More About Series

(entry for 12/13/2024)


Last post we talked about the Overtone Series: how its third, fourth, and fifth steps create the major triad, which is contained within every note that we ever hear.

There are a couple of other important points to get to regarding intervals as defined by the Overtone Series, but before we get there, this would be a good spot to clear up some confusion about the whole ‘Series’ concept. Namely, the difference between the Series of Partials and the Overtone Series.

They’re really the same thing, but they’re numbered differently. Remember back to the idea that each tone in the series is half as loud as the tone before? And remember that we said the first tone, the fundamental, accounts for 50% of the total sound? Well, the Series of Partials starts with that fact. Each step in the series is Part of the whole sound, so each step is a Partial. And Partials include every step in the sound, including the Fundamental (the 50% part).

So in the illustration we gave last post, starting on the A110 which is played by the open string bass string, that pitch, A110 is the ‘first partial,’ because it’s the first ‘part’ of the sound. The A220 then is the second partial, the E330 is the third partial, the A440 is the fourth partial, etc. But the Overtone Series does not include the Fundamental, because the fundamental is not an overtone. So when you’re counting overtones, the first overtone is the first note you come to in the series that is NOT the fundamental. In other words, the first overtone of A110 is A220, made by the half-lengths of the string. The second overtone is the E330, and the third overtone is the A440. If you think about it for a minute, you’ll see that to convert the step number in the Series of Partials to the numbering system of the Overtone Series, you simply subtract ‘1’ from the Partial number. That is, the 5th partial is the 4th overtone. Both numbering systems are correct; they’re just different from each other.

Once you get that concept, it’s pretty easy to figure out the reverse: to get from the Overtone Series number to the equivalent number in the Series of Partials, you add ‘1.’

Which numbering method you choose to use in your own thinking is entirely up to you. As said before, both are correct. But on the other hand you do need to pay close attention to which series is being talked about when you listen to someone talk or read a book or article on the subject. If the book says ‘Partials’ and you’re thinking Overtones,’ you’re always going to be one off!

There is a third way of referring to all this, and that is the words ‘Harmonic Series.’ Here it gets a bit confusing (as though it wasn’t confusing enough already!). If somebody says, “I’m playing the second harmonic,” that means exactly the same thing as if they had said, “I’m playing the second overtone.” Both ways of saying it mean that the third partial is being played, and the third partial (subtract ‘1,’ remember?) is the same thing as either the second overtone or the second harmonic.

However! (Don’t you just love all the ‘howevers’ in music?) If you say the words “Harmonic Series,” meaning the entire slew of pitches contained in a single note, you mean the Series of Partials, including the fundamental. So to make the confusion complete, it follows that if you refer to the 5th harmonic (overtone), you’re also referring to the 6th step in the Harmonic Series (Series of Partials)! Egads! After thinking about that stunner for a moment, I think you’ll agree that we should avoid the word Harmonic altogether and just say Overtone or Partial, depending on which way we’re counting.

(By the way, if you’re already a fretless string instrument player such as a fiddler or doghouse bass performer, you already know this, but for newcomers: If you touch a string very lightly at exactly its half-way point, and don’t create any vibrato, you get a sound called the ‘first harmonic,’ or ‘first overtone.’ This is caused by both halves of the string vibrating at once. If you touch it at one third of its length and play either the shorter or the longer portion, you get the ‘second overtone’ or ‘second harmonic.’ )The string is now vibrating in thirds, but it's vibrating in all three thirds, as opposed to getting only one portion vibrating when you press firmly. Etc. Doing this has an eerie, glassy sound, that is much loved in some styles of music. If you’re VERY careful, you can do the same thing even on a fretted instrument such as a guitar or banjo.)

I like the idea that every A on an A string (except the fundamental) can be thought of as some ‘power’ of two, such as two squared (4), or two cubed (8), etc. To make this work, we have to use the Series of Partials rather than the Overtone Series, so from now on, for the rest of this post and all future posts, whenever I give the number of some step in the series, I’m going to be using the number as it exists in the Series of Partials. If your own preference is for the Overtone Series, just subtract ‘1’ from each number I use. (Doing it my way, for example, the partials used to make the major triad contained in every pitch are the fourth, fifth, and sixth partials. Whereas at the beginning of the post we said those notes came from the third, fourth, and fifth overtones. Both statements are correct, because one uses partials and the other uses overtones.)

To see how this works in real life, here’s the same sample series we used before, but this time with the powers of 2 shown, rather than merely the actual vibration frequency.


You can see here how by using Partials rather than Overtones, all the 'A' pitches can be shown as powers of 2 times the fundamental frequency (which in this case is A110). If you're not into exponents, you probably won't get what I'm taking about with that notion of 'powers,' but you can always look it up in Google or in Wikipedia and see what I'm talking about. Just enter 'mathematical powers' and all will be revealed.

As a parting shot, take a look at the example at the head of this post. This shows that the 4th partial of the A110 string is the same as the 3rd partial on the D146.67 string. (Don't let that decimal thing throw you. It's just 146 and two-thirds. You get that by dividing A220 by 1.5.) We'll get to these kinds of relationships in a later post, but I thought I'd get your musical juices flowing by showing the example here. Those kinds of same-note relationships between series based on different pitches will come into play in a later post regarding 'pivot tones.'  Please remember that term.  It's important!

Next post, we go back and revisit the 7th partial, the one we said wasn't important in Western music.  We'll discover why that is, and where it is important.

Post after next, we’re going to talk about Series once again, and this time explain why certain intervals are as important as they are when we discuss the foundations of harmony. (And that business in the top example about series equivalences will come into play.)


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