# Harmonic series (music)

Harmonic series of a string with terms written as reciprocals (2/1 written as 1/2)
First eight harmonics on C as klang.

A harmonic series is the sequence of sounds[1] where the base frequency[2] of each sound is an integral multiple of the lowest base frequency.[3]

Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves. Interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the typical spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency, and such multiples form the harmonic series (see harmonic series (mathematics)).

The musical pitch of a note is usually perceived as the lowest partial present (the fundamental frequency), which may be the one created by vibration over the full length of the string or air column, or a higher harmonic chosen by the player. The musical timbre of a steady tone from such an instrument is determined by the relative strengths of each harmonic.

## Terminology

### Partial, harmonic, fundamental, inharmonicity, and overtone

A "complex tone" (the sound of a note with a timbre particular to the instrument playing the note) "can be described as a combination of many simple periodic waves (i.e., sine waves) or partials, each with its own frequency of vibration, amplitude, and phase."[4] (See also, Fourier analysis.)

A partial is any of the sine waves (or "simple tones", as Ellis calls them when translating Helmholtz) of which a complex tone is composed.

A harmonic is any member of the harmonic series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The fundamental is also considered a harmonic because it is 1 times itself. A harmonic partial is any real partial component of a complex tone that matches (or nearly matches) an ideal harmonic.[5]

An inharmonic partial is any partial that does not match an ideal harmonic. Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial.[6]

Many pitched acoustic instruments are designed to have partials that are close to being whole-number ratios with very low inharmonicity; therefore, in music theory, and in instrument design, it is convenient, although not strictly accurate, to speak of the partials in those instruments' sounds as "harmonics", even though they have some inharmonicity. Other pitched instruments, especially certain percussion instruments, such as marimba, vibraphone, tubular bells, and timpani, contain mostly inharmonic partials, yet may give the ear a good sense of pitch because of a few strong partials that resemble harmonics. Unpitched, or indefinite-pitched instruments, such as cymbals, gongs, or tam-tams make sounds (produce spectra) that are rich in inharmonic partials (make "noise") and give no impression of implying any particular pitch.

An overtone is any partial except the lowest partial. The term overtone does not imply harmonicity or inharmonicity and has no other special meaning other than to exclude the fundamental. It is the relative strengths of the different overtones that gives an instrument its particular timbre, tone color, or character. When writing or speaking of overtones and partials numerically, care must be taken to designate each correctly to avoid any confusion of one for the other, so the second overtone may not be the third partial, because it is second sound in series.[7]

Some electronic instruments, such as theremins and synthesizers, can play a pure frequency with no overtones (a sine wave). Synthesizers can also combine pure frequencies into more complex tones, such as to simulate other instruments. Certain flutes and ocarinas are very nearly without overtones.

## Frequencies, wavelengths, and musical intervals in example systems

The simplest case to visualise is a vibrating string, as in the illustration; the string has fixed points at each end, and each harmonic mode divides it into 1, 2, 3, 4, etc., equal-sized sections resonating at increasingly higher frequencies.[8] Similar arguments apply to vibrating air columns in wind instruments, although these are complicated by having the possibility of anti-nodes (that is, the air column is closed at one end and open at the other), conical as opposed to cylindrical bores, or end-openings that run the gamut from no flare (bell), cone flare (bell), or exponentially shaped flares (bells).

In most pitched musical instruments, the fundamental (first harmonic) is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency waves occur with varying prominence and give each instrument its characteristic tone quality. The fact that a string is fixed at each end means that the longest allowed wavelength on the string (which gives the fundamental frequency) is twice the length of the string (one round trip, with a half cycle fitting between the nodes at the two ends). Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times that of the fundamental.

Theoretically, these shorter wavelengths correspond to vibrations at frequencies that are 2, 3, 4, 5, 6, etc., times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies. (See inharmonicity and stretched tuning for alterations specific to wire-stringed instruments and certain electric pianos.) However, those alterations are small, and except for precise, highly specialized tuning, it is reasonable to think of the frequencies of the harmonic series as integer multiples of the fundamental frequency.

The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f, 5×f, ...). In terms of frequency (measured in cycles per second, or hertz (Hz) where f is the fundamental frequency), the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because human ears respond to sound nonlinearly, higher harmonics are perceived as "closer together" than lower ones. On the other hand, the octave series is a geometric progression (2×f, 4×f, 8×f, 16×f, ...), and people hear these distances as "the same" in the sense of musical interval. In terms of what one hears, each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals.

The second harmonic, whose frequency is twice of the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher).

An illustration of the harmonic series in musical notation. The numbers above the harmonic indicate the number of cents' difference from equal temperament (rounded to the nearest cent). Blue notes are flat and red notes are sharp.
Harmonics on C, 1 to 32.
Harmonic series as musical notation with intervals between harmonics labeled. Blue notes differ most significantly from equal temperament. One can listen to A2 (110 Hz) and 15 of its partials
Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C. These are "prime harmonics".[9]

As Mersenne says, "the order of the Consonances is natural, and...the way we count them, starting from unity up to the number six and beyond is founded in nature."[10]

## Musical notation

The following section tries to explain differentiation between common "Western" pitch systems and the more natural, ancient Greek Pythagorean system using symbolic graphic symbols. Approximations of the first notes of the harmonic series, can be presented as (without enharmonic, i.e. microtonal, amendments) may be considered as most faithful, for example, a musical notation by

Approximations of the first notes of the harmonic series, written by Kathleen Schlesinger:[11]

The essence of any musical scale, is best analyzed through comparisons with Pythagorean pitches.[12] Such comparison reveals in the set of first 16 overtones a fragment of Pythagorean chainlet with 2 perfect fifths,[13] formed by three overtones with numbers 4, 6, 9; and a subset of the overtones of Pythagorean pitches that contains not only listed overtones, but also those from octave chainlets where these listed are found. This fact evidently demonstrates the relation on the set of matrix structure that reflects the pitches of musical example involving literal notation of Helmholtz where by annexes ${\displaystyle \theta }$ (from Greek: Πυθαγόρας) are marked Pythagorean notes and by arrows ― chainlet of perfect fifths::

${\displaystyle \left\{{\begin{matrix}\vdots \\\theta c^{3}[16]&&\vdots \\\theta c^{2}[8]&&\theta g^{2}[12]&&\vdots \\\theta c^{1}[4]&\leftrightarrows &\theta g^{1}[6]&\leftrightarrows &\theta d^{2}[9]\\\theta c[2]&&\theta g[3]\\\theta C[1]\end{matrix}}\right\}\subset \left\{{\begin{matrix}\vdots &&\vdots &&\vdots \\\theta c^{3}[16]\\b^{2}[15]&&b^{2}[15]\\bes^{2}[14]\\a^{2}[13]\\\theta g^{2}[12]&&\theta g^{2}[12]\\f^{2}[11]\\e^{2}[10]\\\theta d^{2}[9]&&\theta d^{2}[9]&&\theta d^{2}[9]\\\theta c^{2}[8]\\bes^{1}[7]&&&\swarrow \nearrow \\\theta g^{1}[6]&&\theta g^{1}[6]\\e^{1}[5]&\swarrow \nearrow \\\theta c^{1}[4]\\\theta g[3]&&\theta g[3]\\\theta c[2]\\\theta C[1]\end{matrix}}\right\}.}$

A subset of non-Pythagorean overtones remains after removing Pythagorean subset from the set of all overtones of series:

${\displaystyle \left\{{\begin{matrix}\vdots &\vdots &\vdots \\b^{2}[15]&b^{2}[15]\\bes^{2}[14]&&bes^{2}[14]\\a^{2}[13]\\f^{2}[11]\\e^{2}[10]&e^{2}[10]\\bes^{1}[7]&&bes^{1}[7]\\e^{1}[5]&e^{1}[5]\\\end{matrix}}\right\}=\left\{{\begin{matrix}\vdots &\vdots &\vdots \\\theta c^{3}[16]\\b^{2}[15]&b^{2}[15]\\bes^{2}[14]\\a^{2}[13]\\\theta g^{2}[12]&\theta g^{2}[12]\\f^{2}[11]\\e^{2}[10]\\\theta d^{2}[9]&\theta d^{2}[9]&\theta d^{2}[9]\\\theta c^{2}[8]\\bes^{1}[7]\\\theta g^{1}[6]&\theta g^{1}[6]\\e^{1}[5]\\\theta c^{1}[4]\\\theta g[3]&\theta g[3]\\\theta c[2]\\\theta C[1]\end{matrix}}\right\}\setminus \left\{{\begin{matrix}\vdots \\\theta c^{3}[16]&\vdots \\\theta c^{2}[8]&\theta g^{2}[12]&\vdots \\\theta c^{1}[4]&\theta g^{1}[6]&\theta d^{2}[9]\\\theta c[2]&\theta g[3]\\\theta C[1]\end{matrix}}\right\}.}$

If non-Pythagorean pitches are compared with appropriate Pythagorean, the first are distinguished from second on small intervals, called commas, among diversity of which is one of the very famous ― syntonic comma by Didymus [14] (further designation ${\displaystyle \Delta \iota {,}}$ ― from Greek: Δίδυμος ― for sharping and inverted ― ${\displaystyle \iota \Delta {,}}$ ― for flattering):

${\displaystyle {\begin{array}{lclcl}\Delta \iota {,}&=&1200\cdot \log _{2}(81/80)&\approx &+21{,}51{\cancel {\mbox{C}}}[{\mbox{Cent}}];\\\iota \Delta {,}&=&1200\cdot \log _{2}(80/81)&\approx &-21{,}51{\cancel {\mbox{C}}}.\end{array}}}$

Because the non-Pythagorean pitches ${\displaystyle e^{1}[5],e^{2}[10],b^{2}[15]}$ may be obtained from the Pythagorean ${\displaystyle \theta e^{1}[81/16],\theta e^{2}[81/8],\theta b^{2}[243/16]}$ through flattering of latest on ${\displaystyle \iota \Delta {,}[80/81]}$, they need be noted as ${\displaystyle \iota \Delta {,}\theta e^{1}[5];\iota \Delta {,}\theta e^{2}[10];\iota \Delta {,}\theta b^{2}[15]}$. Really:

${\displaystyle {\begin{array}{rcccl}\iota \Delta {,}\theta b^{2}[15]&=&\iota \Delta {,}\theta b^{2}[(80/81)\cdot (243/16)\equiv (80/16)\cdot (243/81)]&=&b^{2}[5\cdot 3\equiv 15];\\\iota \Delta {,}\theta e^{2}[10]&=&\iota \Delta {,}\theta e^{2}[(80/81)\cdot (81/8)\equiv (80/8)\cdot (81/81)]&=&e^{2}[10\cdot 1\equiv 10];\\\iota \Delta {,}\theta e^{1}[5]&=&\iota \Delta {,}\theta e^{1}[(80/81)\cdot (81/16)\equiv (80/16)\cdot (81/81)]&=&e^{1}[5\cdot 1\equiv 5].\end{array}}}$

For the just notation of pitches ${\displaystyle bes^{1}[7],bes^{2}[14]}$ is needed another comma, known as septimal comma by Archytas [15] (further designation ${\displaystyle \mathrm {A} \rho {,}}$ ― from Greek: Αρχύτας ― for sharping and inverted ― ${\displaystyle \rho \mathrm {A} {,}}$ ― for flattering):

${\displaystyle {\begin{array}{lclcl}{\mbox{A}}\rho {,}&=&1200\cdot \log _{2}(64/63)&\approx &+27{,}26{\cancel {\mbox{C}}};\\\rho {\mbox{A}}{,}&=&1200\cdot \log _{2}(63/64)&\approx &-27{,}26{\cancel {\mbox{C}}}.\end{array}}}$

By using the prefixes of flattering on ${\displaystyle \rho {\mbox{A}}{,}[63/64]}$ of Pythagorean pitches ${\displaystyle \theta bes^{1}[64/9],\theta bes^{2}[128/9]}$ notation for just intonation ${\displaystyle bes^{1}[7],bes^{2}[14]}$ gets the form ${\displaystyle \rho {\mbox{A}}{,}\theta bes^{1}[7],\rho {\mbox{A}}{,}\theta bes^{2}[14]}$, the truthfulness of which is easy to check:

${\displaystyle {\begin{array}{rcccl}\rho {\mbox{A}}{,}\theta bes^{2}[14]&=&\rho {\mbox{A}}{,}\theta bes^{2}[(63/64)\cdot (128/9)\equiv (63/9)\cdot (128/64)]&=&bes^{2}[7\cdot 2\equiv 14];\\\rho {\mbox{A}}{,}\theta bes^{1}[7]&=&\rho {\mbox{A}}{,}\theta bes^{1}[(63/64)\cdot (64/9)\equiv (63/9)\cdot (64/64)]&=&bes^{1}[7\cdot 1\equiv 7].\end{array}}}$

The pitch ${\displaystyle f^{2}[11]}$ require undecimal comma by al-Farabi [16] (позначення ${\displaystyle \Phi \alpha {,}}$ ― from Greek: αλ-Φαράμπι ― for sharping and inverted ― ${\displaystyle \alpha \Phi {,}}$ ― for flattering):

${\displaystyle {\begin{array}{lclcl}\Phi \alpha {,}&=&1200\cdot \log _{2}(33/32)&\approx &+53{,}27{\cancel {\mbox{C}}};\\\alpha \Phi {,}&=&1200\cdot \log _{2}(32/33)&\approx &-53{,}27{\cancel {\mbox{C}}}.\end{array}}}$

Prefix of sharping ${\displaystyle \Phi \alpha {,}[33/32]}$ leads Pythagorean notation ${\displaystyle \theta f^{2}[32/3]}$ to form ${\displaystyle \Phi \alpha {,}\theta f^{2}[11]}$, which corresponds the just intonation ${\displaystyle f^{2}[11]}$:

${\displaystyle {\begin{array}{rcccl}\Phi \alpha {,}\theta f^{2}[11]&=&\Phi \alpha {,}\theta f^{2}[(33/32)\cdot (32/3)\equiv (33/3)\cdot (32/32)]&=&f^{2}[11\cdot 1\equiv 11].\end{array}}}$

One else pitch ${\displaystyle a^{2}[13]}$ is required tridecimal comma [17] (designation ${\displaystyle \rho \iota {,}}$ ― from Greek: δεκατρία ― for sharping and inverted ― ${\displaystyle \iota \rho {,}}$ ― for flattering):

${\displaystyle {\begin{array}{lclcl}\rho \iota {,}&=&1200\cdot \log _{2}(27/26)&\approx &+65{,}34{\cancel {\mbox{C}}};\\\iota \rho {,}&=&1200\cdot \log _{2}(26/27)&\approx &-65{,}34{\cancel {\mbox{C}}}.\end{array}}}$

Sharping ${\displaystyle \theta a^{2}[27/2]}$ on ${\displaystyle \iota \rho {,}}$ gives ${\displaystyle \iota \rho {,}\theta a^{2}[13]}$, meaning just intonation ${\displaystyle a^{2}[13]}$:

${\displaystyle {\begin{array}{rcccl}\iota \rho {,}\theta a^{2}[13]&=&\iota \rho {,}\theta a^{2}[(26/27)\cdot (27/2)\equiv (26/2)\cdot (27/27)]&=&a^{2}[13\cdot 1\equiv 13].\end{array}}}$

Must be accounted that the duality of existence multiplicity harmonious[18] and subharmonious,[19] as well as intervals and tones duality,[20][21] was reflected in the dual numbering of pitches (via a slanted, rarer simple, fractional line) of just intonation system. Before (over) the line are written pitch numbers in an overtone series, and after (under) line ― undertone number from which this series was built.[22]

Overtones of musical example by Kathleen Schlesinger are natural numbered, but harmonic series is subsystem of just intonation. Identical renumbering in a dual manner, with the 1 after line clearly express that the whole series was built from the 1st undertone (coinciding with the 1st overtone) and each number before the line indicates its membership namely in overtone series from common fundamental, i.e. from the 1st undertone in a series those from this common fundamental.

Thus if for full definiteness add else the designation of absence of any comma ${\displaystyle \chi {,}}$ (from Greek: χωρίς), the set of first 16 overtones has the form:

${\displaystyle \left\{{\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline _{\chi {,}\theta }&_{\chi {,}\theta }&_{\chi {,}\theta }&_{\chi {,}\theta }&_{\iota \Delta {,}\theta }&_{\chi {,}\theta }&_{\rho \mathrm {A} {,}\theta }&_{\chi {,}\theta }&_{\chi {,}\theta }&_{\iota \Delta {,}\theta }&_{\Phi \alpha {,}\theta }&_{\chi {,}\theta }&_{\iota \rho {,}\theta }&_{\rho \mathrm {A} {,}\theta }&_{\iota \Delta {,}\theta }&_{\chi {,}\theta }\\^{~~C}_{[1/1]}&^{~~c}_{[2/1]}&^{~~g}_{[3/1]}&^{~~c^{1}}_{[4/1]}&^{~~e^{1}}_{[5/1]}&^{~~g^{1}}_{[6/1]}&^{~bes^{1}}_{[7/1]}&^{~~c^{2}}_{[8/1]}&^{~~d^{2}}_{[9/1]}&^{~~e^{2}}_{[10/1]}&^{~~f^{2}}_{[11/1]}&^{~~g^{2}}_{[12/1]}&^{~~a^{2}}_{[13/1]}&^{~bes^{2}}_{[14/1]}&^{~~b^{2}}_{[15/1]}&^{~~c^{3}}_{[16/1]}&^{\cdots }\\\hline \end{array}}\right\}}$

Combining with musical example shows that literal names are fully responsible to it. Therefore, the notation by Kathleen Schlesinger stands out from other known as the most faithful for use to it enharmonic fictas (in this example they are above notes) that prescribe all necessary microtonal pitch bends to reach just intonation.

Harmonic Series in C

${\displaystyle {\begin{array}{|c|c|c|}\hline _{\chi {,}\theta }&_{\chi {,}\theta }&_{\chi {,}\theta }\\\hline \end{array}}~~~~~~~~~~~~{\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline _{\chi {,}\theta }&_{\iota \Delta {,}\theta }&_{\chi {,}\theta }&_{\rho \mathrm {A} {,}\theta }&_{\chi {,}\theta }&_{\chi {,}\theta }&_{\iota \Delta {,}\theta }&_{\Phi \alpha {,}\theta }&_{\chi {,}\theta }&_{\iota \rho {,}\theta }^{~~~\pitchfork }&_{\rho \mathrm {A} {,}\theta }&_{\iota \Delta {,}\theta }&_{\chi {,}\theta }\\\hline \end{array}}}$

${\displaystyle {\begin{array}{|c|c|c|}\hline ^{~~C}_{\left[{\frac {1}{1}}\right]}&^{~~c}_{\left[{\frac {2}{1}}\right]}&^{~~g}_{\left[{\frac {3}{1}}\right]}\\\hline \end{array}}~~~~~~~~~{\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline ^{~~c^{1}}_{\left[{\frac {4}{1}}\right]}&^{~~e^{1}}_{\left[{\frac {5}{1}}\right]}&^{~~g^{1}}_{\left[{\frac {6}{1}}\right]}&^{bes^{1}}_{\left[{\frac {7}{1}}\right]}&^{~~c^{2}}_{\left[{\frac {8}{1}}\right]}&^{~~d^{2}}_{\left[{\frac {9}{1}}\right]}&^{~~e^{2}}_{\left[{\frac {10}{1}}\right]}&^{~~f^{2}}_{\left[{\frac {11}{1}}\right]}&^{~~g^{2}}_{\left[{\frac {12}{1}}\right]}&^{~~a^{2}}_{\left[{\frac {13}{1}}\right]}&^{~bes^{2}}_{\left[{\frac {14}{1}}\right]}&^{~~b^{2}}_{\left[{\frac {15}{1}}\right]}&^{~~c^{3}}_{\left[{\frac {16}{1}}\right]}\\\hline \end{array}}}$

Paying attention to the fact of presence in each enharmonic ficta the symbol ${\displaystyle \theta }$ indicating Pythagorean level of pitch, should be remembered that a bend to the pitch of just intonation a standard tempered pitch, for example, is necessary at first to perform to Pythagorean level, and then whether leave so, if no comma-prefix or is without-comma ${\displaystyle \chi {,}}$ prefix, either perform from Pythagorean level else bend, by comma-prefix specified.

Character ${\displaystyle \pitchfork }$ above ${\displaystyle \theta }$ of pitch ${\displaystyle a^{2}}$ bind its Pythagorean level with standard tuning frequency,[23] what expresses equality:

${\displaystyle {\begin{array}{rcccl}_{\theta }^{\pitchfork }a^{2}[27/2]-(\theta P8[2/1])&=&_{\theta }^{\pitchfork }a^{(2-1\equiv 1)}[(27/2)\cdot (1/2)\equiv (27/4)]&=&{\pitchfork }a^{1}[27/4][440{\mbox{Hz}}].\end{array}}}$

## Harmonics and tuning

If the harmonics are transposed into the span of one octave, some of them are approximated by the notes of what the West has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal semitones, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. In the late 1930s, composer Paul Hindemith ranked musical intervals according to their relative dissonance based on these and similar harmonic relationships.[24]

Below is a comparison between the first 31 harmonics and the intervals of 12-tone equal temperament (12TET), transposed into the span of one octave. Tinted fields highlight differences greater than 5 cents (1/20th of a semitone), which is the human ear's "just noticeable difference" for notes played one after the other (smaller differences are noticeable with notes played simultaneously).

Harmonic 12tET Interval Note Variance cents
1 2 4 8 16 prime (octave) C 0
17 minor second C, D +5
9 18 major second D +4
19 minor third D, E −2
5 10 20 major third E −14
21 fourth F −29
11 22 tritone F, G −49
23 +28
3 6 12 24 fifth G +2
25 minor sixth G, A −27
13 26 +41
27 major sixth A +6
7 14 28 minor seventh A, B −31
29 +30
15 30 major seventh B −12
31 +45

The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals (see just intonation). This objective structure is augmented by psychoacoustic phenomena. For example, a perfect fifth, say 200 and 300 Hz (cycles per second), causes a listener to perceive a combination tone of 100 Hz (the difference between 300 Hz and 200 Hz); that is, an octave below the lower (actual sounding) note. This 100 Hz first-order combination tone then interacts with both notes of the interval to produce second-order combination tones of 200 (300 – 100) and 100 (200 – 100) Hz and all further nth-order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When one contrasts this with a dissonant interval such as a tritone (not tempered) with a frequency ratio of 7:5 one gets, for example, 700 – 500 = 200 (1st order combination tone) and 500 – 200 = 300 (2nd order). The rest of the combination tones are octaves of 100 Hz so the 7:5 interval actually contains 4 notes: 100 Hz (and its octaves), 300 Hz, 500 Hz and 700 Hz. Note that the lowest combination tone (100 Hz) is a 17th (2 octaves and a major third) below the lower (actual sounding) note of the tritone. All the intervals succumb to similar analysis as has been demonstrated by Paul Hindemith in his book The Craft of Musical Composition, although he rejected the use of harmonics from the 7th and beyond.[24]

## Timbre of musical instruments

The relative amplitudes (strengths) of the various harmonics primarily determine the timbre of different instruments and sounds, though onset transients, formants, noises, and inharmonicities also play a role. For example, the clarinet and saxophone have similar mouthpieces and reeds, and both produce sound through resonance of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, the even-numbered harmonics are less present. The saxophone's resonator is conical, which allows the even-numbered harmonics to sound more strongly and thus produces a more complex tone. The inharmonic ringing of the instrument's metal resonator is even more prominent in the sounds of brass instruments.

Human ears tend to group phase-coherent, harmonically-related frequency components into a single sensation. Rather than perceiving the individual partials–harmonic and inharmonic, of a musical tone, humans perceive them together as a tone color or timbre, and the overall pitch is heard as the fundamental of the harmonic series being experienced. If a sound is heard that is made up of even just a few simultaneous sine tones, and if the intervals among those tones form part of a harmonic series, the brain tends to group this input into a sensation of the pitch of the fundamental of that series, even if the fundamental is not present.

Variations in the frequency of harmonics can also affect the perceived fundamental pitch. These variations, most clearly documented in the piano and other stringed instruments but also apparent in brass instruments, are caused by a combination of metal stiffness and the interaction of the vibrating air or string with the resonating body of the instrument.

## Interval strength

David Cope (1997) suggests the concept of interval strength,[25] in which an interval's strength, consonance, or stability (see consonance and dissonance) is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law.

Thus, an equal-tempered perfect fifth ( ) is stronger than an equal-tempered minor third ( ), since they approximate a just perfect fifth ( ) and just minor third ( ), respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3.

## Notes

1. ^
2. ^ IEV 1994, fundamental: http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-30-01
3. ^ IEV 1994, harmonic series of sounds: http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-30-04
4. ^ William Forde Thompson (2008). Music, Thought, and Feeling: Understanding the Psychology of Music. p. 46. ISBN 978-0-19-537707-1.
5. ^ John R. Pierce (2001). "Consonance and Scales". In Perry R. Cook. Music, Cognition, and Computerized Sound. MIT Press. ISBN 978-0-262-53190-0.
6. ^ Martha Goodway and Jay Scott Odell (1987). The Historical Harpsichord Volume Two: The Metallurgy of 17th- and 18th- Century Music Wire. Pendragon Press. ISBN 978-0-918728-54-8.
7. ^ Riemann by Shedlock (1876). Dictionary of Music. Augener & Co., London. p. 143. let it be understood, the second overtone is not the third tone of the series, but the second.
8. ^ Juan G. Roederer (1995). The Physics and Psychophysics of Music. p. 106. ISBN 0-387-94366-8.
9. ^ Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", p.121. Perspectives of New Music 29, no. 2 (Summer): 106–37.
10. ^ Cohen, H.F. (2013). Quantifying Music: The Science of Music at the First Stage of Scientific Revolution 1580–1650, p.103. Springer. ISBN 9789401576864.
11. ^ EB 1911, Valves
12. ^ Волконский 1998, pp. 3-4: «Basic source of both European and non-European musical scales — the spiral by Pythagoras consisting of a chainlet of pure fifths which go to infinity. In evaluating of various European musical scales that have appeared in different times and intervals arising from them should always keep in mind the relationship between the spiral by Pythagoras and natural scale.(Russian: Источник основных как европейских, так и не европейских звукорядов — спираль Пифагора, состоящая из цепочки чистых квинт, уходящих в бесконечность. При оценке различных европейских звукорядов, появившихся в разные эпохи, а также возникающих из них интервалов следует всегда помнить о взаимоотношении между спиралью Пифагора и натуральной гаммой.
13. ^ Coul, List of intervals, 3/2: «perfect fifth»
14. ^ Coul, List of intervals, 81/80: "syntonic comma, Didymus comma"
15. ^ Coul, List of intervals, 64/63: «septimal comma, Archytas' comma»
16. ^ Coul, List of intervals, 33/32: «undecimal comma, al-Farabi's 1/4-tone»
17. ^ Coul, List of intervals, 27/26: «tridecimal comma»
18. ^ IEV 1994, harmonic series of sounds: «fundamental frequency of each of them is an integral multiple of the lowest fundamental frequency» http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-30-04
19. ^ IEV 1994, subharmonic response: «is a submultiple of the excitation frequency» http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-24-25
20. ^ Partch 1974, p. 76: «A system of music is an organization of relationships of pitches, or tones, to one another, and these relationships are inevitably the relationship of numbers. Tone is number, and since a tone in music is always heard in relation to one or several tones — actually heard or implied — one has at least two numbers to deal with: the number of the tone under consideration and the number of the tone heard or implied in relation to the first tone. Hence, the ratio
21. ^ Partch 1974, p. 71: «Interval: a pitch relation between two musical sounds, a ratio. Interval, ratio, tone, are virtually synonymous in this exposition; a ratio is at one and the same time the representative of a tone and of an interval, and a tone always implies a ratio, or interval
22. ^ Partch 1974, p. 67: «In the original manuscript the two numbers of each ratio were shown one above the other, and this form is significant to the exposition in certain cases, the "over" number and the "under" number frequendy having connotations of a very specific nature, as will be seen. The exigencies of typesetting, however, made it difficult to preserve this form where ratios occur in the text. Both numbers of a ratio appear in the same line; therefore, the number preceding the diagonal will be considered "over" and the number following the diagonal will be considered "under." In the diagrams the two numbers are always shown one above the other, so that the "over" and "under" connotations, if applicable, are obvious.»
23. ^ IEV 1994, standard tuning frequency: «for the note LA in the treble octave, of 440 Hz» http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=801-30-18
24. ^ a b Hindemith, Paul (1942). The Craft of Musical Composition: Book 1—Theoretical Part,[page needed]. Translated by Arthur Mendel (London: Schott & Co; New York: Associated Music Publishers. ISBN 0901938300). [1].
25. ^ Cope, David (1997). Techniques of the Contemporary Composer, p. 40–41. New York, New York: Schirmer Books. ISBN 0-02-864737-8.