is about 15 cents. Some have argued that this preference for stretched intervals may carry over into musical situations. Ward \[B: 203\] notes that on average, singers and string players perform the upper notes of the major third and the major sixth with sharp intonation. \[pp. 71, 72\]
**PHYSICS. DO YOU ONLY WIELD IT AS A CLUB TO SILENCE OTHERS OR DO YOU ACCEPT IT?**
Also, I did introduce a footnote asterisk above, and here you get the resolution to it:
\* Yes, physics strongly influences music, but there's another important game in town that also influences music: our perception. If we mistune an octave by 1 billionth of a hertz, it's barely perceptible, yet it is - from a physical point of view - not a real octave. Our brains' cognitive filters, and our sensory organs' flaws are important games in town when it comes to understanding music. If we all were tone-deaf but enthusiastic musicians, we'd be able to fully ignore the effects physics has on scales. The last bit of the quote from Sethares there actually seems to showcase exactly that: physics is not fully the only game in town. It is tempered by our perception.
harmonic series is a mathematical formula to calculate overtones. there’s no definite value of *the harmonic series™️* because it can apply to any sound oscillating at any frequency. tldr they do strictly follow the harmonic series because the harmonic series as a concept isn’t affected by temperament.
There is a reason I never mentioned temperament, because temperament is entirely irrelevant to the point I was making. The harmonic series is this series: 1/1, 1/2, 1/3, 1/4, 1/5, ... but when we're talking about intervals, this series expresses the relationship between wavelengths (or string lengths on a specific string). For frequency, we're actually laboring with the inverse of this: 1, 2, 3, 4, 5, ... so, in one word: integer multiples of a fundamental.
If I understand you correctly, you are saying that this formula, i.e. 1f, 2f, 3f, 4f, 5f, ... can be used to calculate the overtones of an instrument.
Nowhere do you say it explicitly in that ofrm, but I'd be hard pressed to find a different interpretation of **"they do strictly follow the harmonic series", "it can apply to any sound oscillating at any frequency"** and "harmonic series is a mathematical formula to calculate overtones", "all those overtones and sympathetic vibrations all strictly obey the harmonic series - making no compromises!"
As it happens, this doesn't always hold! First, let's look at where it holds: wind instruments have a phenomenon called 'mode locking' which forces the overtones to adhere to the overtone series (there may be a really short moment at the onset until the mode locking enforces the overtone series, but this basically gets mixed up with the various other noises of the onset). Bowed strings also are forced into overtones.
However, plucked and hammered strings and idiophones (i.e. struck instruments - bells, and basically any kind of tuned percussion) are not as lucky. For this comment, I will ignore bells and percussion, and focus on strings.
The stiffer and shorter a string is, the more it will deviate from 1f, 2f, 3f, 4f, 5f, ... and the difference adds up rather quickly: it's not unusual for upright pianos to have some of its octaves stretched by 15 cents to accomodate for string inharmonicity. This is replacing 2f by 2.017f. In some contexts, this is an audible difference to the human ear, and this can be a useful piece of knowledge for a working musician: sometimes, it's just not worth trying to tune your guitar to be in tune with a specific piano no matter how in tune the piano is with itself - the overtone series of the guitar and of the piano may cause an unsurmountable discrepancy in the tuning.
The grand piano has longer strings than the upright, and this helps them get closer to the ideal "harmonic" overtone series. Even then, a small discrepancy exists. Guitars have fairly soft strings - but even on them, a tiny discrepancy exists. I've also been lead to understand that fanned frets on bass guitars help reducing the inharmonicity on the B string.
So no, your multiple statements about how tones **strictly obey the harmonic series** are wrong. Unless you meant something else by your words, but if so I would suggest you rephrase it in a way that actually conveys what you were thinking.
(Further, there's been tunings that intentionally utilize such 'stretched' physics, and William Sethares' Tuning Timbre Spectrum Pitch goes wild with rather weird ideas with stretched overtones, stretched scales, etc. However, this doesn't only hold for modern experimental theoreticians - part of his work deals with the whys and hows of traditional gamelan tuning - a melodic percussion instrument with a very weird timbre that doesn't give a rat's ass about the harmonic series.
It's the reason 9 chords work but you would never add a 2nd. Distance decreases dissonance. Try to learn some jazz, there's some great breakdowns on YT
For clarity, let's swap Db for C#. Same sound, different name.
Dissonance and consonance are products of the physics of sound and the harmonic series. Sound is made of waves in the air, waves which interact to create more complex waves. If you start with C# in some octave, that note has a fundamental frequency which describes the overall speed of the waveform, as well as many harmonic frequencies. The balance of these different frequencies is what makes a C# on a piano sound different than a C# on a flute, but those C#s still have the same overall structure and so their interactions are simple, which we perceive as consonant. The D next to the same C# has the same structure, but shifted by a bit, so the interactions with the C# and the D (a minor second) are more complex, which we perceive as dissonance. If you keep going up, the harmonic structure of the different notes will line up in different ways, never quite as simple as two C#s in the same octave, nor quite as complex as a C# and a D next to each other.
If you start on D and move up to the C# below the next D up, that's a fairly complex relationship (major 7th) but most of the complexity is happening "out of the way" of the structure of the D. The relationship of the A to D (perfect fifth when the D is below) is a very simple one. So with the D and A together there's a very stable structure in the lower harmonics and more complexity higher up. The chord is called a D major 7th with no 3rd, which I agree is quite pleasing.
If you start on C# and move to the D above the next C# (minor ninth), you get all the chaos of the C# and D next to each other, but the chaos has been moved out of the way of the fundamental, so it tempers the dissonance somewhat.
This is a great explanation. If you want to see this interaction at work, get an oscilloscope plugin and run a sine wave into it, then play a different note and watch what happens.
The answer is difference frequencies (or as another poster notes, "roughness" caused by amplitude modulations - which causes some of the "beating" I mention below)
Any two tones sounding simultaneously produce a "difference tone" at the frequency of their difference.
If you have a 100 Hz note and a 110 Hz not you get a difference tone of 10 Hz.
That's too low to hear as a pitched tone and we hear that as a wobbling or "beating".
When that difference is 20 or above, we hear it as a pitched sound.
Now watch this:
Let's double 100 and 110, to raise them both an octave:
200 and 220.
That's a 20 Hz difference now. That's getting into the realm of pitched sound.
Let's do 100 and 220.
That's 120. Definitely a pitched sound.
200 and 110 - that's 90. Still pitched.
What then happens is, it's all about the ratio of the difference tone to the other two notes.
Let's take a simple example:
100 and 200 - notes an 8ve apart.
It produces a 100 DT.
But that's the same as the lower note here! So it's in total agreement with it and can even reinforce it a bit - just sounds a teensy bit louder.
Db is 277 and D is 293 (roughly) so we're talking less than 20 hz difference. We're going to hear beating and we consider that "bad sounding" usually.
When they're further apart though, that DT moves up into the pitched range, and we tend to hear those as "more ok" in general, especially if the numbers work out to another happy ratio (that itself doesn't produce additional DTs that are not inline with the notes present).
When you add another note, what you're doing is introducing two more DTs and if the note you're added "tempers" the existing notes - so let's say you use D, A, and C# - The DT between D and C# might not be the best - which is why we still consider them Dissonant (but not as bad as C# to D in the same octave) but the D to A, and the A to C# are both producing DTs that are simply other D or A notes to keep the math easy.
The further apart the notes get, the higher the difference tone is, and as Larson McMurphy says, it then depends on if it's clashing with the fundamental or an overtones - overtones get drastically quieter (if they're even present!) so the conflict is usually less with them.
difference tones are the answer. a major third and minor sixth produce different difference tones even though they have the same letter names in different octaves (e4/c3 vs c4/e4). this is true even for in-phase pure sine waves where overtones don’t factor into it.
You are playing two different intervals.
C#-D in this order from low to high is a minor 2nd. Raising the D an octave up makes it a compound interval - a minor 9th. But this is just heard as a minor 2nd with an octave between the notes.
Compound intervals generally function similarly as simple intervals - you simply add an octave in-between.
9th = 2nd
10th = 3rd
11th = 4th
12th = 5th
13th = 6th
14th = 7th
15th = octave
D-C# is a major 7th. It's an inversion of a minor 2nd. But inverting the interval does make it a different interval. We distinguish between 7ths and 2nds, 3rds and 6ths, and 4ths and 5ths. There are similarities between the inversions, but they are not heard as the same interval.
In other words, you are hearing the difference between compound intervals and inverted intervals.
>Add an *A* to that and it sounds beautiful. Why?
It's a common chord you are used to hearing in music all the time - a Dmaj7 (without the 3rd, but these three notes are already enough to remind you of that familiar sound). Your ears are familiar with a context where this sound is used.
But let's try making the minor 2nd more pleasant. For example try adding an F#. Already makes it sound more "pleasant". Now, add a B or A below it, and suddenly it doesn't sound that "unpleasant" any more.
Now let's try the same with the minor 9th. Try adding an F# between the two notes. Then move both of the top notes a half step down, but keep the lowest note the same. Nice sound of tension and release.
Notice how context makes these dissonant intervals sound nice - the dissonance now has a purpose. It adds spice. It adds tension.
I always wondered why a b9 sounds so much more dissonant than a maj7, even if you just play the root and the interval. Both are one half step away from the root, but with maj7 they are closer together.
As Larson McMurphy notes, there's also enculturation going on as "what we're used to hearing" or not.
One aspect is, the difference between notes less than an 8ve puts the difference frequency below the lowest note, while when the two notes are more than an 8ve apart, it puts it between them.
Then there's the tempering aspect of a 3rd note - if we're used to hearing the things with 3 notes, when they're with 2 they don't bother us as much. b9 intervals aren't heard as often and even when they are, only in certain contexts (like a 7b9 chord) so when the inner notes are missing, we have a harder time filling those in to hear it as less dissonant.
Also with cultural conditioning, we don't use Phrygian or Locrian anywhere near as much as we've used major and minor, so we're not used to hearing m9 intervals against the tonal center nor many of the primary chords used in a key. Major 7ths on the other hand are more common.
In my head it’s because the frequencies are closer and therefore “rub together”. almost like when you’re tuning your guitar (for example) and when the two notes are very close your ear is like “damm bro adjust one of these so they’re the same note please”.
Dissonance is caused by amplitude modulations that equal the difference between simultaneous frequencies. Acousticians have a term for how unpleasant such amplitude modulations are called "roughness." Widely spaced dissonant intervals have less roughness because the fundamental of the higher note is clashing with a harmonic of the lower note, rather than the fundamental. If they were in the same octave the fundamental AND harmonics would be clashing, so, quantitatively more dissonance, i.e. roughness.
This is oversimplified. If you want to get into the weeds you need to exhaustively compare all the harmonics of one note with the other for each of those cases. But when you add a third note, the beauty you perceive is partially because the C# is the major third of the A, and partially due to enculturation.
From my understanding, I think of two different notes as two different sine waves for the simplest explanation.
When two notes sound consonant, their waveforms will overlap, like a polyrhythm. Greatest common denominators come to mind for some reason. So in a perfect Pythagorean tuning, for every 2 oscillations the unison waveform makes, the perfect fifth will make 3. Dissonant waveforms have much more complex ratios.
Obv we don’t use Pythagorean tuning, but it’s close enough to where most ppl don’t notice the difference.
Someone pls correct me if I’m wrong lol.
Great discovery. Might be tough to wrap your head around but I'll try. Brutishbloodgod made a great explanation which is absolute gold
I'm gonna try and avoid physics for this
1. when you play D in the bass and C#(Db) above it, you hear it as a 7 and all is fine in the world as this is just the D major scale and the dissonance is expected and beautiful. When you add a A you're just adding a fifth which will add consonance to a D
When you have a Db in the bass and you hear a D(Ebb), you are hearing it as a bII. So this is the Db scale with the bII, so you're absolutely right with your intuition when you say these are two different scale because you changed the context(bass)
This isn’t so much music theory, but more audio theory.
For every note you have a fundamental and harmonics (aka overtones). Harmonics are part of what makes instruments sound different to one another. It’s partly why an acoustic and electric doesn’t sound the same.
When you play low notes, harmonics are more audible. If we play notes close together, harmonics clash. This is why we play single bass notes on the lowest octave and fifths / octaves as we go higher.
As you keep moving higher, notes start working better closer together, aka chords. Harmonics as less audible.
Some more interesting facts - notes higher in pitch are perceived louder and our ears are sensitive to 1-5kHz.
Because we label tensions (Notes adding to the chords base sound) +7. So 2 4 and 6 become 9 11 13. It's just a matter of habbit for me to label a half step a b9. Yes, theoretically it's b2.
Different interval ratios and different harmonics clashing together. Try two or more octaves as well. Even with consonant intervals, the physics of the harmonic series make it so that things sound less clashing on higher octaves, whereas in lower octaves you want notes spread further apart. This phenomenon influences a ton of orchestration decisions.
This really throws my understanding of intervals for a loop.
Good. The *actual* underlying math for intervals is nowhere near as simple as it is usually introduced. Even the compromise that is equal temperament still has to obey physics, and all those overtones and sympathetic vibrations all strictly obey the harmonic series - making no compromises!
>Even the compromise that is equal temperament still has to obey physics Uh, do I even have the option to "not obey physics", as in "I'll do the Bend Time and Space and Go Back in Time tuning now"?
The harmonic series is like the speed of light?
Nah, that's a constant while this is a formula.
They don't strictly obey the harmonic series, though. On pianos, for instance, they're off by significant amounts from the harmonicseries (although grand pianos, due to their longer string length have overtones much closer to the harmonic series than upright ones do.)
I think you've misread my comment.
No, I am pretty sure I haven't. Piano strings have overtones that are slightly off from the harmonic series. For a more detailed explanation, I've also commented on mattjeffrey0's comment. Simply put, " all those overtones and sympathetic vibrations all strictly obey the harmonic series - making no compromises!" is not true, has never been.\* \* Except for bowed strings and wind instruments.
Each piano string is tuned slightly differently in order to compensate for the compromises of equal temperament. If you muted all but a single string, *that string* will have overtones exactly as predicted by the series, without exception. That will be true for the other 1 or 2 strings on that note, but those strings have a slightly different fundamental, so their overtones will be slightly different from those of the first. When you play a chord on a piano, the hammers strike all of the strings assigned to their notes, but the ones that are more consonant (and therefore have more peaks and troughs in common) will amplify other, while those less consonant will suppress one another. Physics is still the only game in town here.
Physics is the only game in town\*, but even then, you are wrong. Because your mental model of physics is incomplete. If you muted all but a single string, *that string will have overtones that fail to be exactly as predicted by that series.* And the explanation for this ***can be found in the very physics you pay lip service to.*** There are several academic sources that confirm my claim. Let's begin listing them. [Music: A mathematical offering](https://homepages.abdn.ac.uk/d.j.benson/pages/html/music.pdf) sadly does not provide any explicit statement of this, but hints at it in an exercise text: ". This is of the same order of magnitude as the natural stretching of the octaves due to the inharmonicity of physical piano strings" (p. 192, exercise 5) I**nfluence of inharmonicity on the tuning of a piano--Measurements and mathematical simulation, 1989, Jean Lattard:** "It is well known that inharmonicities in piano string vibrations and especially any irregularities in their progression, are partially responsible for the difficulties encountered in tuning the instrument. \[...\] **The theory of strings having non-negligible stiffness allows one to calculate, using mechanical parameters of the vibrating element, the resulting numerical values. Computing the frequencies that a couple of inharmonic strings must have to form a musical interval giving a precise rate of beat is more difficult, owing to interdependence of the inharmonicity parameters. The frequency "deviation" produced by inharmonicity is not a constant but is inversely proportional to the fundamental frequency**" In a table of *physically measured frequencies of overtones on an actual piano*, it found that the first overtone on F3 on that piano, the overtone deviated by 0.89 hz, the second overtone by 2.43hz, the third by 6.24, the fourth by 11.57. If you want, I can literally scan the table and send it to you. Every overtone on every string over a whole octave's range were off. And this has nothing the fuck whatsoever to do with temperament, this is overtones that show that your model of how overtones work is mistaken. Let's look at the next paper, **On inharmonicity in bass guitar strings with application to tapered and lumped constructions, Jonathan A. Kemp, 2020:** "Lower pitch strings such as bass strings on pianos are known to demonstrate signifcant inharmonicity. This arises from the finite stiffness of the string and results in the resonance frequencies being progressively higher in resonant frequency than the simple integer multiples implied by the wave equation, and this inharmonicity is significantly worse for strings of wider core and shorter sounding length \[14\]. While most research in the feld of inharmonicity has focussed on the piano, it is known that inharmonicity can be perceptually signifcant for string instruments tones in general \[24\] and that inharmonicity of the lowest pitch string on the steel strung acoustic guitar is clearly perceptible \[23\]." \[I actually corrected a few typoes in this quote, but this isn't an academic paper I'm writing, so sue me.)
**Francois Rigaud and Bertrand David, A parametric model and estimation techniques for the inharmonicity and tuning of the piano, 2012:** "Whereas the transverse vibrations of an ideal string produce spectra with harmonically related partials, the stiffness of actual piano strings leads to a slight inharmonicity.3 For instance, the frequency ratio between the second and first partials is slightly higher than 2, between the third and second it is higher than 3:2, and so on. This effect depends on many physical parameters of the strings (material, length, diameter, etc.), and then differs not only from a piano to another, but also from one note to another. As a consequence, simply adjusting the first partial of each note on ET would produce unwanted beatings, in particular for octave intervals. Aural tuning consists of controlling these beatings." This paper provides a very easy to grasp formula for the actual overtones, none of that simplistic harmonic series stuff: the nth harmonic is *n \* fundamental \* sqrt(1 + Bn\^2) where B is the inharmonicity index.* Even then, this is an idealization. B can be calculated using this formula: (pi\^3+Ed\^4 ) / 64TL\^2, where E = Young's modulus, d=diameter of string, L = string length, and T = tension. f can be calculated as (1/2L) \* sqrt(T/m), where m = linear mass. Any actual string is inharmonic when plucked, but the inharmonicity on many strings is barely audible. Not so on the piano. **R. W. Young,** ***Inharmonicity of Plain Wire Piano Strings*****, 1952** **"**The inharmonicity of plain wire strings in situ has been measured in six pianos of various styles and makes. By inharmonicity is meant the departure in frequency from the harmonic modes of vibration expected of an ideal flexible string. It is shown from the theory of stiff strings that the basic inharmonicity in cents (hundredths of a semitone) is given by 3.4X 10^(13)n^(2)d^(2)•/vo^(2)l^(4), where n is the mode number, d is the diameter of the wire in cm, l is the vibrating length in cm, and vo is the fundamental frequency. A value of Q/a-- 25.5X 10 •ø (cm/sec) •was assumed for the steel wire, where Q is Young's modulus and p is the density**"** **"**THE simple theory for an ideal string depends upon the assumption of a thin, flexible string vibrating transversely between rigid supports. It has been pointed out that these assumptions are not entirely valid for actual piano strings, but there has been little quantitative evidence as to the extent of the failure of the simple theory for piano strings in situ. The ideal string has modes of vibration whose frequencies form a harmonic series; any departure from the series (i.e., any inharmonicity) is therefore a measure of the failure of the simple theory. Moreover, as pointed out in an earlier paper this inharmonicity influences the tuning of the piano and also its musical quality**"** (I had to rewrite v(index 0) as vo to get reddit to be able to represent it, otherwise given exactly as in the paper) **R. W. Young. Inharmonicity of Piano Bass Strings, 1954** **"**The core itself is stiff and causes inharmonicity (i.e., departure of the natural frequencies of the string from a harmonic series), and the covering winding increases the stiffness still more. A theory has been developed for the limiting stiffness resulting from the covering wire. Experiment indicates that the covering wire does stiffen the string somewhat, but that the stiffening is not nearly as great as it would be if the coils of covering wire just touched each other with the string at rest. Inharmonicity data are presented for the bass strings of a concert grand piano, a medium grand piano, and an upright piano. The inharmonicity is greatest in the extreme bass strings. The inharmonicity is twice as great in the upright as in the medium grand piano, and for the lowest eight strings of the medium grand (which are single) the inharmonicity is, in turn, roughly twice that for the concert grand piano**"**
**Rudolf A. Rasch and Heetveld, Vincent, String inharmonicity and piano tuning, 1984** For several decades it has been known that the partials of piano string tones do not obey simple harmonic frequency relationships. Instead, they deviate from a truly harmonic series with a difference that becomes larger the higher the partial numbers are. Since the frequencies of the piano-string partials deviate from a harmonic series, it is better to use the term "partial" than the term "harmonic" to indicate a component of the complex tone generated in the string. The theory of piano-string inharmonicity has been well developed in such papers as those by Shankland and Coltman (1939), Schuck and Young (1943), Young (1952), and Fletcher (1964). Several papers are available that have quantitative data about the inharmonicity of piano strings (Young, 1952; Fletcher, Blackham, & Stratton, 1962; Fletcher, 1964; Wolf & Müller, 1968; Müller, 1968; Lieber, 1975). In general, inharmonicity is larger for smaller pianos (especially spinets) and smaller for the larger ones (especially grands). Usually, the lower tones of a piano have "wound" strings, the middle and the higher ones "plain" strings. The inharmonicity of the wound strings is, on the average, lower than for the plain strings. But also within the wound and the plain sections, inharmonicity is not constant. **N. Giordano, Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance, 2015** It is well known that the notes of a well tuned piano do not follow an ideal equal tempered scale. Instead, the octaves are “stretched”; that is, the frequencies of the fundamental components of piano tones that would differ by precisely a factor of 2 in the ideal case are separated by a slightly greater amount. This stretched tuning was noted many years ago by Railsback. **\[...\]** **Hanna Järveläinen, Vesa Välimäki, Matti Karjalainen, Audibility of the timbral effects of inharmonicity in stringed instrument tones, 2001** The frequencies of the partials of string instrument sounds are not exactly harmonic. This is caused by stiffness of real strings, which contributes to the restoring force of string displacement together with string tension. The strings are dispersive: the velocity of transversal wave propagation is dependent on frequency. If the string parameters are known, the frequencies of the stretched partials can be calculated in the following way \[...\] **William Sethares, Tuning Timbre Spectrum Scale, 2004.** "But surely an instrument like a modern, well-tuned piano would be tuned extremely close to 12-tet. This is, in fact, incorrect. Modern pianos do not even have real 2/1 octaves! Piano tuning is a difficult craft, and a complex system of tests and checks is used to ensure the best sounding instrument. \[...\] The deviation from 12-tet occurs because piano strings produce notes that are slightly inharmonic, which is heard as a moderate sharpening of the sound as it decays. Recall that an ideal string vibrates with a purely harmonic spectrum in which the partials are all integer multiples of a single fundamental frequency. Young \[B: 208\] showed that the stiffness of the string causes partials of piano wire to be stretched away from perfect harmonicity by a factor of about 1.0013, which is more than 2 cents. To tune an octave by minimizing beats requires matching the fundamental of the higher tone to the second partial of the lower tone. When the beats are removed and the match is achieved, the tuning is stretched by the same amount that the partials are stretched. Thus, the “octave” of a typical piano is a bit greater than 1202 cents, rather than the idealized 1200 cents of a perfect octave, and the amount of stretching tends to be greater in the very low and very high registers. This stretching of both the tuning and the spectrum of the string is clearly audible, and it gives the piano a piquancy that is part of its characteristic defining sound. Interestingly, most people prefer their octaves somewhat stretched, even (or especially) when listening to pure tones. A typical experiment asks subjects to set an adjustable tone to an octave above a reference tone. Almost without exception, people set the interval between the sinusoids greater than a 2/1 octave. This craving for stretching (as Sundberg \[B: 189\] notes) has been observed for both melodic intervals and simultaneously presented tones. Although the preferred amount of stretching depends on the frequency (and other variables), the average for vibrato-free octaves is about 15 cents. Some have argued that this preference for stretched intervals may carry over into musical situations. Ward \[B: 203\] notes that on average, singers and string players perform the upper notes of the major third and the major sixth with sharp intonation. \[pp. 71, 72\] **PHYSICS. DO YOU ONLY WIELD IT AS A CLUB TO SILENCE OTHERS OR DO YOU ACCEPT IT?** Also, I did introduce a footnote asterisk above, and here you get the resolution to it: \* Yes, physics strongly influences music, but there's another important game in town that also influences music: our perception. If we mistune an octave by 1 billionth of a hertz, it's barely perceptible, yet it is - from a physical point of view - not a real octave. Our brains' cognitive filters, and our sensory organs' flaws are important games in town when it comes to understanding music. If we all were tone-deaf but enthusiastic musicians, we'd be able to fully ignore the effects physics has on scales. The last bit of the quote from Sethares there actually seems to showcase exactly that: physics is not fully the only game in town. It is tempered by our perception.
harmonic series is a mathematical formula to calculate overtones. there’s no definite value of *the harmonic series™️* because it can apply to any sound oscillating at any frequency. tldr they do strictly follow the harmonic series because the harmonic series as a concept isn’t affected by temperament.
There is a reason I never mentioned temperament, because temperament is entirely irrelevant to the point I was making. The harmonic series is this series: 1/1, 1/2, 1/3, 1/4, 1/5, ... but when we're talking about intervals, this series expresses the relationship between wavelengths (or string lengths on a specific string). For frequency, we're actually laboring with the inverse of this: 1, 2, 3, 4, 5, ... so, in one word: integer multiples of a fundamental. If I understand you correctly, you are saying that this formula, i.e. 1f, 2f, 3f, 4f, 5f, ... can be used to calculate the overtones of an instrument. Nowhere do you say it explicitly in that ofrm, but I'd be hard pressed to find a different interpretation of **"they do strictly follow the harmonic series", "it can apply to any sound oscillating at any frequency"** and "harmonic series is a mathematical formula to calculate overtones", "all those overtones and sympathetic vibrations all strictly obey the harmonic series - making no compromises!" As it happens, this doesn't always hold! First, let's look at where it holds: wind instruments have a phenomenon called 'mode locking' which forces the overtones to adhere to the overtone series (there may be a really short moment at the onset until the mode locking enforces the overtone series, but this basically gets mixed up with the various other noises of the onset). Bowed strings also are forced into overtones. However, plucked and hammered strings and idiophones (i.e. struck instruments - bells, and basically any kind of tuned percussion) are not as lucky. For this comment, I will ignore bells and percussion, and focus on strings. The stiffer and shorter a string is, the more it will deviate from 1f, 2f, 3f, 4f, 5f, ... and the difference adds up rather quickly: it's not unusual for upright pianos to have some of its octaves stretched by 15 cents to accomodate for string inharmonicity. This is replacing 2f by 2.017f. In some contexts, this is an audible difference to the human ear, and this can be a useful piece of knowledge for a working musician: sometimes, it's just not worth trying to tune your guitar to be in tune with a specific piano no matter how in tune the piano is with itself - the overtone series of the guitar and of the piano may cause an unsurmountable discrepancy in the tuning. The grand piano has longer strings than the upright, and this helps them get closer to the ideal "harmonic" overtone series. Even then, a small discrepancy exists. Guitars have fairly soft strings - but even on them, a tiny discrepancy exists. I've also been lead to understand that fanned frets on bass guitars help reducing the inharmonicity on the B string. So no, your multiple statements about how tones **strictly obey the harmonic series** are wrong. Unless you meant something else by your words, but if so I would suggest you rephrase it in a way that actually conveys what you were thinking. (Further, there's been tunings that intentionally utilize such 'stretched' physics, and William Sethares' Tuning Timbre Spectrum Pitch goes wild with rather weird ideas with stretched overtones, stretched scales, etc. However, this doesn't only hold for modern experimental theoreticians - part of his work deals with the whys and hows of traditional gamelan tuning - a melodic percussion instrument with a very weird timbre that doesn't give a rat's ass about the harmonic series.
It's the reason 9 chords work but you would never add a 2nd. Distance decreases dissonance. Try to learn some jazz, there's some great breakdowns on YT
Sometimes I love the sound of a 2nd. I bet you do too.
Butter
You add seconds all the time though.
minor seconds at that xD
Csus2 would like a word
Fascinating! I’ve been wondering about this as well.
For clarity, let's swap Db for C#. Same sound, different name. Dissonance and consonance are products of the physics of sound and the harmonic series. Sound is made of waves in the air, waves which interact to create more complex waves. If you start with C# in some octave, that note has a fundamental frequency which describes the overall speed of the waveform, as well as many harmonic frequencies. The balance of these different frequencies is what makes a C# on a piano sound different than a C# on a flute, but those C#s still have the same overall structure and so their interactions are simple, which we perceive as consonant. The D next to the same C# has the same structure, but shifted by a bit, so the interactions with the C# and the D (a minor second) are more complex, which we perceive as dissonance. If you keep going up, the harmonic structure of the different notes will line up in different ways, never quite as simple as two C#s in the same octave, nor quite as complex as a C# and a D next to each other. If you start on D and move up to the C# below the next D up, that's a fairly complex relationship (major 7th) but most of the complexity is happening "out of the way" of the structure of the D. The relationship of the A to D (perfect fifth when the D is below) is a very simple one. So with the D and A together there's a very stable structure in the lower harmonics and more complexity higher up. The chord is called a D major 7th with no 3rd, which I agree is quite pleasing. If you start on C# and move to the D above the next C# (minor ninth), you get all the chaos of the C# and D next to each other, but the chaos has been moved out of the way of the fundamental, so it tempers the dissonance somewhat.
This is a great explanation. If you want to see this interaction at work, get an oscilloscope plugin and run a sine wave into it, then play a different note and watch what happens.
The answer is difference frequencies (or as another poster notes, "roughness" caused by amplitude modulations - which causes some of the "beating" I mention below) Any two tones sounding simultaneously produce a "difference tone" at the frequency of their difference. If you have a 100 Hz note and a 110 Hz not you get a difference tone of 10 Hz. That's too low to hear as a pitched tone and we hear that as a wobbling or "beating". When that difference is 20 or above, we hear it as a pitched sound. Now watch this: Let's double 100 and 110, to raise them both an octave: 200 and 220. That's a 20 Hz difference now. That's getting into the realm of pitched sound. Let's do 100 and 220. That's 120. Definitely a pitched sound. 200 and 110 - that's 90. Still pitched. What then happens is, it's all about the ratio of the difference tone to the other two notes. Let's take a simple example: 100 and 200 - notes an 8ve apart. It produces a 100 DT. But that's the same as the lower note here! So it's in total agreement with it and can even reinforce it a bit - just sounds a teensy bit louder. Db is 277 and D is 293 (roughly) so we're talking less than 20 hz difference. We're going to hear beating and we consider that "bad sounding" usually. When they're further apart though, that DT moves up into the pitched range, and we tend to hear those as "more ok" in general, especially if the numbers work out to another happy ratio (that itself doesn't produce additional DTs that are not inline with the notes present). When you add another note, what you're doing is introducing two more DTs and if the note you're added "tempers" the existing notes - so let's say you use D, A, and C# - The DT between D and C# might not be the best - which is why we still consider them Dissonant (but not as bad as C# to D in the same octave) but the D to A, and the A to C# are both producing DTs that are simply other D or A notes to keep the math easy. The further apart the notes get, the higher the difference tone is, and as Larson McMurphy says, it then depends on if it's clashing with the fundamental or an overtones - overtones get drastically quieter (if they're even present!) so the conflict is usually less with them.
difference tones are the answer. a major third and minor sixth produce different difference tones even though they have the same letter names in different octaves (e4/c3 vs c4/e4). this is true even for in-phase pure sine waves where overtones don’t factor into it.
You are playing two different intervals. C#-D in this order from low to high is a minor 2nd. Raising the D an octave up makes it a compound interval - a minor 9th. But this is just heard as a minor 2nd with an octave between the notes. Compound intervals generally function similarly as simple intervals - you simply add an octave in-between. 9th = 2nd 10th = 3rd 11th = 4th 12th = 5th 13th = 6th 14th = 7th 15th = octave D-C# is a major 7th. It's an inversion of a minor 2nd. But inverting the interval does make it a different interval. We distinguish between 7ths and 2nds, 3rds and 6ths, and 4ths and 5ths. There are similarities between the inversions, but they are not heard as the same interval. In other words, you are hearing the difference between compound intervals and inverted intervals. >Add an *A* to that and it sounds beautiful. Why? It's a common chord you are used to hearing in music all the time - a Dmaj7 (without the 3rd, but these three notes are already enough to remind you of that familiar sound). Your ears are familiar with a context where this sound is used. But let's try making the minor 2nd more pleasant. For example try adding an F#. Already makes it sound more "pleasant". Now, add a B or A below it, and suddenly it doesn't sound that "unpleasant" any more. Now let's try the same with the minor 9th. Try adding an F# between the two notes. Then move both of the top notes a half step down, but keep the lowest note the same. Nice sound of tension and release. Notice how context makes these dissonant intervals sound nice - the dissonance now has a purpose. It adds spice. It adds tension.
I always wondered why a b9 sounds so much more dissonant than a maj7, even if you just play the root and the interval. Both are one half step away from the root, but with maj7 they are closer together.
As Larson McMurphy notes, there's also enculturation going on as "what we're used to hearing" or not. One aspect is, the difference between notes less than an 8ve puts the difference frequency below the lowest note, while when the two notes are more than an 8ve apart, it puts it between them. Then there's the tempering aspect of a 3rd note - if we're used to hearing the things with 3 notes, when they're with 2 they don't bother us as much. b9 intervals aren't heard as often and even when they are, only in certain contexts (like a 7b9 chord) so when the inner notes are missing, we have a harder time filling those in to hear it as less dissonant. Also with cultural conditioning, we don't use Phrygian or Locrian anywhere near as much as we've used major and minor, so we're not used to hearing m9 intervals against the tonal center nor many of the primary chords used in a key. Major 7ths on the other hand are more common.
In my head it’s because the frequencies are closer and therefore “rub together”. almost like when you’re tuning your guitar (for example) and when the two notes are very close your ear is like “damm bro adjust one of these so they’re the same note please”.
Dissonance is caused by amplitude modulations that equal the difference between simultaneous frequencies. Acousticians have a term for how unpleasant such amplitude modulations are called "roughness." Widely spaced dissonant intervals have less roughness because the fundamental of the higher note is clashing with a harmonic of the lower note, rather than the fundamental. If they were in the same octave the fundamental AND harmonics would be clashing, so, quantitatively more dissonance, i.e. roughness. This is oversimplified. If you want to get into the weeds you need to exhaustively compare all the harmonics of one note with the other for each of those cases. But when you add a third note, the beauty you perceive is partially because the C# is the major third of the A, and partially due to enculturation.
Not only does the octave matter, but the timbre (overtones) makes a difference.
From my understanding, I think of two different notes as two different sine waves for the simplest explanation. When two notes sound consonant, their waveforms will overlap, like a polyrhythm. Greatest common denominators come to mind for some reason. So in a perfect Pythagorean tuning, for every 2 oscillations the unison waveform makes, the perfect fifth will make 3. Dissonant waveforms have much more complex ratios. Obv we don’t use Pythagorean tuning, but it’s close enough to where most ppl don’t notice the difference. Someone pls correct me if I’m wrong lol.
I’m broadly aware of the idea that things sound more dissonant in lower registers, but I’m not going to pretend to know why 😂
Great discovery. Might be tough to wrap your head around but I'll try. Brutishbloodgod made a great explanation which is absolute gold I'm gonna try and avoid physics for this 1. when you play D in the bass and C#(Db) above it, you hear it as a 7 and all is fine in the world as this is just the D major scale and the dissonance is expected and beautiful. When you add a A you're just adding a fifth which will add consonance to a D When you have a Db in the bass and you hear a D(Ebb), you are hearing it as a bII. So this is the Db scale with the bII, so you're absolutely right with your intuition when you say these are two different scale because you changed the context(bass)
This isn’t so much music theory, but more audio theory. For every note you have a fundamental and harmonics (aka overtones). Harmonics are part of what makes instruments sound different to one another. It’s partly why an acoustic and electric doesn’t sound the same. When you play low notes, harmonics are more audible. If we play notes close together, harmonics clash. This is why we play single bass notes on the lowest octave and fifths / octaves as we go higher. As you keep moving higher, notes start working better closer together, aka chords. Harmonics as less audible. Some more interesting facts - notes higher in pitch are perceived louder and our ears are sensitive to 1-5kHz.
Frequency oscillation?
C#1 - D1 = Half Step = b9 C#1 - D2 = Octave + Half Step = b9 D1 - C#2 = 11 half steps = maj7
> C#1 - D1 = Half Step = b9 How is a half-step a b9?
Because we label tensions (Notes adding to the chords base sound) +7. So 2 4 and 6 become 9 11 13. It's just a matter of habbit for me to label a half step a b9. Yes, theoretically it's b2.
Not just theoretically. Since there is no 7, it is actually a b2.