# Notes: Pitch, the Harmonic Series, Tuning, and Timbre

This post has little to do with tango music, specifically. Yet, on a fundamental basis, it has everything to do with tango music; everything to do with all music, past and present, in fact. Such is the relevance and importance of the naturally occurring phenomenon called the harmonic series.

The harmonic series is the foundation for ALL the world’s music, in its vast variety and richness, today and in the past. There isn’t any pitch in any music system in any culture that isn’t found in the harmonic series. It just might be informative to learn something about it.

I hope to explain some of the physics of sound, how pitches originate in nature, why Western music uses seven note scales, why there are twelve unique notes, why octaves sound the same, why the tonic (the first note in a scale) and dominant (the fifth note in a scale) and subdominant (the fourth note in a scale) are the three most important notes in our scales, and why the triad is the basis of harmony. I also explain why and how pitches are adjusted, that is, tuned, to enable music to be written in any and modulate, that is change, to any other key within the same piece of music. And finally I briefly explain why the same note (ie. pitch)  played by different instruments or sung by different singers has unique sound qualities. While the pitch is the same, the timbre, the sound quality is different. The harmonic series underlies everything we hear in music.

(I mention modes, which were used for hundreds of years in Western music prior to the diatonic scale system, only in passing, because we have lived in a tonal musical world for three hundred years. Although Art (“classical”) music moved beyond tonality in the late 1800s, all popular music is firmly rooted in the diatonic scale system. Although some forms use modal elements. Nor do I discuss other world music systems, again, except in passing).

Now it certainly isn’t necessary to understand the physics of a vibrating string or air column, or the resulting harmonic series to appreciate music. (But I’m going to explain them nonetheless). Music being aural, musicians composed and played whatever sounded “good” or “pleasant” to their ears. Music was created in accordance with the concepts of consonance and dissonance as perceived at the time and in that cultural environment. It still is. However, the notes, more accurately their pitches and the way they are used singularly and in combination, is rooted in the harmonic series; whether musicians or listeners are aware of it or not.

The single most distinguishing quality of s is their pitch. Most people can accurately judge if a certain pitch is higher or lower than another and identify a single pitch as “high”, “low” or somewhere in between. Many people can tell if a note is in tune or not, based on those sounding before or after it, and can sing melodies from favourite music in tune. Fewer people have relative pitch. Given a pitch they can sing or hear any interval (the distance between two notes) above or below it. Very few have absolute or perfect pitch: the ability to hear a pitch and identify it by name, and know if it is in tune, sharp, slightly higher, or flat, slightly lower, than it should be. Those with absolute pitch can also sing a specific pitch in tune, given its name. The best “ears” can distinguish pitch in microtones, pitch frequencies between semitones. Musicians have absolute or perfect pitch.

What does “high” and “low”, or “in tune” mean?  Low pitches have lower frequencies, meaning they vibrate at a slower rate than higher pitches. Of course we don’t see vibrating sound waves when we listen to music but that natural phenomenon causes us to hear pitch differences and allows us to name and notate them. The staff system (the five lines and four spaces between them) arranges notes in ascending pitch order from the bottom line to the top. It is a visual representation of sound frequencies, inherent in the notes placed on the lines and spaces.

Explaining “in tune” and “out of tune” is more difficult. A note is in or out of tune relative to other pitches used in the melody or the harmonies. When musicians tune up before playing there is a common note whose frequency is agreed upon. All other pitches are referenced to that one. Pitch, the specific frequency the note is tuned to, varies somewhat from country to country and historical period. Today musicians in many parts of the world tune to A at 440Hz.

Tuning is more complicated than that, though. Exactly how are other notes in or out of tune compared to the reference note, and to each other? All notes, their pitches, are tuned in some consistent, systematic manner. The tuning system called equal temperament is the method used for the past 200-250 years, and I’ll have more to say about equal tempered tuning further on.

Pitch is a tone produced by a vibrating string (eg. piano, guitar, violin), a column of air (eg. wind instruments), a stretched hide or membrane (eg. certain drums), or other methods where distinct tones can be sounded.

The simplest example to visualize is a vibrating string, which is fixed at both ends and under tension. When set in motion the string vibrates along its full length, from end to end, in an oscillating sine wave. The tone produced is the fundamental pitch and frequency we hear. Above the fundamental are numerous (theoretically infinite) other wave lengths, each with a unique frequency.

Observe the illustration below. (It is a simplification but correct in the essentials). Looking at the oscillations, notice each is an integer step above the preceding one (actually, the illustration uses reciprocals of the integer: 2/1 written as 1/2). A string vibrates in 1, 2, 3, 4, etc. equally sized segments, and does so simultaneously. Successive wave length oscillations resonate at increasingly higher frequencies, being 2, 3, 4, etc., times the fundamental frequency.

Each pitch we hear and identify as a single note is actually a composite of the frequencies produced by the fundamental and each of the shorter oscillating string lengths. The harmonic series is the ascending order in which these frequencies occur.

As noted, each pitch is a composite of many other frequencies. These other pitches are called harmonics or harmonic partials by scientists and overtones by musicians. (There is a numbering difference. In harmonic terms the first note, the fundamental tone, is the first harmonic partial. Using overtone terminology the second tone is the first overtone.) We predominantly hear the fundamental tone, the note being played. But harmonics are very much part of the sound we hear, the pitch and the resulting timbre.

If you have access to an acoustic piano the existence of harmonics is easily verified. (Do not use the sustain pedal). Lightly press and hold the down – not enough to sound the note – C an octave below middle C. Then play C an octave below that one, pressing the key quickly and firmly and instantly releasing it. The upper C will clearly ring. Next, silently press down the G below middle C and strike the low C again. The G will clearly ring. That note is an octave and a perfect 5th above the fundamental (or, a perfect 5th above the second harmonic, the C below middle C previously pressed down).  Silently press down middle C, the G and C an octave below middle C. Strike low C and all three held down notes will be clearly heard. These three notes are the first 3 overtones above low C. Now add the E above middle C. That will require a third hand, so just hold down the G, middle C and the E above middle C.  Strike low C and the notes G-C-E will ring. Notice the notes form a major triad. This triad on the fundamental (the tonic in scale terms) is the basis for the major scale and harmony. And it is built into every individual tone produced.

As we observed with the vibrating string, harmonic partials progress in an ascending integer series. Each successive harmonic partial’s frequency is an integer multiple of the fundamental frequency (f): 1f-2f-3f-4f-5f, etc. The frequency for low C (C2) in the following example is 65.41Hz, given A = 440Hz. The second partial  is another C, exactly an octave higher, at 2 times the fundamental frequency, 130.82Hz. The third partial is a G, 3 times the fundamental at 196.23Hz. And the fourth partial is another C, two octaves above the fundamental and 4 times it’s frequency, at 261.64Hz.

 Note Multiple Hertz C2 1f 65.41 C3 2f 130.82 G 3f 196.23 C4 4f 261.64

The first four harmonics on low C (C2):

Before moving on to the next few pitches in the series, I should say a bit about the octave.

Notice the first overtone (the second harmonic partial) is an octave above the fundamental. It is exactly double its frequency: 2f. That is an important observation. Notes are repeated exactly an octave above on each doubling of the integer multiplier, in an exponential series: 1f, 2f, 4f, 8f, 16f, etc. Notice the fundamental note, the C, occurs three times in the first four partials. That fact, and the exponential relationship between octave frequencies are the reasons we hear notes separated by an octave (or multiple octaves) as the same note.

It might take an untrained ear some time to hear octaves as the same note. They are the same note in a different register. One should hear octaves as “higher” or “lower”, but not “different”. That is why they have the same letter name, which appears to break the rule that each note has a unique frequency, but doesn’t really. We add an octave number to the name. For example, the lowest C in the example below is C2, then C3, middle C is C4, followed by C5, and the highest note in the example, C6.

C in four octaves, from C2 through C6:

Octaves, especially when several of them are sounded simultaneously as above, have a hollow sound – like something is missing. Compare them to the next sample (a couple paragraphs down), which has a perfect 5th and major 3rd: a major chord.

Referring back to the first four harmonics, notice the G, an octave plus a perfect 5th above the fundamental. (Called a compound perfect 5th because of the octave separation). It is the only unique note in the first four partials. That’s important. In our tonal scale system, the first note, called the tonic, and the perfect fifth above it, called the dominant, are the two most important notes. These notes and the triads (chords) built on them establish the tonality and create movement and tension-resolution qualities. The relationship between these two notes is the basis of tonal music and functional harmony. That cannot be overstated.

Looking at the next harmonic partial, it is a new note, an E. The E is two octaves plus a major 3rd above the fundamental C. (Again, called a compound major third, this time separated by two octaves from the fundamental). The 3rd interval above the fundamental (the tonic in scale terms) is the defining interval in scales and harmonies. And in the harmonic series it is the major 3rd which naturally occurs. (But not exactly. We’ll see that it is not quite a major 3rd to our modern ears – it is flat).

Arranging the fundamental and the two different notes in ascending order on the staff produces a major triad on the fundamental, when played simultaneously: C-E-G. Every pitch contains within it a major triad.

C major triad (with notes transposed):

Continuing up the series, the next harmonic is another G, then a new note, a B flat, and the 8th harmonic partial is another C.

The first eight harmonic partials on the note C2:

Notice these notes form a dominant 7th chord. But not quite. All the examples with music use equal tempered tuning. Several of the overtones are not in tune, according to our concepts of consonance and dissonance. The E is actually flatter than we are accustomed to hearing as sounded in these examples. And the B flat is actually not quite a B flat nor an A sharp. It is in between and cannot be played on the piano. This is easily confirmed by repeating the piano experiment. Silently hold down the B flat and strike the low C. The B flat does not ring. Repeat with the A; it does not ring either. If raised to B flat then the first eight harmonic partials form a dominant 7th chord.

The next sample is the harmonic series up to five octaves above a very low C (C1). Note that after the 27th one all 12 notes in the chromatic scale are included in the series. And from the 23rd one standard notation begins to collapse as each pitch gets closer to the one before. Standard notation cannot represent most of the upper tones because they are microtones, falling between the semitone. They cannot be explained using tone-semitone terminology, nor can they be played on a piano. (There are markings used to indicate quarter or microtones – sort of, not precisely – but I didn’t use them; they are a relatively recent tool. And Noteflight isn’t able to play them anyway). Notice partial 22 is an F sharp and partial 23 a G flat. In standard notation these notes are enharmonic, meaning they are the same pitch. Subsequent notes are similarly written enharmonicaly. In nature, in the harmonic series, these notes have different frequencies and cannot be notated.

The number below each note is the fundamental frequency multiple (and harmonic partial number).

Here is the above segment, expanded an octave to C7, with pitches as they naturally occur in the harmonic series. The segment is not terribly pleasant in the upper registers, but it is well worth hearing the pitches sounding closer and closer together until there is barely an audible difference between them. Maybe turn down the volume and click the stop icon if you’ve had enough.

Harmonic partials, 1-64, C1-C7.

By the 27th partial all 12 notes within an octave have been sounded. Many of them twice and the first three unique tones – C-G-E – three or four times each. I wrote “within the octave” which is only apparent after transposing, adjusting, the tone’s octaves and placing the notes in consecutive semitone order. Once done (and tuned), they form the 12 note chromatic scale.

Now there is no necessary reason to stop there, at 12 semitones. The Western system could have introduced the microtones coming after the 22nd partial. Other world music systems use them; ours doesn’t. Why not? Because composers and listeners in the Western tradition liked the sound and interrelationship between pitches without them. (Some modern “classical” music uses microtones. Modern meaning music composed after the so called “common practice period”, post 1900).

The next sample is the transposed and re-ordered first 12 unique harmonic series pitches, excluding microtones, forming a C chromatic scale; in equal tempered tuning. Frequency multiples on the fundamental, which are also the harmonic partial numbers, are identified under each note.

Other world cultures and systems of music recognize the octave as the essential range used to build scales within its compass. The main difference is how the octave is divided into individual notes. Some have up to 22 notes (Indian). As we know, the West divides the octave into 12 individual notes. But until relatively recently, the past hundred years, Western music did not use the chromatic scale as the basis for composition. Only seven of the notes were used and formed into scales. Similarly, the 22 note Indian division is the basis for several scales and is not used as the basis for individual pieces of music.

Seven note scales are typical, almost universal or at least known in other cultures. Even the Oriental pentatonic (five note) scale recognizes two additional notes, which are used as passing tones to the 1st and 5th notes. The eight church modes once commonly used in Western music prior to the early 1600s (from Gregorian chant through Renaissance polyphony) are seven note scales. Two modes were added in the mid 1500s, the Ionian and Aeolian. These are the major and natural minor scale respectively.

Specifically regarding Western music, there are numerous theories regarding why only 7 of the 12 available notes are used to form scales. I’ll explain the one I prefer.

First take another look and listen to the first five partials.

As the harmonic series rises the pitches become less and less audible. The first few unique pitches above the fundamental are plainly and clearly heard and these are vital in creating a seven note scale. Identifying the intervals separating each successive note in the series, they are: an octave (C to C), a perfect 5th (C to G), a perfect 4th (G to C), a major 3rd (C to E). The relationship between C and G is most important both because G is the first different note in the series and because it falls between two Cs, making the intervals of a perfect 5th then perfect 4th. These intervals matter because they are so audible in the series.

A seven note scaled is derived as follows:

From the fundamental we get the notes C-G-E. Using the perfect 5th, G, as a fundamental the first three unique notes are G-D-B. Using the perfect 4th, F, as a fundamental, the first three unique notes are F-C-A. (Notice C is a perfect 5th above the F).

There is now a full seven note scale, once these pitches are ordered alphabetically: C-D-E-F-G-A-B.

The notes in the first five partials of each harmonic series are: C-C-G-C-E; G-G-D-G-B; F-F-C-F-A. Observing which ones dominate, those occurring most frequently, we see C and G occur four times each and F three times. All others just once. The fact F has C as its first different note, being the perfect 5th above, is important: that connection to C (in a C scale) has significance in the tonal system. It is the sub-dominant (IV) and is the third most important note in tonality, after the tonic (the fundamental, I) and dominant (V).

C major scale in equal temperament tuning:

Audio samples:
C major scale, using harmonic series pitches.
C major scale, using equal tempered pitches.

The naturally occurring pitches found in the harmonic series are in tune with themselves but not with other notes based on a different fundamental. That is a major issue for pitched musical instruments, which need to be tuned. When an instrument is tuned to one specific series it cannot play any other in tune unless it is re-tuned to that series. To compensate, many tuning systems have been developed, some still used to this day in world music. I won’t discuss them here but you are encouraged to read about some of the others: just, meantone, and well-tempered tuning in particular.

The concept of equal temperament has been known for many centuries, but few instruments were able to implement it until the mid 1700s. The system gradually became the standard and it is no co-incidence music became more varied and adventurous and much more complicated since its broad acceptance, expanding the choice of keys and modulations and harmonies.

Equal temperament makes slight adjustments to the naturally occurring frequencies in order to be able to play in any key and modulate to any other key within the same piece of music, use harmonies built on each scale degree and expand them beyond the triad (7th, 9th, 13th, for example). No other tuning system permits all these things.

Equal temperament gets its name from the process used to adjust the frequencies: the ratio between each of the 12 notes in the chromatic scale is “equaled”. Consecutive pitches are adjusted by a common factor, specifically the twelfth root of 2, since there are twelve different notes:

$\Large\sqrt[12]{2} = 1.05946.....$

The system works brilliantly and, as noted its implementation vastly expanded the range of possibilities for composers. But it is a compromise. Of course some of the tempered pitches are not in tune at all, based on the naturally occurring pitches in the harmonic series. They are sharper or flatter. However, the pitches are close enough and we have heard them played in equal temperament for so long they sound correct. But to well-trained ears the difference is noticeable.

The piano is an equal tempered instrument. It is tuned in order to play in any key, all of them somewhat out of tune. Many other instruments are not equal tempered – bowed strings and voice, for example  – and many musicians prefer the tuning of pitches and the resulting intervals using just or meantone tuning. An a capella vocal ensemble (no instruments) and a classical string quartet (two violins, viola, cello) are not bound by equal tempered tuning. Such ensembles are able to slightly adjust pitches to match the more natural ones.

Below are two tables. The first is a C chromatic scale with the harmonic series and equal tempered frequencies, with the differences between the natural and tempered notes identified in the right column.

The second table is the harmonic series, up to the 27th partial, comparing frequencies in the harmonic series, the second column, to their equal tempered ones in the third column.

In both tables the frequency for C2 was calculated using A 440Hz as the reference pitch. The harmonic series frequencies were then calculated from C2 as 1f, 2f, 3f, etc. and the equivalent equal tempered frequencies calculated using the 12th root of 2, 1.05946.

The average person can identify pitch differences of slightly less than a single herz, about 1/20th of a semitone; trained musicians less than that.

C chromatic scale with frequencies as found (and transposed) in the harmonic series, their tempered pitches, and the amount the harmonic series differs from equal temperament. Based on A 440Hz. Differences greater than one hertz are in bold. These differences are quite clearly audible.

 Note Harmonic Series Equal Tempered Difference C4 261.64 261.63 .01 C# 277.99 277.18 .81 D 294.35 293.64 .68 D# 310.68 311.13 -.45 E 327.05 329.63 -2.58 F 343.40 349.23 -5.83 F# 359.76 369.99 -10.24 G 392.46 392.00 .46 G# 408.81 415.30 -6.49 A 440.00 440.00 0.00 A# 457.84 466.16 -8.32 B 490.58 493.88 –3.31 C5 523.25 523.25 0.22

Audio samples:

C chromatic scale, using harmonic series pitches.
C chromatic scale, using equal tempered pitches.

Naturally occurring pitch frequencies in the harmonic series and equal temperament. Based on A 440Hz:

 Note Multiple Harmonic Partial Equal Tempered C2 1f 65.41 65.41 C3 2f 130.82 130.81 G 3f 196.23 196.00 C4 4f 261.64 261.63 E 5f 327.05 329.63 G 6f 392.46 392.00 Bb 7f 457.84 466.16 C5 8f 523.28 554.36 D 9f 588.69 587.32 E 10f 654.10 659.26 F# 11f 719.51 739.99 G 12f 784.92 783.99 Ab 13f 850.33 830.61 Bb 14f 915.74 932.33 B 15f 981.15 987.77 C6 16f 1046.56 1046.50 C# 17f 1111.97 1108.73 D 18f 1177.38 1174.66 D# 19f 1242.79 1244.51 E 20f 1308.20 1318.51 F 21f 1373.61 1396.91 F# 22f 1439.02 1479.97 Gb 23f 1504.43 1479.97 G 24f 1569.84 1567.97 G# 25f 1635.25 1661.20 Ab 26f 1700.66 1661.20 A 27f 1766.07 1759.98

Harmonic series on C2, partials 1-27.

A brief word about timbre, the sound qualities notes have when played by an instrument or sung.

Notes sound different when played on different instruments because the timbre is different. Not the pitch; if the instruments are tuned to the same frequency it is the same. Each instrument captures and feeds back the overtone series to our ears differently. A lush, rich sounding note captures many complementary overtones. A dry dull one does not. There is a range when our ears respond to the fundamentals and overtones better than others. It is the mid-to-low-range. At the ends of the spectrum a very high pitched fundamental has less audible overtones and sounds “tinny” and a very low pitched one has too many and sounds “muddy”.

Certain instruments are grouped together or play at the same time because their sound is complimentary or the contrast is pleasing. Some combinations are avoided because they clash. is about balancing the various timbres instruments posses.

A synthesizer electronically produces pure pitch frequencies, sine waves with no overtones. That is why sound samples music notation and MIDI software create are bland, without warmth or brilliance. Software algorithms attempt to create the timbre of an instrument by adding overtones to the fundamental. Attempt is the key word here. The result is only a vague hint of the timbre the instruments create when played by a human. Software algorithms will never match the warmth and range of a masterfully built acoustic instrument played by a master musician. (I’d say algorithms can’t match even a poorly made acoustic instrument played by a relative beginner).

The key specifies the first note in the scale and the interval, or distance, between successive notes. Keys are either major or minor and have strikingly different musical qualities. Major keys are "bright", "happy". Minor keys are "sad", "melancholic". (Very generally speaking!)

Keys are identified by the key signature, the sharps (#) or flats (b) (or lack of them in C major/a minor), that are positioned on the lines and spaces at the beginning of each staff, after the clef sign. The key signature alone tells us only which major or minor key the music is in. Not until we read or hear the music is it possible to know whether the music in a major or minor key, because every major key has a relative minor and vice versa. That is, relative keys - one always major, the other always minor - share a common key signature.

A note is a sound or tone having two aspects:
1) The primary, auditory one, is pitch. Each note has a unique pitch, with a sound wave frequency measurable in hertz. In many parts of the world instruments are tuned to A at 440Hz.

2) The secondary, temporal one, is duration, called "time value" or "note value". When written or played each note has a specific duration, how long it lasts relative to the beat.

When pitch and duration are combined we get melodic shape and rhythm. Pitch creates melody and gives it direction, the melodic shape; duration provides the melodic rhythm.

Orchestration or instrumentation is how the instruments are used; which instruments are playing at any given time and what is their function, such as melodic, accompaniment, creating the pulse, linking phrases (fills).

### 10 Responses to Notes: Pitch, the Harmonic Series, Tuning, and Timbre

1. clivemusic says:

Fantastic Blog – so much useful information and AMAZING! Thank you.

2. Mikael says:

Thank you for this. The clearest explanation of these concepts I have found online

• tangomonkey says:

You’re welcome. It was a fun post to write.

3. Tom Hopwood says:

For me, that was very smooth reading from start to finish. My particular interest is where the 7 notes of the octave came from. I’ve searched a few years now, and most answers are so complicated.
If you ever get the time, would you please comment on this theory of the 7 notes (which is very, very similar to yours):
Focus on middle C. It doesn’t exist alone, but as a Triad with background overtones E and G.
Still focusing on middle C, it goes home to F (which also exists as a Triad), from which C sprang as a perfect 5th in the harmonic series of F.
Still focusing on middle C, G goes home to C in exactly the same way that C went home to F.
Just like a child has a home to go to, and someday is a home to child. The major scale is a family of notes.
Or even more simply (in just one sentence), sit at a piano, focus on middle C, and play/say/LISTEN: C triad goes home to F triad, G triad goes home to C triad.
Voila—the major scale.
I can’t find this anywhere on the web.
Thanks for putting up with it.

Tom Hopwood

• tangomonkey says:

Thanks Tom. There area few ways to approach the derivation of 7 note scales and the relationship between tonic (I), sub-dominant (IV) and dominant (V) found in the harmonic series. I prefer the version I described, and I really don’t have more to say than what I already wrote… Thanks again.

4. Ken says:

A superb explanation. I studied music and acoustics at Sacramento State and have always been amazed that pretty much anything that can vibrate creates the major triad through the first four overtones; it makes sense why we hear the major triad as “home, warm, content, complete, light, bright, restful” etc.

The reason I was reading your post is that I was looking for a good explanation as to where the other tones in the major scale came from; using the dominant’s major triad and the sub dominant’s makes complete sense.

Thank you!

5. Kelly Johnson says:

But if you listen to the harmonic series, it is basically the V7 chord. In its most discernible form, It does not contain two very important chords: the minor triad, and the major 7th chord. You might want to read up on psychoacoustics and music psychology, such as in articles by Richard Parncutt.

• tangomonkey says:

Um, no that’s not so. It is basically octaves, 5ths and 4ths and a major triad. Note the 3rd is not clearly a major 3rd until it is adjusted upwards by equal temperament. If adjusted down instead it becomes a minor 3rd – the 3rd is somewhat ambiguous in the natural frequency. The major-minor 7th comes after those intervals, with the seventh tone in the series. The major 7th chord is not a factor in core tonal and harmonic systems, and is not present in the tones of the harmonic series.

Thanks but I’ll pass on that offer. Parncutt states the obvious, in an obscurantist way with obtuse terminology. He is not saying anything new or anything not obvious to a seasoned musician. He gets many things right, yet there is much nonsense.