# Does the difference in harmonic series between instruments have a significant effect on the consonance of the sound?

We all know that an instrument's timbre is uniquely determined by its harmonic series. Harmonic series are also very effective in describing whether an interval is consonant or not. For example, the perfect fifth (say C to G) is consonant because the harmonic series of C and G have a very large overlap. (Actually, they almost coincides!)

Having those two ideas in mind, here is what I am thinking about: if the relative loudness of the harmonic series of two instruments are different (of course the notes in it are the same), then when they play the same interval, one of them might sound more consonant than the other. By "relative loudness of the harmonic series", I mean something like this (taken from this pdf):

For example, trumpets have a very loud third harmonic (this may be not right; just to illustrate what I mean), so when that third harmonic happens to be a very unpleasant note after combining with the rest of the orchestra, trumpets should not be used. Perhaps we should instead use an oboe or something.

Question: Are there any theories about what I mentioned above? Is it an important thing to consider in Orchestration?

• If you want to research for theories about orchestration in respect to your question you might look for timbre and formants and orchestration. (s. my 2. answer!) you'll find e.g. this pdf: asa.scitation.org/doi/pdf/10.1121/1.4799391 Commented Dec 31, 2019 at 11:07

Is it an important thing to consider in Orchestration?

You have in fact stumbled onto the very foundation (and art) of orchestration. Orchestration is about not only knowing how each instrument sounds, but how to blend those sounds together to get the effects / textures you're looking for.

Composers generally don't think about blend in terms of harmonic series, but as Tim alluded to, we think about character. To illustrate my point, I'll give a couple brief examples.

To extend your (the OP's) point, not only does each instrument have a unique base timbre, but that timbre also changes not only throughout the instrument's pitch range but dynamic as well.

Let's, for a moment, think like an orchestrator:

EXAMPLE 1

• The clarinet has 3 registers: chalumeau, clarion, and altissimo, each with its own characteristics: the first is dark and velvety, the 2nd is focused and clear, and the 3rd is piercing but smooth. Though I haven't tested it, I'm quite confident that the harmonic spread of these registers would differ.

All of this is wonderful to know, but it's also important to know that clarinets also get buried easily. If we wanted the sound to cut through the ensemble, you could pair the clarinet with oboe, trumpet, or piccolo, for example. If you wanted a softer, fuzzier sound, you would blend the clarinet with flute, especially in the flute's lowest register.

Every instrument has unique characteristics dynamically (in terms of the dynamic range they are able to produce, which in fact does change throughout the instrument's range; such as the oboe at the bottom of its range!), timbrally, and unique characteristics due to the nature of how the instrument makes sound (for example, woodwinds have a more difficult time playing fast, repeated notes, where brass and strings would not).

Composers and orchestrators have go-to combinations that are tried and true and deliver confident results (ever wonder why "movie" music sounds a certain way different than other types of orchestral music?) There are too many to list here, but there are many, many books written on the subject.

• One could argue that a brass instrument has many registers. For a particular valve position, there is a lowest resonant frequency, but my lip tension controls which harmonic is the lowest you hear. Better players can play more than the six I can do—and one of the ones I can do is lower than what I was taught is the lowest possible. Commented Dec 30, 2019 at 18:08
• @WGroleau of course, I myself also play brass. As I said above "every instrument has unique characteristics" - I simply pointed to clarinet as a single example, but brass instruments have just as many unique characteristics as all of the other instruments. Commented Dec 30, 2019 at 19:18
• Did you intend to add more examples after "EXAMPLE 1?" Commented Dec 31, 2019 at 3:27
• @Bladewood Yeah I thought about it but then quickly realized I'd spend hours and hours typing so I didn't. Then I just didn't think about changing it. Commented Dec 31, 2019 at 15:28
• Also note that the human ear only picks up so many harmonics. So not only the physical vibration of the instrument, but also a human ear's receptability will affect the relative strengths of the harmonics, and thus the timbre. This is one of the effects that makes an instrument sound differently in different registers, particularily the highest instruments, like piccolo flutes, where the highest note only has 3-4 harmonics that we can hear at all. Commented Jan 2, 2020 at 15:06

Other answers so far make good points -- matching timbres (and sound spectra) is actually essential to orchestration, and composers have been noticing these patterns (and using them in orchestration) even before analysis of harmonic spectra was possible.

I would add one other related issue to answering the title question about "differences in harmonic series" in general. First off, there's an entire subgenre of modern art music, commonly called spectral music or spectralism that is based on manipulation of timbre and harmonic spectra. There are all sorts of well-known effects that can be created with consonance, dissonance, and other musical parameters through manipulation of spectra, as many of these compositions have explored.

For an intro on how this sort of spectral manipulation can work, I might suggest a classic book: William Sethares's Tuning, Timbre, Spectrum, Scale. (There are more recent books on this topic, but Sethares gives a reasonable intro to many of the issues, including timbral manipulation and its effects on consonance and dissonance.)

Sethares has posted a few musical examples that accompanied his book here. It's one thing to consider instruments that have stronger or weaker harmonics, but what if we stretched or compressed the sound spectrum of instruments artificially, so that they no longer would produce consonance at the normal intervals?

Take a listen to his examples of this phenomenon:

• First, a normal example in standard 12-tone equal temperament
• Now, the same tune played with all the overtones "stretched" so they no longer correspond to the harmonic series: standard music intervals now sound dissonant
• But what if we stretch the scale underlying the tune too? We get this, which sounds weird, but at least with something resembling "consonant"-like intervals.
• Note that if we fix the timbre back to "normal" (with overtones following the harmonic series), that last example with a stretched scale sounds dissonant and horrible too.

Basically, consonance and dissonance are fundamentally (no pun intended) wrapped up with the structure of overtones in sounds. One can make a major seventh sound somewhat "consonant" by manipulating the overtones, and one can make an octave sound quite dissonant by changing the overtones as well.

This spectrum manipulation, by the way, is also really important to instrument building and construction. Strings and long tubes are relatively "one-dimensional" and thus tend to have relatively "harmonic" spectra, i.e., with overtones resembling the harmonic series. But other types of instruments (bells, drums, etc.) do not have overtones that fall nicely into that harmonic series patterns. There's a whole ancient lore about church bells that basically involves casting patterns and shaving off metal in various places to try to make the bells sound more "harmonic" rather than dissonant (both with other bells and even "clangy" and unfocused in tone when rung alone). And if you watch a professional timpanist who is making fine adjustments to "clear" a drum head and tune it, what the timpanist is really doing is manipulating the drumhead to produce a more harmonic sound (again, with overtones corresponding closer to the harmonic series, even though the sound is produced by a two-dimensional vibrating object with different modes of vibration than a string or air column would have), so it will have a clearer tone and will blend better with other instruments.

Does the difference in harmonic series between instruments have a significant effect on the consonance of the sound?

Absolutely - and not only between instruments. Different ranges of the same instrument have different harmonic structures - a commonly-given example is the 'muddy' sound at the bottom end of the piano, caused partly by relatively weak lower harmonics.

Are there any theories about what I mentioned above?

One of the most general theories relating to this comes from Plomp & Levelt's work on consonance. The basic idea is that the dissonance of a sound at any time can be worked out from the pitch relationships of all sinusoidal pairs.

I think it's fair to say that once you really dig into the topic of what consonance and dissonance is, you'll get various different notions and ideas of what those words really mean - there isn't a single, specific, well-agreed-upon definition, as far as I am aware.

I'm sure different composers and arrangers have produced useful specific guidelines over the years - hopefully someone else can give some examples. But it may not be possible to make rules as simple as 'don't use this instrument in this situation', for an instrument's harmonic structure also varies over time, and depending on how it is played. In fact it's often down to the player to mould the timbre they produce to suit the composition.

• I expect a lot of composers are at least sub-conciously aware of this, and mix accordingly. It could even be a 'trademark' of some of those composers.
– Tim
Commented Dec 30, 2019 at 12:45
• Even in the Brass band scene it is possible to identify different composers by their typical instrumentation and the „sound“. Commented Dec 30, 2019 at 20:56

It would since the very nature of consonance vs dissonance is dependent on the interference of harmonics (in theory). So it stands to reason that if a particular instrument had missing harmonics there would be fewer opportunities for dissonance with when playing intervals on one instrument (string, or piano) and harmony with other players. However that is not to say that all opportunities for dissonance disappear when harmonics are absent. The ear creates aural harmonics. So even in the presence of two "pure tones" a listener will perceive standard consonance/dissonance as explained by standard theory. Whether or not the strength of each harmonic has a significant effect on the perception of consonance/dissonance is an other issue. I am not sure if the judgement of this quality is binary or even subjective. Attempts to relate this to the physics of waves go back to Helmholtz but there are always outliers to this type of phenomenon.

I disagree that "muddiness" is the same issue or related to this issue but it might be. There is a limit to pitch discrimination in humans and this is frequency dependent. This is why large intervals are placed in the bass and smaller intervals in the higher register. Once you reach the limit of pitch discrimination it almost doesn't make sense to say the interval is consonant or dissonant since you will probably hear the sum tone with a difference tone envelope.

Keep in mind that an instrument does NOT have a single spectrum. The spectrum depends as much on the attack as on the physics of the instrument itself. So you can adjust the attack to accentuate certain harmonics. This is very easy to do on the guitar and gives it a characteristic versatility. Perhaps not at easy for brass or woodwind but I am not sure as I don't play those families of instruments. While holding a chord for a long time with 3 or 4 trumpets it would be interesting to hear how the harmony changes as the players adjust their embouchure gradually. If that is easy enough to do.

Lastly, most models of musical instrument spectrum we have follow the harmonic series fn = n*f0. It could be said that we modify the materials and construction to get close to this series. But the fact is that not all physics of vibrating system following this series. The stiff rod or plate with boundary conditions can have dissonant harmonics. By that I mean the ratio of a harmonic to the fundamental can be an augmented 5th +/- a few cents, or some other note that isn't evne in the just or 12TET scale. These instruments will sound intrinsically "out" and when played in harmony with other instrument would be quite a bit more dissonant. Bells are an example, the basic xylophone bars can exhibit this, but they are usually modified with a variable cross section to "tune" them..

Does the difference in harmonic series between instruments have a significant effect on the consonance of the sound?

My personal experience (working with synthesizers (ADSR), Finale, VST and wavelab. There I encounter often the problems you are mentioning in your question.

The sound of different instruments is strong related to the question of timbre and formants.

So you probably know what timbre and formants mean.

If not ... these 2 wiki sites give a lot information about those terms.

https://en.wikipedia.org/wiki/Formant

(while there seems to be great differing in definition of timbre!)

Timbre has been called "...the psychoacoustician's multidimensional waste-basket category for everything that cannot be labeled pitch or loudness."

This article (wiki Timbre) discusses In music history also the examples I've mentioned in my first answer: Wagner and Debussy!

Instrumental timbre played an increasing role in the practice of orchestration during the eighteenth and nineteenth centuries. Berlioz (Macdonald 1969, 51) and Wagner (Latham) made significant contributions to its development during the nineteenth century. For example, Wagner’s “Sleep motif” from Act 3 of his opera Die Walküre, features a descending chromatic scale that passes through a gamut of orchestral timbres. First the woodwind (flute, followed by oboe), then the massed sound of strings with the violins carrying the melody, and finally the brass (French horns).

http://musicweb.ucsd.edu/~trsmyth/timbre175/Musical_Instrument_Formants.html

while the English wiki site is focussing on human voice and vocals the German wiki site gives a lot more of information (you can translate it to English. If I do it here it would be to broad and it is not the core of your question!)

And in the bibliography of the wiki sites you'll find links to literature about Theories for Orchestration.

http://musicweb.ucsd.edu/~trsmyth/timbre175/Musical_Instrument_Formants.html

(here you find also Pdf's)

e.g. this one

http://newt.phys.unsw.edu.au/jw/formant.html

Are there any theories about what I mentioned above? Is it an important thing to consider in Orchestration?

https://asa.scitation.org/doi/10.1121/1.4806102

Most of the music we enjoy uses the musical qualities of different instruments to create specific perceptual and emotional effects that composers sculpt over time. Timbre is the auditory attribute that distinguishes different instruments. Research on timbre perception has demonstrated that it is multifaceted and contributes in many ways to the perceptual organization of musical structures. The art of structuring music with timbre is orchestration. A survey of orchestration treatises reveals the dearth of underlying theory, in sharp contrast to other traditional areas such as harmony and counterpoint, which have long theoretical traditions. We seek to develop a theoretical ground for orchestration practice starting with the structuring role that timbre can play in music. Many aspects of musical structuring are achieved by auditory scene analysis, the perceptual processes that result in unified events, integrated streams of events, groups of events segmented into phrases and sections, and larger-scale units extended over time that we call orchestral gestures. The roles that timbre plays in the manifestation of these principles in orchestration practice will be considered as potential elements of a theory of orchestration. How such principles might be incorporated into computer-aided orchestration systems and computer-aided orchestral rendering systems will also be examined.

https://de.scribd.com/doc/39643485/Walter-Piston-Orchestration

https://ccrma.stanford.edu/~jmccarty/formant.htm

http://www.sengpielaudio.com/FormantenPraegenDieKlangfarbe.pdf this pdf I've translated by Google for you:

Note: Formants are resonance-like amplified frequency ranges of partial tones in voices and musical instruments, the position of which is preserved, regardless of the fundamental frequency of the sound. They represent acoustic filters with low resonance circuit quality - a not too narrow-band increase (fifth to octave) - which are particularly noticeable with vowels. Partial tones = partials, harmonics - but also overtones.

....

Formant areas of woodwind instruments: *)

Instrument main area secondary area

around

Flute 810 Hz

Oboe 1400 Hz 2960 Hz

English horn 950 Hz 1350 Hz

Clarinet 1180 Hz 2700 Hz

Bassoon 440 Hz 1180 Hz

Contrabassoon 250 Hz 450 Hz

Double reed instruments have particularly distinctive formants.

Formant areas of brass instruments:

Instrument main area secondary area around

Horn 340 Hz 750 Hz

Trumpet 1200 Hz 2200 Hz

Trombone 520 Hz 1500 Hz

Bass trombone 370 Hz 720 Hz

Tuba 230 Hz 400 Hz

The formation of the formants in brass instruments is caused by the mouthpiece.

Formant areas of the strings:

Instrument main area secondary area

Violin 400 Hz 1000 Hz

Viola 220/350 Hz 600/1600 Hz

Violoncello 250/400 Hz 600/900 Hz

Double bass 70 to 250 Hz 400 Hz

The sound spectra of the string instruments are very individual due to the large differences in design. The areas of the formants are highlighted by the resonance of the resonance body and the enclosed air volume from the spectrum of the vibrating strings.

Directional tapes according to Jens Blauert (Blauertsche tapes):

Sensation main area secondary area around

present, front 3150 Hz 315 Hz

diffuse, rear 1000 Hz (10,000 Hz)

*) based on: Paul Heinrich Mertens, "Die Schumannschen Klangfarbengesetze und ihre Bedeutung für die Übertragung von Sprache und Musik", Verlag "Das Musikinstrument" Erwin Bochinsky, Frankfurt/M, 1975. ISBN 3-920-11254-7

• Calling timbre a 'waste-basket' seems a bit silly.... sure, it's a general term for the character of a sound, and as such is indeed multifaceted (as your other quote states). Commented Dec 31, 2019 at 21:30

It’s essential to understand the difference between overtones and harmonics for one who wants to understand orchestration from a physical point of view. The easiest to understand are harmonics, which was originally a mathematical term associated with a harmonic series, in which each term has a whole number ratio to all other terms. Overtones rather are associated with a physical system and the term describes the different ways a vibrating system vibrates.

The piano is a good example. Its sound contains a fundamental and overtones. The overtones are close to, but not harmonic. This is because no real string is infinitely flexible, complications because of string length and tension, and the fact that the piano tone is a transient, not a steady periodic motion. But bear with me, and you will understand where I'm going.

Thus “overtones” can be characteristic of a single tone, with different frequency components making up the tone. Another example of overtones would be to excite a steel piano string with, say a periodic magnetic field. If you did it at the frequency of the fundamental, the string would vibrate at that frequency, and there would be very little other vibratory components – no overtones. Then you could excite it at the first overtone frequency that occurs during the conventional vibration. Again, the string will vibrate at that overtone frequency, with no fundamental and very little other components. You can excite the string at any frequency, but if that frequency doesn’t match the fundamental or some overtone, the string will vibrate with little amplitude. This is called resonance. Stay with me.

Any steady, periodic tone must have overtones that are harmonic. If that weren’t the case, the vibration would not be periodic. That’s because the motion must repeat exactly every time, meaning all overtones must also repeat. It’s mathematically impossible for the overtones to repeat as a multiple of the fundamental if they are not some whole number ratio of the fundamental frequency. We come to the difference between the piano and the violin. The piano tone is not steady or periodic. It’s a transient, made by striking the string, and there's no requirement that the vibration repeat exactly for all cycles. It doesn't, because of physics, not mathematics, and in addition the amplitude of all components during one cycle is less than the corresponding amplitude in the previous cycle. With the steady periodical vibration of other instruments, such as the violin, the vibration is "forced" or maintained by the action of the bow. That explains the periodic nature of the sound. Energy is steadily being transferred from the bow to the string.

So we have an example of the same vibrating sound source – a string – behaving in very different ways. With the violin, the tone contains overtones that are harmonics, and with the piano, the overtones are not harmonic. Of course, the violin can be played in transient ways, like other instruments can, using various techniques other than a steady draw with the bow.

To summarize, all musical tones that are steady and periodic have overtones that are harmonic, and all musical tones that are transients have overtones that are not harmonic, although some can be very close to harmonic.

What does this mean for orchestration? Well, first, I point out that some here seem to think that dissonance – non-harmonicity – is always bad. That’s not true. The dissonance of a cymbal is a favored sound. The 20th century jazz bands of Stan Kenton and Maynard Ferguson are now forever in orchestral annals of examples of how to use dissonance to blood-boiling acoustic effect. Then we also have the later orchestral composers of the modern age. Also, the transient sounds of bells and chimes contain non-harmonic overtones, and it's an art to tune them for maximum effect. The "bigness" of the sound of church bells is enabled by non-harmonic partials. Another example is the equal tempered scale, which has enormous advantages, but some people actually prefer it over other "natural" scales because of the inherent dissonance it provides.

It’s worth looking with more detail at the “upper partials” of a musical tone in the example of the piano. As I mentioned, because the strings are not perfectly flexible, and because string vibrations in some cases produce tensile fluctuations that begin to be at the level of string tension, the overtones in the transient tone are not harmonic, but “stretched,” meaning that the overtones are consistently larger than what the harmonic calculation would predict. And that characteristic makes the piano a beautifully sounding instrument. The stretched overtones mean that individual strings can be more easily discerned from each other and from other instruments, so that chords are vibrant and well identified. Compare that to the steady chords played by an accordion, which are harmonic, due to the steady nature of the sound source. With the accordion, chords are muddled, mushy, because of the exact harmonics. Say you play two notes an octave apart on the accordion. The upper note plays the exact same overtones as the lower note, with the only difference being their fundamentals. It’s very difficult to discern the two notes, and when you could, it’s usually because one note is out of tune.

I could explain much more about the beautiful piano tone and stretched harmonics, the effect of three strings playing together, and the interaction with the sound board. All of that involves overtones in very complicated ways. But I’d rather make corrections to the OP’s comments.

“We all know that an instrument's timbre is uniquely determined by its harmonic series.” This is incorrect because not all instruments have overtones that are a harmonic series. Additionally, the questioner may be confused in believing that we can identify the source of a musical tone only by its timbre. We also need to hear the “start transient.” There are huge differences in start transients between the piano and violin, the accordion, and the wind instruments. The percussive start of the piano – or guitar – tone help us identify the instrument. In fact, there have been scads of experiments wherein people cannot tell the difference between musical tones from different instruments when the start transient is eliminated. Even with instruments that we know have significantly different overtone/harmonic structure. Many people think the clarinet contains ONLY odd harmonics, though that’s not accurate, and when its musical tone is played without a start transient, it’s easy to confuse it with the other steady tones.

I’d like to draw closer to what the OP wants to know, but there are too many misconceptions and the questions don’t really make sense. The OP seems to believe that a harmonic overtone in one instrument can clash with a harmonic overtone in another instrument. Why so?

“if the relative loudness of the harmonic series of two instruments are different (of course the notes in it are the same), then when they play the same interval, one of them might sound more consonant than the other.”

Why would one harmonic frequency be more consonant than another? Or why would the additional volume when a third harmonic of one instrument is added to a harmonic of another? It's pretty much by definition that any harmonic is consonant with any other harmonic. I suggest that the OP needs to understand the difference between overtones that are harmonic and overtones that are not. The simplistic view by the OP is also evident in the steady tone spectrum examples shown. These are steady periodic tones that are not often found in actual music. In practice, there are many dynamics and transients in the playing of any instrument and thus very few tones conform to the information contained in those Fourier spectrums.

The OP is worried that a predominant third harmonic of the trumpet might clash because, “when that third harmonic happens to be a very unpleasant note after combining with the rest of the orchestra, trumpets should not be used.” Is the OP wondering about the clash only when the trumpet player is assigned “a very unpleasant note”? I can’t make heads or tails out of what the OP wants to know.

“Question: Are there any theories about what I mentioned above? Is it an important thing to consider in Orchestration?”

I suggest the OP re-write this question, using hopefully a better understanding of the role of overtones and harmonics in music. Roughly speaking, and I’m not an orchestral composer, my guess is that composers attempt no theoretical applications concerning overtones, but rather rely on their experience of the total sound of each instrument and how they blend together in a very dynamic environment. It's bottom line experience and it's very subjective. It would surprise me greatly if composers consulted graphs of steady-state Fourier series during composition.

In addition to the good answers you‘ve already got I want to add that Ravels Bolero, Shostakovich‘s ostinato in his 7th Symphony (1. movement) or Lohengrin by Wagner are great studies of orchestration.

There are also interesting personal styles of composers: you can recognize like e.g. Schumann leading the flutes parallel (or voicing identically) with the violins, that makes it easy to identify a piece as typical by his.

Are there any theories about what I mentioned above? Is it an important thing to consider in Orchestration?

There are surely theories and books about orchestration but a good method will be to experiment by yourself and learn by listening to orchestra music, analyzing and studying full scores, arranging and experimenting like we know of Debussy who was changing the parts even during the final rehearsal of his après-midi d‘un faune.

With your question you are on a right way.

Does the difference in harmonic series between instruments have a significant effect on the consonance of the sound?

No, not in the real world. In the real world, judgments about consonance and dissonance are based on culture, experience, and context. The psychoacoustic facts only have an indirect effect in most cases, because they set some parameters within which culture has to work.

Some of the earliest careful empirical work and detailed mathematical modeling on this was done by Kameoka and Kuriyagawa in 1969. They tested audio engineers who were musically untrained, and they tested them in artificial circumstances (designed, e.g., to avoid harmonic distortion) and without any musical context. They did find that consonance and dissonance were judged by these musically naive subjects to be well reproduced by a mathematical model involving clashing harmonics within a certain critical bandwidth of frequency (about 1-10%). For instance, pure sine waves in frequency ratios normally considered dissonant (e.g., a tritone) were judged by these subjects to be consonant.

However, none of this corresponds in any detail to how people actually perceive music if either (a) there is some musical context, or (b) the person is musically trained. For example, an equal-tempered minor third is a frequency ratio of 1.189. So if we have a minor third like A with C, the 5th harmonic of A and the 4th harmonic of C differ by only 5%. This lies right in the middle of the critical bandwidth, and therefore according to this type of model should produce a strong sensation of dissonance. Of course people raised in the modern Western musical tradition consider this interval to be sweet and consonant.

• When using sounds that have very strong third and fifth harmonics (e.g. an organ pipe mixture that features 2 2/3' and 1 3/5' ranks), playing two notes a minor sixth apart can yield a rather unpleasant beat frequency which wouldn't be present when using other sounds a minor sixth apart. Commented Dec 30, 2019 at 22:52
• Yes, and being used to Western music doesn't prevent one from perceiving Western harmony as dissonant when it's played on instruments where this harmony doesn't physically work. The easiest example are bells: play a piano piece on carillion, and you get a pretty unbearable cacophony. Vice versa, many Western people seem to quite like Gamelan music, whose harmony is tailored to the instruments with very different harmonics. Commented Jan 1, 2020 at 15:04