The Harmonic Series

The basis of all musical harmony can be found within the Harmonic Series.

Harmonic Series on C (First 10)
Harmonic Interval Note Cents
4thDouble OctaveC″
6thDominant's OctaveG″+2¢
7thSeptimal SeventhBb″-31¢
8thTriple OctaveC‴
9thMajor ToneD‴+4¢
10thMediant's OctaveE‴-14¢

1st Harmonic (Western Root)

This is the fundamental frequency of oscillation. In western music theory this is called the "Root".
The word root is a metaphor for a tree where the fundamental frequency is the tree roots and all other frequencies are the branches growing out of the trunk. In physics it is much more common to just call this the fundamental frequency.

2nd Harmonic (Western Octave)

The relationship between a fundamental frequency and its 2nd harmonic (first in-tune overtone) is so strong that most cultures perceive it as being of "sameness". This strong harmonic relationship combined with the human psycho-acoustic phenomenon of the Missing Fundamental allows for the 2nd harmonic intervals to be treated as the same as the fundamental.

This is called Octave Equivalency and while this is not found in all cultures, it is more of a norm across many cultures and not the exception.

3rd Harmonic (Western Perfect Fifth)

The relationship between the fundamental frequency and its 3rd harmonic is very strong. This is considered the first true new tone in the western musical systems. In most cultures you will find this interval used. Because of it's strong quality it is called the Dominant.

Most instruments and systems of oscillation will produce this overtone. Also given that it's only a multiple of 3 it shows up well within the hearing range. For example for the note A4 = 440hz the 3rd harmonic will be E6 = 1,320Hz which is the second highest E on a grand piano.

4th Harmonic (Western Double Octave)

An interesting repeating mathematical phenomenon begins to happen here at the 4th harmonic.
If the 2nd harmonic was an Octave. Because any frequency that is double has such a strong relationship most cultures considered it the same. What happens when you take the second harmonic of the second harmonic (2 x 2 = 4)? You get a double octave.
All powers of two (1,2,4,8,16...) are octaves when you use octave equivalence as a rule.

5th Harmonic (Western Major Third)

This harmonic is something of huge interest in modern western music harmony.
In the western music system the 5th harmonic is the center piece of music theory and provides the basis for chordal harmony. The major chord is the first 5 overtones stacked and arranged in varies voicing orders.

The 5th harmonic is seen as having a bright but stable harmonious sound that we call a "Major Third".

6th Harmonic (Western Perfect Fifth's Octave)

Using the previously established rule of octave equivalence. This harmonic is twice the frequency of the (Dominant) 3rd harmonic (3 x 2 = 6). So the 3rd harmonic being its own unique tone called the Dominant. The 6th harmonic is the 2nd harmonic of the 3rd harmonic, therefore the Dominant's Octave.

7th Harmonic (The Forgotten Seventh)

Things start to get interesting here. The 7th harmonic is not accurately played in the modern western tuning. But is a timbral quality that show up in a lot of instruments. Once very popular in folk and early music. You only hear an approximation of this interval in modern 12EDO Equal Temperament. The harmonic Septimal 7th chord is sometimes called the lost or forgotten chord in music theory that got dropped when we moved to equal temperament.

It is the basis for the Mixolydian church mode which is found in world, folk, jazz, blues and rock music. It is likely one of the oldest musical scales.

It is commonly mislabeled as a Dominant 7th interval. But that is only correct when it is used on the dominant chord because in the modern major scale the dominant scale degree has a subdominant a whole step under it. (996 cents VS 969 cents)

8th Harmonic (Western Triple Octave)

The second harmonic's, second harmonic's, second harmonic... (2 x 2 x 2 = 8)
Hello again octave equivalence.

9th Harmonic (Western Major Second)

The next harmonic is known as the Major Second in western music. This harmonic is also the Dominant's Dominant, or 3rd harmonic's 3rd harmonic. (3 x 3 = 9)
You find this harmonic used in most musical scales either by using the 9th harmonic of a fundamental (as in Mixolydian and Lydian), or more commonly in modern music by using the 3rd harmonic of the dominant (as in the major scale). The 9th harmonic is commonly described as having an uplifting bright quality but historically was considered dissonant when used in harmony because of its closeness between the Octave and Mediant (Major 3rd).

10th Harmonic (Western Major Third's Octave)

The next harmonic and last on this list is the Major Third's Octave or 5th harmonic's 2nd harmonic. (5 x 2 = 10)
Hello again octave equivalence.

Rules of thumb

Unique ratios from the harmonic series start out consonant and become progressively more dissonant as you go higher in the series.

Imposing limits on dissonances

Because odd order harmonics get more dissonant and closer together as we go up the series.
Most of modern music theory is build around using just the first 5 harmonics as a Major Triad. Example: (C), G, E
Notes not in the first five harmonics are merging in from neighboring harmonic series to get more variety. For example in the Key of C major. You can merge in the series on the 3rd harmonic (Dominant G) and get the notes (G), D, B.
Or you can work backwards and use a new harmonic series that includes your old fundamental as a harmonic. Like for example using the (Subdominant F) series and get the notes (F), A, C.

The note F is not part of the C harmonic series. BUT!!! C is the 3rd harmonic of a series built on F. So they are related. They are called musical relatives. C is a mathematical subset of F. Another way of looking at this relationship is in family terms. C is a child and F is the parent, likewise G would be the grandchild. (F -> C -> G)

This is also where the term Relative Minor comes from. The chord A minor is the relative to C major. They aren't from the same harmonic series. But share a common parent. The F harmonic series.

How many harmonics?

Even though modern music theory stops at just the first five harmonics. You'll find historical and ethnic scales that use more of the series. For example the Mixolydian mode includes the 7th harmonic commonly miscalled as a "natural/flat/minor/dominant" 7th. Those terms are incorrect though and each of those have their own slightly different pitch and function.

Some exotic music scales include the 11th harmonic. It is approximated as an Augmented 4th in our 12 tone equal temperament system. Though this is also a labeling quirk as the modern notation system starts to break. The 11th harmonic is almost exactly dead middle of the perfect 4th and the augmented 4th. Sometimes called a semi-augmented interval or undecimal tritone. It is heard in the Lydian ♭7 scale also called the Acoustic Scale.

The upper limits

A violin playing a low G3 (196hz) has noticeable harmonics all the way to 10khz which is the first 50 harmonics! Even though the highest harmonics are more then -24dB attenuated in amplitude as compared to the fundamental. Rolling off the highs on a Violin above 8khz and you will notice the sound as being "less bright" or "muddy".

After doing some experiments I was able to hear the 50th harmonic 10khz even when it was -48dB down from the fundamental. That translates to it still being audible even at 0.4% the amplitude.

Also remember while the 50th harmonic without any context would seem like a really high pitch and above the range of human hearing... In reality it is only an octave about the 25th harmonic. Notice how the intervals between harmonics get smaller as you go up the series:

━━━━ Harmonic Series Graphed ━━━━
│                6  ...          
│             5 ┏━━┛ Minor Third 
│            ┏━━┛ Major Third    
│          4 ┃                   
│         ┏━━┛ Perfect Fourth    
│         ┃                      
│       3 ┃                      
│      ┏━━┛ Perfect Fifth        
│      ┃                         
│      ┃                         
│    2 ┃                         
│   ┏━━┛ Octave                  
│   ┃                            
│   ┃                            
│   ┃                            
│   ┃                            
│   ┃                            
│ 1 ┃                            
│━━━┛ Unison                     

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