Harmonics

What are harmonics? Well, according to a reputable university website that is heavily based on music and science it means, "The spectrum of a single note from a musical instrument usually has a set of peaks at (approximately) harmonic ratios. That is, if the fundamental frequency is f, there are peaks at f, and also at (about) 2f, 3f, 4f, etc. For many, but not all, instruments, these ratios are close to exact: the second partial is at 2.00 times the first, etc." (Wolfe, 2006).

According to a simple google search, it can mean,

(Harmonics, 2019)

Harmonics are overtones. When a string vibrates or when a key is pressed on an organ, the resulting sound consists of a fundamental frequency plus a series of overtones. While the note consciously heard by us is fundamental, these overtones can subtly clash with notes but enough to cause the music or sound to be better or not in a given context. For example, if I isolated and amplified the 7th harmonic of a note and someone played a minor 7th plus 3 octaves above my original note, it can produce a beating effect.

These overtones are not always easy to pick out with our ears unless we train extensively. People have made music solely based on the harmonic series, and it sounds fascinating. For example, Tuvan throat singers make all the different harmonics of a single low note "stand out" one at a time, so clearly that you can hear a melody in them. A few Western singers have also sung using the overtone series (Paul Erlich, personal communication, June 18th, 2019).

Overtones are not always precisely harmonic. There are instruments whose sounds have overtones that are pure harmonics – integer multiples of the fundamental frequency – through slip-stick or periodic motion, such as bowed instruments, the human singing voice, or instruments that you blow through using a reed or lips to drive the vibration. However, plucked-string instruments are slightly inharmonic, and percussion instruments are very inharmonic (thus, it is challenging to figure out the "pitch" of a drum kit) (Wolfe, 2006).

Can one play overtones individually on an instrument without hearing the fundamental or other overtones at the same time? Yes, for example, rock and metal musicians use the so-called "pinch harmonic" technique very often to lend a little 'spice' to their soloing or another section that might be a prominent part in a song.

Here is a diagram of the modes of vibration of a string:

(The nodes of a vibrating string are harmonics, 2010)

As shown at the top of the diagram, the fundamental corresponds to a movement of the entire string up, then down, then back up, and so on. The overtones, shown below that, correspond to vibrations of fractions of the string in alternating directions. A string that is plucked or struck to vibrate will execute a combination of most of these vibrations at the same time, generating an overtone series of frequencies of sound. What are the “notes” that the overtone series consists? Are they the same as the fundamental frequencies available on our instrument? The answer is often “not really.”

For simplicity, I will neglect the inharmonicity of the overtones of the string – which one can ignore for an infinitely thin or zero-stiffness string. Therefore, I will refer to the overtones as harmonics.

The notes below show the first sixteen harmonics of the pitch C2; the blue numbers above show how much they are “off” by compared to the 12-edo system on most modern instruments (as explained in my previous blog).

(Giedraitis, 2018)

The harmonic series can go infinitely high, but after a certain point, we stop being able to hear it unless we octave reduce it. For example, the interval from the fundamental to the 700th harmonic is 11341.4533 cents (more on how I calculate this later), but it’s often much too high for me to hear. However, if I bring up the fundamental in octaves (powers of 2) -- 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16 all the way to 2^9 = 512 since 512 is below the 700th harmonic -- then 700:512 would be = 541.453334 cents, an interval we can hear. Octave-reduction is also a way of separating the interval into some integer number of octaves plus some smaller interval less than an octave, for easier comprehension. 512 = 2^9, so the 700th harmonic is 9 octaves plus 541.453334.

For some choices of fundamental, the 700th harmonic is still audible: 25 Hz * 700 = 17500 Hz, which is audible. However, if we had our fundamental at 440 Hz, then the 700th harmonic is 440 * 700 = 308000 Hz, too high to be audible by humans. However 440 * 700/512 = 601.5625 Hz, something we can hear.

(Table of Harmonics, 2009)

[On the orange line of the G string that is below “1/4”, the note should be G and not E].

These are the notes that occur in the first 16 harmonics, the blue numbers above show how much they are “off” by compared to 12-edo (as explained in my previous blog). Some musicians have gone uptown 128 harmonics for their music, Johnny Reinhard.

"Harmonic timbres" are musical instrument sounds where a single tone's spectrum is a harmonic series. If you use such a timbre, then 3:5 is the tuning of the major sixth that "locks in" by ear and seems most "blending," "smooth," and "beatless." That's why people who study tuning call it a "pure" interval. The term is generally applied to intervals where the two frequencies form very simple ratios with one another. Say, for example, you have a saxophone and bowed violin. On the sax, play the note that is exactly in tune to the third harmonic of some reference note, and on the violin, the note that is exactly in tune to the fifth harmonic of the same reference note. If you play these two notes at the same time, you create a "pure major sixth" which is 884 cents (Paul Erlich, personal communication, June 18th, 2019).

Each tone's spectrum is a harmonic series, as discussed earlier. Under these conditions, the two instruments, if they play two notes forming a simple-integer frequency ratio, produce an interval with the qualities referred to as "pure." If the interval is off from pure tuning, you will hear beating, and the beating will slow to a stop as you tune to interval to pure (IF your timbres are harmonic). In the example above, the 5th harmonic of the saxophone and the 3rd harmonic of the violin will beat against one another unless the major sixth they are playing is tuned purely (Wolfe, 2006).

What makes different instruments sound different? Mainly, their spectra -- which overtones are present and how loud they are relative to one another, and how the loudness of each overtone changes through time (Wolfe, 2006).

The bass guitar (my instrument) timbre is almost harmonic but not precisely (stretched harmonic series) since it is a plucked string instrument. Moreover, as also discussed earlier, instruments that require blowing such as woodwinds or brass instruments have harmonic timbres. If you play a single note on a woodwind or brass instrument, you usually get a set of frequencies that are exact multiples of the fundamental frequency. All these frequencies sound at once, and you hear it as a single note partly because our brains are designed to listen to a harmonic series as a single sound; this is the case partly because we evolved to listen to human voices, which are also harmonic-spectrum timbres. The inharmonicity of bass or piano overtones is not sufficient to destroy the sensation of a single note, but for bells and gongs, one gets the sensation of multiple notes, and for drums, there is sometimes no clear sensation of note at all (Wolfe, 2006).

Harmonics can go much deeper with many physics that include, resonance, inharmonicity, instruments that we can think of that are inharmonic or harmonic in nature, stretch and equal harmonics. The pages from UNSW have helped me understand, learn, and educate others on the topics of sound. I will be adding plenty of resources in my appendix for this blog, be sure to check it out. I was also helped by my friend and mentor, who is a fantastic modern-microtonalist and physicist, Paul Erlich.

REFERENCES:

Giedraitis, K. (2018). The harmonic series in C [Image]. Retrieved from http://www.tallkite.com/AlternativeTunings.html

Harmonic. (2019). Retrieved from https://www.encyclopedia.com/reference/encyclopedias-almanacs-transcripts-and-maps/harmonic

Table of Harmonics [Image]. (2009). Retrieved from https://upload.wikimedia.org/wikipedia/commons/2/21/Table_of_Harmonics.svg

The nodes of a vibrating string are harmonics [Image]. (2010). Retrieved from https://upload.wikimedia.org/wikipedia/commons/2/2f/Moodswingerscale.svg

Wolfe, J. (2006). How harmonic are harmonics? Retrieved from https://newt.phys.unsw.edu.au/jw/harmonics.html

APPENDIX:

http://newt.phys.unsw.edu.au/music/

https://microtonal.m.miraheze.org/wiki/Harmonic

https://newt.phys.unsw.edu.au/jw/Bows.html

https://newt.phys.unsw.edu.au/jw/sound.spectrum.html

https://newt.phys.unsw.edu.au/jw/strings.html#music

https://newt.phys.unsw.edu.au/jw/voice.html

https://www.youtube.com/watch?v=GpfHfZvYZEM (Canto Difonico Tuvano - Dag Kargyraa (Throat singing - sub-harmonic singing))

https://www.youtube.com/watch?v=UHTF1-IhuC0 (Polyphonic overtone singing - explained visually)

https://www.youtube.com/watch?v=V76psBrEypg (Tuvan Throat Singing | Alash | TEDxBaltimore)

https://www.youtube.com/watch?v=vC9Qh709gas (Polyphonic overtone singing - Anna-Maria Hefele)