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If I play out the harmonic series on the piano, the notes are far apart over multiple octaves and I was wondering why we squish everything into one octave to create chords/scales?

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  • Our music did not originate from the harmonic series to which you are referring. People played what sounded good and what was within their ability to play well. Subsequently, it was discovered that some musical systems - for example our western musical tradition using certain tuning conventions (not all musical systems throughout the world, and not all of our tuning conventions, except with a bit of fudging) had a satisfying scientific basis.

    There is no evidence to support the notion that the math of the harmonic series came first, and music followed. On the contrary -people were making music long before it was discovered. See: Prevailing theories about discovery of harmonic intervals - and they were not about to change the way they made music because Pythagoras (or some other mathematician...) discovered the harmonic series.

  • Music which spanned many octaves would not sound good - it would sound disjointed and unsatisfying. It would also be very difficult to play. Even if we say that some systems were developed based on the discovery of the overtones series, that would only be after the chromatic octave was condensed and organized into its present form. Playing music and listening to music in the manner you are suggesting just doesn't work.

Bottom Line: Our taste and ears, and our ability to play our music are the determining factors, not a mathematical algorithm for building the chromatic octave using the harmonic series, discovered by a mathematician with musical curiosity.

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I don't agree with your premise. Many songs have melodies that range over an octave. On the piano, you play a chord with your right hand, and that is all on the same octave (can you stretch your hand much more than that? I can't), but what is your left hand doing? Playing notes in a different octave. I could go on: choirs, symphonies, your favorite band all have ranges greater than one octave.

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    True, but each section plays within its own limited range of a few octaves at most. All the chords and scales used are "squished" within their own context. Many songs have melodies that range over an octave The harmonic sequence uses 12 octaves to build the chromatic scale, going from perfect 5th to perfect 5th until the cycle of 5ths is completed. (afterwards they are condensed into one contiguous chromatic octave) You will hard pressed to find a song that spans that range. – Stinkfoot Mar 5 '18 at 17:00
  • @Stinkfoot I would not call the circle of fifths a harmonic series. But you have a point that there is some "squishing," even if it isn't "squished" all the way down to a single octave in all cases. – David K Mar 6 '18 at 2:43
  • @DavidK - note that I did not say one octave only in my comment. The question as stated is talking about something else entirely, not a piece where the range for a particular section is two or three contiguous octaves. – Stinkfoot Mar 6 '18 at 4:56
  • @Stinkfoot Yes, and I think your comment makes more sense than the question with regard to how many octaves people actually use. – David K Mar 6 '18 at 4:59
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We don't. But notes in close harmony do work well together, probably better than a 1 in one octave, 3 in the next and 5 in the next to that. And do try singing that sort of sequence! A lot of instruments only have a three octave or so range anyway.

You mention scales. Can't imagine how they would be played if the notes weren't pretty close to each other, can you?

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Notes in a scale are chosen such that the pitches of those notes (and their harmonic partials) have interesting relationships with each other.

It is entirely possible that some people might like a scale made of notes whose fundamental pitches matched the harmonic series - there's nothing wrong with that idea as one way to make a scale.

However, there are lots of ways to make a scale that allows interesting frequency relationships between notes. It simply isn't logical to think that would require the notes to directly follow the harmonic series, and indeed in the diatonic scale that Western music has adopted as the primary scale, they don't (even though there are some clear correspondences).

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It's helpful to remember that we're not playing just fundamentals. Every time we strike a key or blow a note, we get the fundamental and a whole set of overtones. Psychoachoustically speaking, our ears are used to this and they capture all of these frequencies together as a pitch and a quality of some sort.

If we play notes from the harmonic series across the entire range of the piano, they sound disconnected. They don't sound like a chord. This is because our brain is smart. It noticed that the set of overtones it heard did not line up well with what it learned things sound like over the last few million years. The loudness of each harmonic isn't quite right for it to be a single source. Thus, the ear quickly breaks the sound into multiple series of harmonics, and you get the disjoint sound.

When you play a chord where the notes are closer together, the loudnesses of the overtones are closer to what we learned things sound like millions of years ago. We start to hear the sound as more of a single sound and less like a set of individual sounds. Qualitatively, we find the notes of the chord form one unit that acts together. When the notes are far apart, we don't get that qualitative feel.

Interestingly enough, throat singers do the exact opposite. A throat singer starts with a very rich set of overtones (like what the human voice can create), and then carefully crafts their vocal tract to be resonant with one of the overtones. When our ears hear this, they do not expect to see one resonant frequency sticking out over the top of what is otherwise a well behaved set of overtones. The result is that it sounds like one is "singing two notes at once," because our ears decouple that odd overtone from the rest of the sound, treating it as though it's a separate sound source.

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Only the low notes of the harmonic series are far apart. As they go higher, the notes of the harmonic series get closer and closer together. The fifth octave of the harmonic series already contains (approximately) all the notes of the chromatic scale.

So, I would say that squishing isn't the right word for what's going on. Instead, why do we use this fifth octave of the harmonic series instead of a lower (or higher) part?

Well, sometimes we do use the lower part. For example, bugle calls (like Taps and Reveille) use the second and third octaves of the harmonic series since those are the notes that a bugle can (easily) play. But perhaps these are not the most interesting songs.

Ultimately it does all come down to aesthetics, but there are possibly some mathematical reasons why the fifth octave is a good choice. If you care about harmonics, then you may also care about harmonics of harmonics (ie the circle of fifths). The diatonic fifth (the third harmonic and first non-octave harmonic) needs a diatonic fourth to get back to the octave. But the harmonic used for the diatonic fourth doesn't come until the 21st harmonic which is in the fifth octave.

If we insist on thinking of squishing, then scales are almost squished by definition. A scale is usually just what we call a set of notes all squished into one octave and then put in order.

For chords, the situation is different. We often think of a chord as existing in one octave (or two when stacking thirds) because there are prototypical instances of each note all within a single octave. But in practice, the situation is different. Consider instruments that actually play chords. A piano player playing two-handed chords will often span three octaves limited only by hand span and finger count. A guitar player will typically play two octave chords, limited by hand span and string count. When multiple performers play at the same time, things get even more spread out.

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There's no "squishing" -- an "octave" is just the difference between one note and another note with double the frequency of the first note. For example, the octave of A with frequency of 440Hz is another A with a frequency of 880Hz.

The word "octave" implies that the frequency range is divided into eight divisions, but that is just an artifact of the way much of Western music has historically been organized. There is nothing to prevent anyone from dividing that frequency range into any number of divisions, nor is there any rule that the divisions must be equal. (And other musical traditions, for instance Indian music, use different divisions).

Having said that, there are certain intervals that are generally perceived as having certain qualities (e.g., a "minor third" sounding "sad"), and Western music uses these qualities to convey emotion.

In the end, it's all about frequencies and math.

Last but not least, the practice of using "tempered" scales in Western music is relatively recent (e.g., Bach composed the "Well-Tempered Clavier" pieces to illustrate the use of the tempered scale). For more on this see Why are there both sharps and flats?.

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