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What Does "Harmonic" Mean?

19K views 137 replies 15 participants last post by  MAXSWAGGER 
#1 ·
What, exactly and completely in all its senses, does the term "harmonic" mean?
 
#2 ·
No takers? Come on, live dangerously!
 
#5 ·
A harmonic is any member of the harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves. A harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is also called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz.
 
#6 · (Edited)
That's correct, but incomplete. What is Bwv 1080 missing?

Harmonic? Harmonic what? It's an adjective, isn't it, so it needs a noun to go with it.
So far, MacLeod is the only one to recognize the term "harmonic" as having meaning as something other than a noun.

Adjectives don't necessarily "need" nouns; they can be descriptive terms on their own, decdribing a quality that remains unspecified.
 
#7 ·
Furthermore, "harmonic" as a noun can be used as an adjective, with a meaning more connected to its meaning as a noun than with its more common adjective meaning. The term "harmonic model" is an example of this 'new' adjective meaning.
 
#10 · (Edited)
A harmonic progression is where the word is used as an adjective instead of a noun. It's a vertical arrangement of notes. ;)
Is it? I thought "harmonic progression" described a progression of chords, or a sequence. The "harmonic" adjective refers to "harmony," not any vertical entity such as an upper partial or overtone.
 
#11 ·
The harmonic series sums up terms like 1/k, where k is an integer.

A Fourier series sums up terms like sin (kwt), where k is an integer and w is the fundamental (k=1). k>1 are the overtones.

A harmonic oscillator is a classic problem in physics where a mass in subject to a force proportional to its displacement.
 
#16 ·
How does the human brain sense the arithmetical relationships in the intervals up to the major third? A child can do it.

At the minor third the uncertainty and sadness begins because the brain can't sense (calculate) the difference between 6 and 7 times the fundamental?
 
#17 ·
The brain senses things relatively, so the difference in thirds is minimal. I proved this to myself by constructing a wind chime set using the Thai scale: 7 notes equally spaced across an octave, called 7-note equal temperament.

When I played melodies on it, it sounded surprisingly diatonic, because it had seven notes.
 
#18 ·
I think Bwv 1080's definition is more than complete enough. "Harmonic" as an adjective should refer in some way to the concept of a frequency or frequencies that are a positive integer multiple of a fundamental frequency. It is the relationship of two or more frequencies that is (or isn't) harmonic.

In western music, there is a significant "fudge" factor in the concept of an harmonic relationship, as the equal-tempered scale has become standard, and although in that scale the "perfect" fifth is very close to a true harmonic fifth (just under 2 cents off), other intervals generally considered to be harmonic diverge much further. Our western ears have been trained to accept these as sounding "right", and traditional music of other cultures tuned to different scales typically sounds odd or dissonant to us. But it is worth remembering that there is nothing inherently natural or correct about equal-tempered harmonic intervals or relationships.
 
#19 · (Edited)
I think Bwv 1080's definition is more than complete enough. "Harmonic" as an adjective should refer in some way to the concept of a frequency or frequencies that are a positive integer multiple of a fundamental frequency. It is the relationship of two or more frequencies that is (or isn't) harmonic.
So, it appears you are prepared to deviate from the idea of the 'natural' harmonic series, and accept any positive integer multiple as a "harmonic." Some literalists would disapprove, saying that "harmonic" should properly only refer to the natural overtone series.

But that would exclude the commonly and frequently-used terms "harmonic progression" and "harmonic analysis," both of which refer to chord progressions and "harmony".

As I said in post #10, "harmonic" as an adjective in this case refers to "harmony," not any vertical entity such as an upper partial or overtone.

 
#21 · (Edited)
How is that any different from what I posted, or bwv 1080, for that matter? No big secret.

So, it appears you are prepared to deviate from the idea of the 'natural' harmonic series, and accept any positive integer multiple as a "harmonic." Some literalists would disapprove, saying that "harmonic" should properly only refer to the natural overtone series.

But that would exclude the commonly and frequently-used terms "harmonic progression" and "harmonic analysis," both of which refer to chord progressions and "harmony".

As I said in post #10, "harmonic" as an adjective in this case refers to "harmony," not any vertical entity such as an upper partial or overtone.
One of the ways verbal language works is that descriptive terms do extra duty, i.e., are given broader or comparable meanings. Western harmony is an outgrowth or development of the harmonic series, making special use first of the unison and octave, then of the fifth, and finally of the triad. Other scales are possible, of course, but the fifth is very useful because in the twelve-tone equal tempered scale, it corresponds so closely to the natural harmonic. The total "correction" required is pretty small, only about 23 cents. But the true 12-tone equal tempered scale is a very modern development in western music, as each step in the scale is higher than the preceding one by the twelfth root of two, and only relatively recently was it possible to calculate that.

So, harmonic progression and analysis, or harmony, as applied to the equal tempered 12-tone scale, still fall under the same general umbrella as the harmonic series. These are related ideas.

All of which you already know.
 
#26 ·
Yes, for literalists like you. Other music can create "harmonic models" by using scales. These scales can contain any notes, which do not need to reflect the harmonic series except in the way that they create an hierarchy of relations to the "fundamental" or key note of the scale.
Is anybody here capable of that level of abstract thought? A scale is a harmonic model.
 
#45 · (Edited)
Millionrainbows -- To use your terminology, "key areas" are established in western music not just by "root movement", but also by patterns and repetition. Rhythm, dynamics and timbre are also used by sophisticated composers to reinforce and complement the tonal system and point the way to the tonal center, often with great subtlety. It's true that over a span of six centuries or so, harmonic progressions played an ever-increasing and more dominant role in western music. But in the early 20th century, there was suddenly a bit of a roll back in that area.

Schoenberg famously demonstrated that music need not be based on establishing tonal hierarchies at all, and some composers followed that idea in many directions. But even composers who rejected that path understood that their music did not have to be dominated by root movement away from and towards a tonal center around the circle of fifths, or at least not as dominated by that principle as most western music had become by the late 19th century. Those composers rely more on structural elements such as patterns, symmetries and repetition, and elements of music not directly related to pitch such as rhythm, dynamics and timbre, to complement their use of harmonic progressions. They also often use dissonance more extensively, and in some cases microtones, pitch bending, and tonal manipulations made possible with modern technology.

To me, the most important leader of western music away from its slavish obsession with harmonic progression or what you call root movement was not Schoenberg, as important as he was, but Stravinsky, who was to music what his good friend Picasso was to visual art. He demonstrated that, while it is possible to abandon traditional western harmony entirely in favor of serialism (and later, musique concrete, indeterminacy, and other alternative systems), much could be accomplished by simply reducing, even if only slightly, the dominant role harmonic progressions had reached in western music by the late 19th century in favor of other elements, many of which could be made to work well with, and even enhance, harmonic progressions along the circle of fifths.

That is why so many contemporary composers owe so much, almost everything, really, to Stravinsky. Even composers who quite consciously took Schoenberg's path like Boulez (who was such a disciple he didn't think he could advance his own ideas without first proclaiming "Schoenberg is dead!") very much show the influence of Stravinsky.
 
#46 ·
Millionrainbows -- To use your terminology, "key areas" are established in western music not just by "root movement", but also by patterns and repetition. Rhythm, dynamics and timbre are also used by sophisticated composers to reinforce and complement the tonal system and point the way to the tonal center, often with great subtlety. It's true that over a span of six centuries or so, harmonic progressions played an ever-increasing and more dominant role in western music. But in the early 20th century, there was suddenly a bit of a roll back in that area.
Harmonic progressions, as strictly defined by CP Western harmony, are derived from the major/minor scales. If a modern composer deviated from that, and used other 'exotic' scales, diminished scales, whole tone, and the kind of things you might see in Slonimsky's Thesaurus of Scales, I don't see what's so "innovative" about that, except that it has expanded the possibilities from the limitations of the CP major/minor system. The principles I have outlined would still apply to those scales which lie outside the purview of CP tonality, and that's why I think they're important, since they are harmonic principles which can apply to areas well outside CP harmonic practice.

To me, the most important leader of western music away from its slavish obsession with harmonic progression or what you call root movement was not Schoenberg, as important as he was, but Stravinsky, who was to music what his good friend Picasso was to visual art. He demonstrated that, while it is possible to abandon traditional western harmony entirely in favor of serialism (and later, musique concrete, indeterminacy, and other alternative systems), much could be accomplished by simply reducing, even if only slightly, the dominant role harmonic progressions had reached in western music by the late 19th century in favor of other elements, many of which could be made to work well with, and even enhance, harmonic progressions along the circle of fifths.

That is why so many contemporary composers owe so much, almost everything, really, to Stravinsky. Even composers who quite consciously took Schoenberg's path like Boulez (who was such a disciple he didn't think he could advance his own ideas without first proclaiming "Schoenberg is dead!") very much show the influence of Stravinsky.
Stravinsky was an innovator, without a doubt, and I think there were also many other innovators, like Slonimsky for his work in scales.
In Boulez' defense, I think he saw Schoenberg as an impediment to musical progress in the serial area, since Schoenberg openly acknowledged that he was a conservative in the late Romantic tradition. Boulez saw the possibilities for expansion of Schoenberg's direction, whereas Schoenberg's time was over, and he was, indeed, dead by this time.

I don't see Stravinsky as being as great an innovator as you do. Schoenberg and Boulez' ideas were more directed away from harmony itself, and in that sense are more radical than Stravinsky, who remained an expanded tonalist/harmonicist until the time of his dabbling with serialism in "Mouvements."

So I see all "harmonic" music as being tonal, or as using principles of tonality, and these are all traceable to "harmonic models" or scales of one kind or another. It doesn't matter if it's Stravinsky or jazz or forms of ethnic music. I trust you see what I am "after" now.
 
#64 · (Edited)
Many scholars argue that the Well-tempered Clavier was not composed with equal temperament in mind. Some suggest Bach intended the use of Werckmeister III, where the Pythagorian comma is equally distributed over four neighboring fifths (rooted on C, G, D and B).
I'm going with the ideas of Dr. Bradley Lehman (larips.com) who deciphered Bach's well-tempered tuning from a decorative flourish on the cover page of the original manuscript. It was basically an attempt at ET, in which all keys sounded good.

...but the piano is designed to play equal-tempered, diatonic scales, most easily C major and nearby keys.
What you say is true, especially in light of key signatures, but Chopin later used distant keys like A flat because of ergonomics: the hand sits naturally on the piano when the middle fingers are covering black keys, and the thumb and pinkies cover white keys. He said that C major was the most difficult scale to play because of the thumb cross-under, and started his students out with other scales like A flat. See book "Natural Fingering."
 
#70 ·
I'm going with the ideas of Dr. Bradley Lehman (larips.com) who deciphered Bach's well-tempered tuning from a decorative flourish on the cover page of the original manuscript. It was basically an attempt at ET, in which all keys sounded good.

What you say is true, especially in light of key signatures, but Chopin later used distant keys like A flat because of ergonomics: the hand sits naturally on the piano when the middle fingers are covering black keys, and the thumb and pinkies cover white keys. He said that C major was the most difficult scale to play because of the thumb cross-under, and started his students out with other scales like A flat. See book "Natural Fingering."
Yes, I read about that analysis of the cover page of the WTC manuscript. Very interesting. I was really only making a general statement that while equal temperament was around by the early 18th century, by Beethoven's time it was well on its way to becoming standard, and the dominance of the piano from the mid-19th century on helped seal the deal. And all this isn't some amazing new insight by me. Historians know all about it.

It's appropriate that you mention Chopin. He was a pioneer in expanding piano technique, and that included playing ferocious virtuoso passages and entire pieces in any key. And in fact, one of the most important, if not the most important, features of equal temperament is that every interval is identical in every key. So any or all 12 keys can be used, in the same piece even, with no odd sounding intervals or chords. Again, no special insight by me. I would think it's obvious how this has had a profound impact on western music and how we hear it.
 
#48 · (Edited)
for the math people in here, here’s what I get;

derived from the C fundamental with A = 440 hz

when they're tempered to be in the equidistant series —— and then from Nature

196.22 G 196

327.03 E 329.63

392.44 G 392

457.84 A 440

457.84 Bb 466.16

588.66 D 587.33



19th Nervous Breakdown - it’s surprising to me that these notes below are so close

1242.72 Eb 1244.51


---------

the rest of the higher notes in the harmonic series included below

654.06 E 659.26

719.47 F 698.46

784.88 G 783.99

850.28 Ab 830.61

850.28 A 880

981.1 B 987.77

1111.91 C# 1108.73

1177.32 D 1174.66
 
#67 ·
I'd like to hear serial music played in 43-tone tuning.
 
#69 ·
Many developers for Kontakt (and similar engines) absolutely always autotune the samples, because many customers (check vi-control forum for many such threads from the last decade) complain about unplayable keys. And many of these libraries record the best orchestral musicians available (Vienna, London, L.A.).

Something funny:

""
 
#74 ·
Many developers for Kontakt (and similar engines) absolutely always autotune the samples, because many customers (check vi-control forum for many such threads from the last decade) complain about unplayable keys. And many of these libraries record the best orchestral musicians available (Vienna, London, L.A.).
That's fascinating info about auto-tune.
 
#88 ·
438 to 440 hz is approximately 7.88 cents.

Maybe you will find this interesting:

"Music production applications

In music production, a single change in a property of sound which is below the JND does not affect perception of the sound. For amplitude, the JND for humans is around 1 dB (Middlebrooks & Green, 1991; Mills, 1960).

The just-noticeable difference (JND) (the threshold at which a change is perceived) depends on the tone's frequency content. Below 500 Hz, the JND is about 3 Hz for sine waves, and 1 Hz for complex tones; above 1000 Hz, the JND for sine waves is about 0.6% (about 10 cents).[3] The JND is typically tested by playing two tones in quick succession with the listener asked if there was a difference in their pitches.[4] The JND becomes smaller if the two tones are played simultaneously as the listener is then able to discern beat frequencies. The total number of perceptible pitch steps in the range of human hearing is about 1,400; the total number of notes in the equal-tempered scale, from 16 to 16,000 Hz, is 120"

https://en.wikipedia.org/wiki/Just-noticeable_difference

Still, all this is irrelevant to harmony discussions. Such small intervals are considered useless in actual music compositions, that's why equal and unequal temperaments are used.
 
#89 ·
438 to 440 hz is approximately 7.88 cents.

[ .... ]

Still, all this is irrelevant to harmony discussions. Such small intervals are considered useless in actual music compositions, that's why equal and unequal temperaments are used.
Thanks, that was a very interesting and useful post. However, these "small intervals" aren't entirely irrelevant once we start playing around with these other scales, especially those that aren't equal tempered, in actual musical compositions, right? That's why I was interested in your opinion of the Well-tempered Clavier played in a non-equal tempered tuning like Werckmeister III.
 
#92 · (Edited)
I was tested, with another guy, and we both could hear a difference of 2 cents, on fairly high-pitched wind chimes, probably A=880. The difference was identifiable, but it really wasn't pitch I was hearing; it was that the higher note sounded ever-so-slightly "brighter."

I got to do all kinds of experiments when I was working at the chime factory. I cut a 17-note per octave Arabic chime, a 7-note ET chime (which is Thai tuning), A "harmonic" bass tuning which had a harmonic seventh, just fifth and third and some kind of second, a solfeggio tuning, a septimal whole tone scale based on a septimal second, all sorts of stuff.
It was valuable, because it showed me that sound is linked to physics, arithmetic, and actual materials.
 
#94 · (Edited)
One advantage of creating other ET octave divisions on a normal keyboard (by modulating/"stretching" the keyboard voltage) is that you can see some connections with normal tuning. In 19-tone, tune the octave past the "C to C" 12-note ET octave to "C-G", the G past the octave C. If you compose melodies in 19, then convert them back to 12, it gives a diminished tonality.

!7 note ET is octave on F; 22 tone ET is C-Bb (flat seven).

You begin to see that these other ETs are based on overtones of the 12 ET, and that is why certain ET tunings are favored, such as 17 (C-F fourth), 19 (C-G fifth), 21 (C-A relative minor), and 22 (C-Bb flat seven).
 
#95 · (Edited)
Well, BabyGiraffe, I guess I and a lot of people I know have exceptional hearing, as we can easily tell the difference between equal temperament and various forms of just intonation in the right contexts. In his Well-tempered Clavier, Bach wrote a prelude and fugue for all 24 major and minor keys of the 12-tone scale. As only equal temperament gives you identical intervals and chords in every key, the differences there are especially clear. Have you listened yourself?

Or take a famous traditional Japanese melody such as Sakura (cherry blossoms). This uses a pentatonic scale roughly similar to the Phrygian mode. Compare a recording made with traditional Japanese instruments such as the koto with a westernized recording. This should be very easy to do. The famous French flutist Jean-Pierre Rampal made a series of recordings of traditional Japanese melodies, including Sakura, all in westernized arrangements with equal-tempered tuning. I think in most the flute is accompanied by a western harp, although the koto and other instruments are also used. Listen and compare:

 
#96 ·
...In his Well-tempered Clavier, Bach wrote a prelude and fugue for all 24 major and minor keys of the 12-tone scale. As only equal temperament gives you identical intervals and chords in every key, the differences there are especially clear. Have you listened yourself?
In the case of Bach, who used his own "ET"-ish tuning, there may be slight differences in fifths and thirds which influenced the way he actually composed each prelude/fugue for the WTC. For example, if the key had a really good sounding major third, he might emphasize that note more. If it had a bad minor third, he might not linger on that note, etc.
 
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#102 ·
This tome should clear up some questions about the Pythagorean method and scales, and clarify things a bit.

View attachment 126142
 
#103 ·
Or this one I just found, which is written in a more non-technical, non-mathematical style, and really only summarizes and simplifies what musicologists have written about with great precision and in great detail, but looks like a fun read for those who don't want to get too technical. He discusses what I was trying to talk about, i.e., the imperfections of equal temperament and why it isn't necessarily the best-sounding scale in all circumstances, but he also gives more general information. I do not go along with the idea that equal temperament "ruined" harmony, and I don't think the author does either. He's just trying to sell books by using a provocative title.

Musical instrument Guitar accessory String instrument accessory Font String instrument
 
G
#108 ·
You gain and you loose. In equal temperament you can use any key and so a broader range of modulation is available, but you lose the unique "sound" of each key. Probably we should use whatever tuning the composer was familiar with, if only we knew. Wouldn't it be wonderful if Bach had told someone how he liked his harpsichord tuned? :)
 
#109 · (Edited)
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