# Need help understanding harmonic series and intervals

So I need some clarification as this is confusing me terribly. What are the first 12 harmonic series of a fundamental (Lets say A (440hz)?

Do those harmonic series of the fundamental pertain to every note in an equal temperament scale? Such as if we use A as the fundamental, do the notes B,C,D,E,F,G exist in the harmonic series of A?

Could you explain to me in a simplified manner, how we take a fundamentals harmonic series and turn them into intervals? How does that process work exactly?

## 3 Answers

What are the first 12 harmonic series of a fundamental (Lets say A (440hz)?

You probably mean (as other answers indicate) the first 12 harmonics of A. I list the frequencies and the pitch name according to the western music scale in scientific notation.

• 1st: A4 (440Hz)
• 2nd: A5 (880Hz)
• 3rd: E6 (1320Hz)
• 4th: A6 (1760Hz)
• 5th: C#7 (2200Hz)
• 6th: E7 (2640Hz)
• 7th: (none) (3080Hz), this is between F#7 and G7
• 8th: A7 (3520Hz)
• 9th: B7 (3960Hz)
• 10th: C#8 (4400Hz)
• 11th: (none) (4840Hz), this is between D8 and D#8
• 12th: E8 (5280Hz)

Do those harmonic series of the fundamental pertain to every note in an equal temperament scale?

You can calculate the harmonic series of every fundamental frequency. But you do not obtain all 12 pitches of the equal temperament scale from the series. In fact, you do not obtain any pitch from the equal temperament except for the different octaves of A exactly, because the ratio between different pitches is irrational (as it is powers of the twelth root of 2), but the harmonic series produces rational ratios. The harmonics listed above coincide with just tuning based on A. The frequencies of the named pitches are different in equal temperament tuning, but they are quite close.

Such as if we use A as the fundamental, do the notes B,C,D,E,F,G exist in the harmonic series of A?

As you see, the harmonic series only has A, B, C# and E (and some frequencies that sound out-of-tune for the western ear). You can obtain further frequencies that can sound in-tune with A by dividing instead of multiplying by integers. In that case you are not looking for frequencies that are in the harmonic series of A, but frequencies that contain A in their own harmonic series:

• The 3rd harmonic of D3 is A4
• The 5th harmonic of F2 is A4
• The 9th harmonic of G1 is A4

Still, this does not provide all pitches from the A major scale (F# and G# missing) or from the A minor scale (C natural missing).

and last question, could you explain to me in a simplified manner, how we take a fundamentals harmonic series and turn them into intervals? How does that process work exactly?

Intervals are frequency ratios. In all western scales, we treat pitches with a ratio of 2 (or a power of two) as "nearly identical", and call a ratio of two an "octave". The exact value of the other intervals depends on the tuning you use. If you base the tuning on harmonics, the intervals will be simple fractions, whereas if you use equal temperament, the intervals are powers of the twelth root of two. The names from the intervals are counting how many steps in a typical western scale they cover, so you first need to settle down on a scale before you can understand the names of the intervals. Yet you can identify frequency ratios you expect to appear in the scale. As western scales contain 7 pitches (within each octave), and interval names are counting the number of steps they take in the scale, with one indicating the first note, i.e. no step, the first interval, with a ratio of 2, is called the octave. If you continue to look into the series, you discover the following ratios that are of use:

• Ratio 3: A4 to E6. If you step down 1 octave, it's 3/2 for A4 to E5. The (perfect) fifth.
• Ratio 5: A4 to C#7. If you step down 2 octaves, it's 5/4 for A4 to C#5. The major third.
• Ratio 9: A4 to B7. If you step down 3 octaves, it's 9/8 for A4 to B4. The major second.

You can step down another octave (i.e. divide by 2), so the intervals are no longer going up, but going down:

• The perfect fifth upwards, A4 to E5 gets the perfect fourth downwards, A4 to E4 with a ratio of 3/4, so the pure fourth upwards from E4 to A4 has a ratio of 4/3.
• The major third upwards, A4 to C#5 gets the minor sixth downwards, A4 to C#4 with a ratio of 5/8, so the minor sixth upwards has a ratio of 8/5.
• The major second upwards, A4 to B4 gets the minor seventh downwards, A4 to B3, with a ratio of 9/16, so the major seventh upwards has a ratio of 16/9.

Note that I did not include any interval based on the 7th harmonic. This ommision is not inherent to how the harmonic series works, but this interval just isn't used in western music and does not appear (even approximately) in the equal temperament scale. This does not mean there are no other scales that do include that interval.

In the end, you need to be aware the just intonation you obtain from the harmonic series and the equal-temperament scale are different. In just intonation, not every major second is equal. We defined that A4-to-B4 is 9/8, and A4-to-C#5 is 5/4. This makes B4-to-C#5 to be 5/4 divided by 9/8, which is 10/9. We call both A4-to-B4 and B4-to-C#5 a "major second", but they turn out a different ratios! Equal temperament is designed as an approximation of the just intonation (which actually sounds more pure in the key it is base on) where the ratio of neighbouring pitches is always exactly the same, so in equal temperament, A4-to-B4 and B4-to-C#5 in fact has the same ratio, which is neither 9/8 nor 10/9, but something inbetween.

• "The frequencies I listed above are from just tuning": I would say that the pitch names you listed are from just tuning, since the frequencies are the harmonics, and those just come from physics. That is, the 3rd harmonic of 440 Hz is 1320 Hz, no matter what. But whether 1320 Hz is the note E depends on your temperament or tuning system; in equal temperament E is 1318.51 Hz. Also, it is a myth that just intonation allows one to tune a keyboard that sounds good in one key. Even to sound good in one key, the keyboard must be tempered. – phoog Apr 29 '19 at 14:42
• @phoog I understand your concern. What I meant to say was that 1320Hz is the frequency of E6 in (5-limit or 3-limit) just tuning based on A440, whilst in equal temperament on A440 the frequency of E6 is not the frequency I listed. I check how I can rework that sentence to be more clear on that. – Michael Karcher Apr 29 '19 at 16:42
• The ones you listed as "none", I might explain which two notes they're in-between. +1 – user45266 May 1 '19 at 16:08
• Thanks for the suggestion. I intentionally didn't list the pitches, because all the pitches I did list could be defined from the harmonic series. The 7th and 11th harmonic do not define any pitch of the western scale in any common tuning. I still added the pitches that are in-between as a coarse guide. To support this distinction, I intentionally left it being "none" in the list, but added the "hint" at the end of the rows. – Michael Karcher May 2 '19 at 15:52

What are the first 12 harmonic series of a fundamental (Lets say A (440hz)?

I assume you mean the first 12 harmonics in the harmonic series.

The theoretical 'perfect' harmonic series is simply the multiples of 440 - 440, 880, 1320... and so on. 12 * 440 is 5280. (For fuller detail, please see Michael Karcher's excellent answer!)

In real life, the few instruments actually have overtones following a perfect harmonic series - partly though design, as it sounds very boring and flat.

Do those harmonic series of the fundamental pertain to every note in an equal temperament scale?

The overtone structure of a note determines what the timbre of the sound is. Different notes played on the same instruments often have slightly different overtone structures. In most cases, they will have a strong correlation with the theoretical 'perfect' harmonic series, but this depends on all sorts of factors (such as how the instrument is played).

Such as if we use A as the fundamental, do the notes B,C,D,E,F,G exist in the harmonic series of A?

They do, but in some cases at frequencies much higher than the fundamental. In a note played by a real instrument, those overtones might be entirely absent, or very quiet. On the other hand, there might be lots of overtones present in the sound of a note that don't relate directly to the frequencies of other notes in the scale. The harmonic series effectively goes on forever and will contain frequencies relating to so many notes that it would be hard to make a discrete-pitched instrument (such as a piano, or fretted guitar) that could play them all.

Could you explain to me in a simplified manner, how we take a fundamentals harmonic series and turn them into intervals?

It sounds like you've got the impression that musical scales and intervals are derived in a direct way from the harmonic series, but that isn't really the case. It is true that the important notes in some scales correlate with the frequencies of what are often strong overtones, but other harmonics will be present that don't correlate with notes in the scale.

So, how are scales formed?

Generally, I would say that they're formed to allow both satisfying consonances, and interesting dissonances, to occur between the notes in the scale.

Very simply speaking,

• Consonances occur when the harmonics of the notes coincide (have the same, or similar frequencies). If enough of the harmonics coincide, the ear may hear two or more notes 'blending' together.
• Dissonances occur when the harmonics of the notes don't coincide - although they are strongest when the differences between strong harmonics are still within a critical bandwidth, such that beat frequencies occur.

So it is a question of harmonics / overtones, and it is related to the harmonic series. it's just not quite as simple as finding the notes in the harmonic series and using those notes, because:

• We want to allow consonances between pairs or groups of notes that don't include the tonic note. The 7th note in the major scale, for example, is very dissonant with the tonic, but very consonant with the fifth. It's these relationships between notes other than the tonic that allow interesting harmonic motions in the music to occur.
• We want to allow dissonances, as well as consonances, to create tension and resolution with the music
• What consonances and dissonances occur is partly a question of what instrument is playing them; What consonances and dissonances are satisfying is somewhat subjective, and a question of musical culture.

Thinking of it from another angle: The harmonic series is the same in every country in the world. But as Albrecht points out in his answer, different musical cultures have come up with many different scales from which music can be made! Thinking of it this way will provide further evidence that scales aren't directly derived from the harmonic series.

• Understood. So how then are scales formed from a physics standpoint? I was under the impression that scales were created from the harmonic series but you don't believe thats the case? My goal here basically is to understand the source of scale creation. – Seery Apr 29 '19 at 1:39
• @Seery many people ascribe to this theory that the scale is derived from the harmonic series, but that is a huge overstatement. If you're taking a theory class, though, the teacher might want you to learn this theory. Just be aware that it really doesn't explain everything. For example, if you're building a C-major scale, F is not in the harmonic series of C; rather, C is in the harmonic series of F. – phoog Apr 29 '19 at 1:42

I hope by this answer your next question will be enlightened too.

A scale or (sound) scale in the music is a series of the pitch of ordered notes, bounded by frame tones beyond which the tone row is usually repeatable.

In most cases, a scale has the circumference of one octave and in many cases follows a heptatonic tone scale construction. How a scale is built up is determined in the sound system.

The most common European and non-European scales are based on five or seven notes within the octave, which are called tone levels. Widely used are diatonic scales in major and minor or the church modes. Scales are defined by pitches. The tones contained in the concrete scale are called conductor-specific tones.

In non-European music such as classical Arabic or Indian music, there are sound systems and scales that divide the sound space differently. So there are scales that contain more than seven fixed tone levels, such as Mugam, Maqam or Raga.

https://en.wikipedia.org/wiki/Scale_(music)