How often does each interval appear in the harmonic series (relative to the fundamental)?

Question

For practical purposes, lets assume a limited harmonic series.

How many harmonics are a fifth? How many a minor third? And so on.

What intervals are commonly found in the harmonic series? What intervals are under represented?

Answer 1

The harmonic series extends to infinity, so we need to limit the analysis to the first n harmonics. The results, then, will depend on how many harmonics you analyze and how much value you give to each harmonic (maybe weight each harmonic with its amplitude).

I made a small program that analyses that relationship between harmonics and intervals. You can find it here. It has these specifications:

• The program calculates for each harmonic which interval it is the closest to in cents.
• This analysis is performed using the harmonic series found in a sawtooth wave.
• The weighted total uses the weighting 1 / harmonic number, so each harmonic gets a value depending on its amplitude in relation to the fundamental.
• It uses both the ratio of 2^(1/12) (12-tone equal temperament) and Just Intonation (Pythagorean Tuning) to calculate the frequency of the intervals.

Let's see the results for the first 50 harmonics analyzed with this program. There are two tables: the first one shows the frequency each interval was found, the second one shows the weighted sum for each interval.

First 50 harmonics analyzed:

(Results were the same in both 12-TET and Just Intonation)
Ordered by total:
fifth           7.0
root            6.0
tritone         6.0
major third     5.0
major second    4.0
minor third     4.0
minor sixth     4.0
minor seventh   4.0
minor second    3.0
fourth          3.0
major seventh   3.0
major sixth     1.0

Ordered by weighted total:
root            1.97 
fifth           0.688
major third     0.399
minor seventh   0.284
tritone         0.247
major second    0.223
minor sixth     0.175
major seventh   0.132
minor third     0.132
minor second    0.119
fourth          0.0947
major sixth     0.037

We can see that the most common intervals found in the harmonic series are unison and perfect fifth. The less common intervals found in the harmonic series are the major sixth, major seventh, and perfect fourth.

The weighted values give similar results, the difference is that the perfect fourth scores even lower. I wonder if there is a better weighting value or dynamic? I might be giving too little value to the higher harmonics.

Differences between harmonics and their closest interval.

Someone commented that it would be interesting to know what is the difference between the harmonics and their closest interval. After some modifications to the program, here are the results:

Average difference for the first 50 harmonics, in cents:

Average difference:

Interval      12-TET    JI
root          0.000     0.000
minor second  18.879    22.138
major second  14.148    12.193
minor third   24.028    24.474
major third   16.762    21.454
fourth        23.319    22.667
tritone       35.395    32.136
fifth         11.703    10.306
minor sixth   33.950    33.950
major sixth   5.865     0.000
minor seventh 30.775    28.820
major seventh 22.833    26.091

Complete list of differences for the first 50 harmonics, in cents:

Differences for 12-TET:
root          ['0.000', '0.000', '0.000', '0.000', '0.000', '0.000']
minor second  ['4.955', '46.727', '4.955']
major second  ['3.910', '3.910', '44.860', '3.910']
minor third   ['2.487', '48.656', '2.487', '42.483']
major third   ['13.686', '13.686', '13.686', '13.686', '29.062']
fourth        ['29.219', '29.219', '11.518']
tritone       ['48.682', '48.682', '28.274', '48.682', '9.776', '28.274']
fifth         ['1.955', '1.955', '1.955', '1.955', '34.493', '1.955', '37.652']
minor sixth   ['40.528', '27.373', '40.528', '27.373']
major sixth   ['5.865']
minor seventh ['31.174', '31.174', '31.174', '29.577']
major seventh ['11.731', '11.731', '45.036']

Differences for Just Intonation:
root          ['0.000', '0.000', '0.000', '0.000', '0.000', '0.000']
minor second  ['14.730', '36.952', '14.730']
major second  ['0.000', '0.000', '48.770', '0.000']
minor third   ['3.378', '42.791', '3.378', '48.348']
major third   ['21.506', '21.506', '21.506', '21.506', '21.242']
fourth        ['27.264', '27.264', '13.473']
tritone       ['36.952', '36.952', '40.004', '36.952', '1.954', '40.004']
fifth         ['0.000', '0.000', '0.000', '0.000', '36.448', '0.000', '35.697']
minor sixth   ['48.348', '19.553', '48.348', '19.553']
major sixth   ['0.000']
minor seventh ['27.264', '27.264', '27.264', '33.487']
major seventh ['21.506', '21.506', '35.261']

Answer 2

Technically speaking, the answer is infinity for all intervals.

This is because for any resonant harmonic of a fundamental, a harmonic exists at twice the frequency.

There is an order in which these intervals appear, and that is easily found by looking at the harmonic series. You do need to know, of course, that the 12-tone equal temperament that we use today is derived from the harmonic series by adjusting all of the pitches so that they are an equal distance apart, and the notes in the harmonic series quickly decrease in size until they are less than a semitone apart.

---

Edit:

I'm really not satisfied with the idea of using a 12-TET/common practice definition of "interval" when discussing the harmonic series. With the exception of the octave, all intervals created by the harmonic series must be adjusted to conform to an interval name (like "minor 7th") that is common usage. The correct way to refer to acoustic intervals is by way of a ratio, relating the harmonic to the fundamental. When thinking of it this way, the harmonic series is incredibly simple. Let's build a harmonic series on 100 Hz:

(P = partial -- harmonic frequency is always fundamental frequency x partial number)

P | Hz  | Interval
1 | 100 | 1/1
2 | 200 | 2/1
3 | 300 | 3/1
4 | 400 | 4/1
5 | 500 | 5/1
6 | 600 | 6/1
7 | 700 | 7/1
8 | 800 | 8/1
9 | 900 | 9/1

In order to find octave equivalences, we halve each interval ratio (causing a halving of the resulting frequency) until it is less than 2/1. (Mathematically, this just means we double the "denominator" of the interval.)

P | Hz  | Interval
1 | 100 | 1/1
2 | 200 | 2/2
3 | 300 | 3/2
4 | 400 | 4/4
5 | 500 | 5/4
6 | 600 | 6/4
7 | 700 | 7/4
8 | 800 | 8/8
9 | 900 | 9/8

Then simplify:

P | Hz  | Interval
1 | 100 | 1/1
2 | 200 | 1/1
3 | 300 | 3/2
4 | 400 | 1/1
5 | 500 | 5/4
6 | 600 | 3/2
7 | 700 | 7/4
8 | 800 | 1/1
9 | 900 | 9/8

I included the middle step so it is easier to see that the number of intervals per octave grows exponentially: 2^n, where n is the number of octaves above the fundamental. Every next octave contains all the previous octave's intervals plus 2^(n-1) new intervals.

So, within a finite range of complete octaves and after simplifying all interval ratios, any interval with denominator 2^n is going to have:

• exactly one more occurrence than any intervals with denominator 2^(n+1)
• exactly one less occurrence than any intervals with denominator 2^(n-1)
• exactly the same number of occurrences as any other intervals with the same denominator

It should be fairly easy from this point to create a generalized case for frequency of occurrence given an interval and a range, should you desire.

So, that's how the harmonic series works acoustically and mathematically. If you know how the harmonic series looks tonally, you should be able to match up the ratios listed above with your favorite tonal intervals: 3/2 is the perfect 5th, 5/4 is your major third, 7/4 is the minor 7th, and 9/8 is the major 2nd.

Personally, I don't believe any intervals beyond these should be considered "equivalent" beyond what coincidentally may be the case. Sure, you can get a "minor 3rd" by shooting up to the 19th harmonic, but it's much easier to derive using the distance between the 5th and 6th harmonics instead. Just Intonation uses exactly this for the minor third, 6/5. The perfect fourth is 4/3, major 6th is 5/3, and minor 6th is 8/5. Small whole-number ratios are perceived as consonance.

Some resources:

• Very useful table of the first 32 harmonics
• Wikipedia: Harmonic (lots of awesome charts)
• Some good information about Just Intonation

Answer 3

JC's program goes through and counts, which is one approach to the problem that certainly gives an answer. But what it doesn't provide is insight into the pattern that causes notes repeat in the overtone sequence. I'm going to take the opposite approach, of showing the pattern, without necessarily giving an answer. :)

The key, of course, is octaves, which occur whenever the frequency doubles. If we define units so that the fundamental frequency is '1', then the frequency of the harmonics are simply the integers. But any power of 2 will be some number of octaves above the fundamental:

unison/octaves: 1, 2, 4, 8, 16, 32, ...

In fact, any even numbered harmonic is a repeat of a previously-heard note, since it can be divided in half to find a harmonic one octave lower. So only odd numbers will give us 'new' notes, and every harmonic can be written as an odd number times a power of two.

The next available number in sequence is 3, which in the harmonic series corresponds to a perfect fifth:

fifths: 3, 6, 12, 24, ...

Next is the 5th harmonic, which corresponds to an interval of a major 3rd:

major thirds: 5, 10, 20, 40, ...

Then the 7th harmonic, corresponding to an out-of-tune minor 7th:

minor sevenths: 7, 14, 28, ...

And the 9th harmonic, corresponding to a major 9th (or a major 2nd):

major seconds: 9, 18, 36, ...

The 11th harmonic is another one that's out of tune with 12TET, lying somewhere between a tritone and a perfect fourth:

tritones: 11, 22, ...

From this point on, the harmonics correlate increasingly poorly to the 12TET scale, and need to start being rounded, per JC's program. But you can see the pattern here. If, for example, we're limited to the first 40 harmonics, then all the odd harmonics between 11 and 19 will occur twice (19x2 = 38), and beginning with harmonic 21, they will only occur once. Just to reiterate the point from earlier: every odd harmonic will be a totally new pitch in the series, and will not be an octave of a previous pitch.

Another way to think about harmonics is in terms of a prime number decomposition (every composite number is a product of primes). For example, consider the 60th harmonic: 60 = 2^2 * 3 * 5

Since there are two factors of '2', this is two octaves above harmonic 15 (= 3 * 5). Since there is a factor of '3', this is a perfect fifth above harmonic 5, which is a major third. So we can deduce that harmonic 60 is a major 3rd above a perfect 5th, i.e. a major 7th.