7th harmonic and scale length

Harmony & Theory
#1

Lots of you guys know this, probably, but for me this was news, so…

When I was first learning guitar I noticed there was one harmonic horribly out of tune. I talked about it with a friend who studied music but he said I must be wrong. That was still the 80s and no way to google anything.

Yesterday, with my newfound interest in all things music (thanks CtC), I was reading about overtones that sound together with the strings fundamental frequency and there it was: the seventh overtone is flat by a mile!

https://pages.mtu.edu/~suits/overtone.html

The interesting thing is that when you pick a string on a position of an overtone, that overtone is killed. It remains stationary while the string vibrates. So to kill the flat seventh, you must pick the string 1/7 of the string length from the bridge.

On a 25.5 inch scale that is 3.6 inches or 9.3 cm (93 mm) from the bridge. I checked my pick position and it is exactly at that spot, not even 1 millimeter wrong. This was on sixth string. Might be a bit more on the first ones.

Pickups, their placement and many other things understandably affect what gets heard, but if you consider playing 22.5, 27 or whatever inch guitar, there is a remarkably good, natural reason to use exactly the classic 25.5 inch scale.

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#2

I’m not so sure about this. I agree with the article you linked to, but I see some issues with your conclusion.

For one thing, do you actually hear a negative impact from moving your pick from that spot? Picking in different locations does perceptibly alter the resulting frequency content, but many people use that fact artistically. I do this a lot myself, moving the pick toward the bridge for a thinner sound and toward the neck for a fuller sound, and I haven’t heard that flat overtone ruining anything. (Maybe I just haven’t noticed it, but this overtone is present in every plucked string unless it’s specifically damped. A similar harmonic will also be present in all wind instruments, where it can’t be undamped. How bad can it be? Do all wind instruments sound bad?)

Also, each inch that you increase or decrease the scale length moves the node of the 7th harmonic by 3.6 mm, which is pretty tiny. If picking directly on the node is actually necessary, it should be very easy to appropriately adjust one’s picking position without major changes to one’s general technique, for a wide range of scale lengths.

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#3

Didn’t read the article. Anyway.
Equal Temperament (ET) which is used in modern instruments (like guitars) is a kind of compromise between sound quality and versatility. Disadvanatage of ET is that now there are no ‘clean’ or ‘natural’ intervals in the scale.
Which means that natural harmonics are out of tuning on a modern guitar. For example, natural third (ratio 5/4) differs from ET third (2^[4/12]) in about 0.7% which is audible to human ear.
The only exception is octave: natural eighth = ET eighth = 2/1.

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#4

I think there’s something to the OP’s idea. I’ve always noticed that on a Strat, the neck pickup is right on some big harmonics, and that cancellation of harmonics is what gives the neck pickup such a distinctive sound. I’m not sure of the science behind it.
I remember reading somewhere that a steel string gave false harmonics anyway. Whoever wrote this said a steel string was halfway between a string and a metal rod.
This is what piano tuners try to tame when they “stretch tune” a piano, especially shorter scale spinnet pianos, where the problem is worse than on longer pianos. So maybe a longer scale length for guitar is also desirable for this reason. Okay, now tear me apart. :grin:

#5

If you want to hear what a really flat seventh sounds like, listen to Terry Riley’s “Sri Camel.”

I think what he is using is called a “harmonic seventh.”

This kind of seventh is also used in some African musics, which influenced blues. Since the harmonic seventh was so flat, it did not produce a tendency to want to “resolve” like a Western ET seventh, so it got used in blues as the I, IV, and V. Ever notice how blues uses all dom7 chords, even on I? I think it’s a neat theory.

WIK: The harmonic seventh note is about ⅓ semitone (≈ 31 cents) flatter than an equal-tempered minor seventh. When this flatter seventh is used, the dominant seventh chord’s “need to resolve” down a fifth is weak or non-existent. This chord is often used on the tonic (written as I7 ) and functions as a “fully resolved” final chord.

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#6

No man, the 7th overtone is perfect. It’s the rest of your guitar that’s “out of tune”. :wink: As the other answer discussed, the reason for the “dissonance” of the 7th harmonic is because the guitar uses equal temperament so that you can play in any key without sounding “more” out of tune in some keys than others.

But barbershop vocalists deliberately uses pitches based on the actual harmonic series to make their harmonies sound sweeter. Check out more info than you want on the topic below:

And some smartasses in experimental music divide the octave into more than 12 pitches in pursuit of “less compromised” intervals. One even went so far as to say “why should we keep the octave perfect at the expense of other intervals?”:

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#7

Wikipedia high five! :wink:

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#8

But what does the OP mean when he says “the seventh harmonic is horribly out of tune?” We should be able to get both kinds of sevenths on an open string harmonic, 7:4 as well as 9:5, unless, as I mentioned earlier, steel strings are giving out false harmonics. ET would only apply to fretted notes.

#9

The 7:4 seventh arises early in the harmonic series, with nodes that equally divide the string into 7 equal parts, and even then, its amplitude is still quite low. Without doing the specific math, the fact that we know the first 7 open string harmonics are spoken for means that the 9:5 7th could only be a higher order harmonic, and would thus have even lower amplitude than the 7:4 7th.

(Amplitude limits based on what we can practically observe from making “equivalent” pickstrokes on different natural harmonics, which I guess is based on the ratio of node separation vs “muting fingertip” width: the higher order the harmonic, the more the string’s vibration immediately to each side of the node is impinged by the presence of the muting fingertip. Maybe there are more reasons as well, e.g. the fact that as the order of the harmonic is increased, the number of overtones muted is, by definition, increasing)

Edit: For giggles, I churned out a spreadsheet with the ratios for the first 2048 terms of the harmonic series. As mentioned before, the harmonic 7th comes in at 7:4, or coefficient of 1.75. The 9:5 7th can be viewed as a coefficient of 1.80.

The coefficient for 7:4 (1.75) is 0.05 less than the coefficient for 9:5 (1.80). In the first 2048 terms of the harmonic series, the lowest order terms whose coefficient over the preceding octave has an absolute difference from 1.80 of less than 0.05 have the following ratios:
29:16, (1.8125)
57:32, (1.7813)
59:32, (1.8438)
113:64 (1.7656)
115:64 (1.7969)
117:64 (1.8281)
225:128 (1.7578)
227:128 (1.7734)
229:128 (1.7891)
231:128 (1.8047)
233:128 (1.8203)
235:128 (1.8359)
(and a bunch of N:256 terms)

The closest terms in the first 2048 are:
1843:1024 (1.7998)
461:256 (1.8008)
921:512 (1.7988)
1845:1024 (1.8018)
1841:1024 (1.7979)
923:512 (1.8027)
115:64 (1.7969)

We could argue that 29:16 (1.8125) is a decent lower order approximation for 1.80 (since 9:5 over the fundamental or any of its octaves doesn’t occur in the harmonic series), but even 29:16 is high enough order not to be a natural harmonic we’d use in practice on the guitar.

Edit 2: From wiki, there is a another ratio, 16:9 (1.7778) also used for the minor 7th in just intonation. The closest low order ratio for a prospective open-string harmonic for that one is 57:32 (1.7813).

More on the minor 7th in just intonation here:

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#10

Is this the thread to discuss the Wolf Fifth?



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#11

They’re certainly not allowed in my household. :rofl:

“Sorry kids, only rhythm section instruments in this family. You can choose any instrument you want, as long as it’s guitar, bass, drums, or piano.”

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#12

I am not going to stroll too much deeper into this, I just was happy to realise that the sound is made of overtones, some very dissonant, to give the overall timbre of a plucked string. It does not make a string sound “bad” to have overtones out of tune, they are always there, but this is the reason why the hammer hits piano strings on 1/7 of the length. Guitar playing involves frequent use of open strings and it does not hurt to place your pick there.

I planned a project to create series of sine waves according to the harmonic series and check what comes out. But then this is the year 2018. I found the solution in under 10 seconds. It will be a saw wave:

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#13

well, you could just use fourier transform :wink:

#14

If we want to keep fifth (god of intervals) in our scale then 12 notes per octave is a good approach. Next would be 17 notes per octave, 29 is nice too ))

#15

Single sine waves hit the ears like an icepick when at a volume matching the perceived loudness of other harmonic rich sounds. Folks are going to want to watch their levels watching that video.

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