# Pitch (music)

Last updated

Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, [1] or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies. [2] Pitch can be determined only in sounds that have a frequency that is clear and stable enough to distinguish from noise. [3] Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre. [4]

## Contents

Pitch may be quantified as a frequency, but pitch is not a purely objective physical property; it is a subjective psychoacoustical attribute of sound. Historically, the study of pitch and pitch perception has been a central problem in psychoacoustics, and has been instrumental in forming and testing theories of sound representation, processing, and perception in the auditory system. [5]

## Perception

### Pitch and frequency

Pitch is an auditory sensation in which a listener assigns musical tones to relative positions on a musical scale based primarily on their perception of the frequency of vibration. [6] Pitch is closely related to frequency, but the two are not equivalent. Frequency is an objective, scientific attribute that can be measured. Pitch is each person's subjective perception of a sound wave, which cannot be directly measured. However, this does not necessarily mean that most people won't agree on which notes are higher and lower.

The oscillations of sound waves can often be characterized in terms of frequency. Pitches are usually associated with, and thus quantified as, frequencies (in cycles per second, or hertz), by comparing the sounds being assessed against sounds with pure tones (ones with periodic, sinusoidal waveforms). Complex and aperiodic sound waves can often be assigned a pitch by this method. [7] [8] [9]

According to the American National Standards Institute, pitch is the auditory attribute of sound according to which sounds can be ordered on a scale from low to high. Since pitch is such a close proxy for frequency, it is almost entirely determined by how quickly the sound wave is making the air vibrate and has almost nothing to do with the intensity, or amplitude, of the wave. That is, "high" pitch means very rapid oscillation, and "low" pitch corresponds to slower oscillation. Despite that, the idiom relating vertical height to sound pitch is shared by most languages. [10] At least in English, it is just one of many deep conceptual metaphors that involve up/down. The exact etymological history of the musical sense of high and low pitch is still unclear. There is evidence that humans do actually perceive that the source of a sound is slightly higher or lower in vertical space when the sound frequency is increased or reduced. [10]

In most cases, the pitch of complex sounds such as speech and musical notes corresponds very nearly to the repetition rate of periodic or nearly-periodic sounds, or to the reciprocal of the time interval between repeating similar events in the sound waveform. [8] [9]

The pitch of complex tones can be ambiguous, meaning that two or more different pitches can be perceived, depending upon the observer. [5] When the actual fundamental frequency can be precisely determined through physical measurement, it may differ from the perceived pitch because of overtones, also known as upper partials, harmonic or otherwise. A complex tone composed of two sine waves of 1000 and 1200 Hz may sometimes be heard as up to three pitches: two spectral pitches at 1000 and 1200 Hz, derived from the physical frequencies of the pure tones, and the combination tone at 200 Hz, corresponding to the repetition rate of the waveform. In a situation like this, the percept at 200 Hz is commonly referred to as the missing fundamental, which is often the greatest common divisor of the frequencies present. [11]

Pitch depends to a lesser degree on the sound pressure level (loudness, volume) of the tone, especially at frequencies below 1,000 Hz and above 2,000 Hz. The pitch of lower tones gets lower as sound pressure increases. For instance, a tone of 200 Hz that is very loud seems one semitone lower in pitch than if it is just barely audible. Above 2,000 Hz, the pitch gets higher as the sound gets louder. [12] These results were obtained in the pioneering works by S. Stevens [13] and W. Snow. [14] Later investigations, i.e. by A. Cohen, had shown that in most cases the apparent pitch shifts were not significantly different from pitch‐matching errors. When averaged, the remaining shifts followed the directions of Stevens' curves but were small (2% or less by frequency, i.e. not more than a semitone). [15]

### Theories of pitch perception

Theories of pitch perception try to explain how the physical sound and specific physiology of the auditory system work together to yield the experience of pitch. In general, pitch perception theories can be divided into place coding and temporal coding. Place theory holds that the perception of pitch is determined by the place of maximum excitation on the basilar membrane.

A place code, taking advantage of the tonotopy in the auditory system, must be in effect for the perception of high frequencies, since neurons have an upper limit on how fast they can phase-lock their action potentials. [6] However, a purely place-based theory cannot account for the accuracy of pitch perception in the low and middle frequency ranges. Moreover, there is some evidence that some non-human primates lack auditory cortex responses to pitch despite having clear tonotopic maps in auditory cortex, showing that tonotopic place codes are not sufficient for pitch responses. [16]

Temporal theories offer an alternative that appeals to the temporal structure of action potentials, mostly the phase-locking and mode-locking of action potentials to frequencies in a stimulus. The precise way this temporal structure helps code for pitch at higher levels is still debated, but the processing seems to be based on an autocorrelation of action potentials in the auditory nerve. [17] However, it has long been noted that a neural mechanism that may accomplish a delay—a necessary operation of a true autocorrelation—has not been found. [6] At least one model shows that a temporal delay is unnecessary to produce an autocorrelation model of pitch perception, appealing to phase shifts between cochlear filters; [18] however, earlier work has shown that certain sounds with a prominent peak in their autocorrelation function do not elicit a corresponding pitch percept, [19] [20] and that certain sounds without a peak in their autocorrelation function nevertheless elicit a pitch. [21] [22] To be a more complete model, autocorrelation must therefore apply to signals that represent the output of the cochlea, as via auditory-nerve interspike-interval histograms. [20] Some theories of pitch perception hold that pitch has inherent octave ambiguities, and therefore is best decomposed into a pitch chroma, a periodic value around the octave, like the note names in western music—and a pitch height, which may be ambiguous, that indicates the octave the pitch is in. [5]

### Just-noticeable difference

The just-noticeable difference (jnd) (the threshold at which a change is perceived) depends on the tone's frequency content. Below 500 Hz, the jnd is about 3 Hz for sine waves, and 1 Hz for complex tones; above 1000 Hz, the jnd for sine waves is about 0.6% (about 10 cents). [23] The jnd is typically tested by playing two tones in quick succession with the listener asked if there was a difference in their pitches. [12] The jnd becomes smaller if the two tones are played simultaneously as the listener is then able to discern beat frequencies. The total number of perceptible pitch steps in the range of human hearing is about 1,400; the total number of notes in the equal-tempered scale, from 16 to 16,000 Hz, is 120. [12]

### Aural illusions

The relative perception of pitch can be fooled, resulting in aural illusions . There are several of these, such as the tritone paradox, but most notably the Shepard scale, where a continuous or discrete sequence of specially formed tones can be made to sound as if the sequence continues ascending or descending forever.

## Definite and indefinite pitch

Not all musical instruments make notes with a clear pitch. The unpitched percussion instrument (a class of percussion instrument) does not produce particular pitches. A sound or note of definite pitch is one where a listener can possibly (or relatively easily) discern the pitch. Sounds with definite pitch have harmonic frequency spectra or close to harmonic spectra. [12]

A sound generated on any instrument produces many modes of vibration that occur simultaneously. A listener hears numerous frequencies at once. The vibration with the lowest frequency is called the fundamental frequency ; the other frequencies are overtones . [24] Harmonics are an important class of overtones with frequencies that are integer multiples of the fundamental. Whether or not the higher frequencies are integer multiples, they are collectively called the partials, referring to the different parts that make up the total spectrum.

A sound or note of indefinite pitch is one that a listener finds impossible or relatively difficult to identify as to pitch. Sounds with indefinite pitch do not have harmonic spectra or have altered harmonic spectra—a characteristic known as inharmonicity.

It is still possible for two sounds of indefinite pitch to clearly be higher or lower than one another. For instance, a snare drum sounds higher pitched than a bass drum though both have indefinite pitch, because its sound contains higher frequencies. In other words, it is possible and often easy to roughly discern the relative pitches of two sounds of indefinite pitch, but sounds of indefinite pitch do not neatly correspond to any specific pitch.

## Pitch standards and standard pitch

A pitch standard (also concert pitch) is the conventional pitch reference a group of musical instruments are tuned to for a performance. Concert pitch may vary from ensemble to ensemble, and has varied widely over musical history.

Standard pitch is a more widely accepted convention. The A above middle C is usually set at 440 Hz (often written as "A = 440 Hz" or sometimes "A440"), although other frequencies, such as 442 Hz, are also often used as variants. Another standard pitch, the so-called Baroque pitch, has been set in the 20th century as A = 415 Hz—approximately an equal-tempered semitone lower than A440 to facilitate transposition. The Classical pitch can be set to either 427 Hz (about halfway between A415 and A440) or 430 Hz (also between A415 and A440 but slightly sharper than the quarter tone). And ensembles specializing in authentic performance set the A above middle C to 432 Hz or 435 Hz when performing repertoire from the Romantic era.

Transposing instruments have their origin in the variety of pitch standards. In modern times, they conventionally have their parts transposed into different keys from voices and other instruments (and even from each other). As a result, musicians need a way to refer to a particular pitch in an unambiguous manner when talking to each other.

For example, the most common type of clarinet or trumpet, when playing a note written in their part as C, sounds a pitch that is called B on a non-transposing instrument like a violin (which indicates that at one time these wind instruments played at a standard pitch a tone lower than violin pitch). To refer to that pitch unambiguously, a musician calls it concert B, meaning, "...the pitch that someone playing a non-transposing instrument like a violin calls B."

## Labeling pitches

Pitches are labeled using:

For example, one might refer to the A above middle C as a′, A4, or 440 Hz. In standard Western equal temperament, the notion of pitch is insensitive to "spelling": the description "G4 double sharp" refers to the same pitch as A4; in other temperaments, these may be distinct pitches. Human perception of musical intervals is approximately logarithmic with respect to fundamental frequency: the perceived interval between the pitches "A220" and "A440" is the same as the perceived interval between the pitches A440 and A880. Motivated by this logarithmic perception, music theorists sometimes represent pitches using a numerical scale based on the logarithm of fundamental frequency. For example, one can adopt the widely used MIDI standard to map fundamental frequency, f, to a real number, p, as follows

${\displaystyle p=69+12\times \log _{2}{\left({\frac {f}{440{\mbox{ Hz}}}}\right)}}$

This creates a linear pitch space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and A440 is assigned the number 69. (See Frequencies of notes.) Distance in this space corresponds to musical intervals as understood by musicians. An equal-tempered semitone is subdivided into 100 cents. The system is flexible enough to include "microtones" not found on standard piano keyboards. For example, the pitch halfway between C (60) and C (61) can be labeled 60.5.

The following table shows frequencies in Hertz for notes in various octaves, named according to the "German method" of octave nomenclature:

NoteSub-contraContraGreatSmallOne-linedTwo-linedThree-linedFour-linedFive-lined
B/C16.3532.7065.41130.81261.63523.251046.502093.004186.01
C/D17.3234.6569.30138.59277.18554.371108.732217.464434.92
D18.3536.7173.42146.83293.66587.331174.662349.324698.64
D/E19.4538.8977.78155.56311.13622.251244.512489.024978.03
E/F20.6041.2082.41164.81329.63659.261318.512637.025274.04
E/F21.8343.6587.31174.61349.23698.461396.912793.835587.65
F/G23.1246.2592.50185.00369.99739.991479.982959.965919.91
G24.5049.0098.00196.00392.00783.991567.993135.966271.93
G/A25.9651.91103.83207.65415.30830.611661.223322.446644.88
A27.5055.00110.00220.00440.00880.001760.003520.007040.00
A/B29.1458.27116.54233.08466.16932.331864.663729.317458.62
B/C30.8761.74123.47246.94493.88987.771975.533951.077902.13

## Scales

The relative pitches of individual notes in a scale may be determined by one of a number of tuning systems. In the west, the twelve-note chromatic scale is the most common method of organization, with equal temperament now the most widely used method of tuning that scale. In it, the pitch ratio between any two successive notes of the scale is exactly the twelfth root of two (or about 1.05946). In well-tempered systems (as used in the time of Johann Sebastian Bach, for example), different methods of musical tuning were used.

In almost all of these systems interval of the octave doubles the frequency of a note; for example, an octave above A440 is 880 Hz. If however the first overtone is sharp due to inharmonicity, as in the extremes of the piano, tuners resort to octave stretching.

## Other musical meanings of pitch

In atonal, twelve tone, or musical set theory a "pitch" is a specific frequency while a pitch class is all the octaves of a frequency. In many analytic discussions of atonal and post-tonal music, pitches are named with integers because of octave and enharmonic equivalency (for example, in a serial system, C and D are considered the same pitch, while C4 and C5 are functionally the same, one octave apart).

Discrete pitches, rather than continuously variable pitches, are virtually universal, with exceptions including "tumbling strains" [27] and "indeterminate-pitch chants". [28] Gliding pitches are used in most cultures, but are related to the discrete pitches they reference or embellish. [29]

## Related Research Articles

An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.

A harmonic series is the sequence of frequencies, musical tones, or pure tones in which each frequency is an integer multiple of a fundamental.

In music, there are two common meanings for tuning:

In music, a note is a symbol denoting a musical sound. In English usage a note is also the sound itself.

In music, an octave or perfect octave is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". The interval between the first and second harmonics of the harmonic series is an octave.

In music, harmony is the process by which the composition of individual sounds, or superpositions of sounds, is analysed by hearing. Usually, this means simultaneously occurring frequencies, pitches, or chords.

In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

In music, timbre, also known as tone color or tone quality, is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and musical instruments. It also enables listeners to distinguish different instruments in the same category.

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be perceived between successive notes.

In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency.

A harmonic sound is said to have a missing fundamental, suppressed fundamental, or phantom fundamental when its overtones suggest a fundamental frequency but the sound lacks a component at the fundamental frequency itself. The brain perceives the pitch of a tone not only by its fundamental frequency, but also by the periodicity implied by the relationship between the higher harmonics; we may perceive the same pitch even if the fundamental frequency is missing from a tone.

The tritone paradox is an auditory illusion in which a sequentially played pair of Shepard tones separated by an interval of a tritone, or half octave, is heard as ascending by some people and as descending by others. Different populations tend to favor one of a limited set of different spots around the chromatic circle as central to the set of "higher" tones. Roger Shepard in 1963 had argued that such tone pairs would be heard ambiguously as either ascending or descending. However, psychology of music researcher Diana Deutsch in 1986 discovered that when the judgments of individual listeners were considered separately, their judgments depended on the positions of the tones along the chromatic circle. For example, one listener would hear the tone pair C–F as ascending and the tone pair G–C as descending. Yet another listener would hear the tone pair C–F as descending and the tone pair G–C as ascending. Furthermore, the way these tone pairs were perceived varied depending on the listener's language or dialect.

Piano tuning is the act of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the vibration interaction among notes, which is different for every piano, thus in practice requiring slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament.

Scientific pitch notation is a method of specifying musical pitch by combining a musical note name and a number identifying the pitch's octave.

The twelfth root of two or is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio of a semitone in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals as consisting of different numbers of a single interval, the equal tempered semitone. A semitone itself is divided into 100 cents.

Musical acoustics or music acoustics is a multidisciplinary field that combines knowledge from physics, psychophysics, organology, physiology, music theory, ethnomusicology, signal processing and instrument building, among other disciplines. As a branch of acoustics, it is concerned with researching and describing the physics of music – how sounds are employed to make music. Examples of areas of study are the function of musical instruments, the human voice, computer analysis of melody, and in the clinical use of music in music therapy.

Twelve-tone equal temperament is the musical system that divides the octave into 12 parts, all of which are equally tempered on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, ​112 the width of an octave, is called a semitone or half step.

Music theory has no axiomatic foundation in modern mathematics, although some interesting work has recently been done in this direction, yet the basis of musical sound can be described mathematically and exhibits "a remarkable array of number properties". Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.

Pitch circularity is a fixed series of tones that appear to ascend or descend endlessly in pitch.

## References

1. Anssi Klapuri, "Introduction to Music Transcription", in Signal Processing Methods for Music Transcription, edited by Anssi Klapuri and Manuel Davy, 1–20 (New York: Springer, 2006): p. 8. ISBN   978-0-387-30667-4.
2. Plack, Christopher J.; Andrew J. Oxenham; Richard R. Fay, eds. (2005). Pitch: Neural Coding and Perception. New York: Springer. ISBN   978-0-387-23472-4. For the purposes of this book we decided to take a conservative approach, and to focus on the relationship between pitch and musical melodies. Following the earlier ASA definition, we define pitch as 'that attribute of sensation whose variation is associated with musical melodies.' Although some might find this too restrictive, an advantage of this definition is that it provides a clear procedure for testing whether or not a stimulus evokes a pitch, and a clear limitation on the range of stimuli that we need to consider in our discussions.
3. Harold S. Powers, "Melody", The Harvard Dictionary of Music, fourth edition, edited by Don Michael Randel, 499–502 (Cambridge: Belknap Press for Harvard University Press, 2003) ISBN   978-0-674-01163-2. "Melody: In the most general case, a coherent succession of pitches. Here pitch means a stretch of sound whose frequency is clear and stable enough to be heard as not noise; succession means that several pitches occur; and coherent means that the succession of pitches is accepted as belonging together" (p. 499).
4. Roy D. Patterson; Etienne Gaudrain & Thomas C. Walters (2010). "The Perception of Family and Register in Musical Tones". In Mari Riess Jones; Richard R. Fay & Arthur N. Popper (eds.). Music Perception. Springer. pp. 37–38. ISBN   978-1-4419-6113-6.
5. Hartmann, William Morris (1997). Signals, Sound, and Sensation. Springer. pp. 145, 284, 287. ISBN   978-1-56396-283-7.
6. Plack, Christopher J.; Andrew J. Oxenham; Richard R. Fay, eds. (2005). Pitch: Neural Coding and Perception. Springer. ISBN   978-0-387-23472-4.
7. Robert A. Dobie & Susan B. Van Hemel (2005). Hearing Loss: Determining Eligibility for Social Security Benefits. National Academies Press. pp. 50–51. ISBN   978-0-309-09296-8.
8. E. Bruce Goldstein (2001). Blackwell Handbook of Perception (4th ed.). Wiley-Blackwell. p. 381. ISBN   978-0-631-20683-5.
9. Richard Lyon & Shihab Shamma (1996). "Auditory Representation of Timbre and Pitch". In Harold L. Hawkins & Teresa A. McMullen (eds.). Auditory Computation. Springer. pp. 221–23. ISBN   978-0-387-97843-7.
10. Carroll C. Pratt, "The Spatial Character of High and Low Tones", Journal of Experimental Psychology 13 (1930): 278–85.
11. Schwartz, David A.; Dale Purves (May 2004). "Pitch Is Determined by Naturally Occurring Periodic Sounds". Hearing Research. 194 (1–2): 31–46. doi:10.1016/j.heares.2004.01.019. PMID   15276674. S2CID   40608136.
12. Olson, Harry F. (1967). Music, Physics and Engineering. Dover Publications. pp. 171, 248–251. ISBN   978-0-486-21769-7.
13. Stevens S. S. The relation of pitch to intensity//J. Acoust. Soc. Amer. 1935. Vol. 6. P. 150-154.
14. Snow W. B. (1936) Change of Pitch with Loudness at Low Frequencies. J. Acoust. Soc. Am/ 8:14–19.
15. Cohen, A. (1961). Further investigation of the effects of intensity upon the pitch of pure tones. Journal of the Acoustical Society of America, 33, 1363–1376. https://dx.doi.org/10.1121/1.1908441
16. Norman-Haignere, S.V.; Kanwisher, N.G.; McDermott, J.; Conway, B.R. (10 June 2019). "Divergence in the functional organization of human and macaque auditory cortex revealed by fMRI responses to harmonic tones". Nature Neuroscience. 22 (7): 1057–1060. doi:10.1038/s41593-019-0410-7. ISSN   1097-6256. PMC  . PMID   31182868.
17. Cariani, P.A.; Delgutte, B. (September 1996). "Neural Correlates of the Pitch of Complex Tones. I. Pitch and Pitch Salience" (PDF). Journal of Neurophysiology. 76 (3): 1698–1716. doi:10.1152/jn.1996.76.3.1698. PMID   8890286 . Retrieved 13 November 2012.
18. Cheveigné, A. de; Pressnitzer, D. (June 2006). "The Case of the Missing Delay Lines: Synthetic Delays Obtained by Cross-channel Phase Interaction" (PDF). Journal of the Acoustical Society of America. 119 (6): 3908–3918. Bibcode:2006ASAJ..119.3908D. doi:10.1121/1.2195291. PMID   16838534 . Retrieved 13 November 2012.
19. Kaernbach, C.; Demany, L. (October 1998). "Psychophysical Evidence Against the Autocorrelation Theory of Auditory Temporal Processing". Journal of the Acoustical Society of America. 104 (4): 2298–2306. Bibcode:1998ASAJ..104.2298K. doi:10.1121/1.423742. PMID   10491694.
20. Pressnitzer, D.; Cheveigné, A. de; Winter, I.M. (January 2002). "Perceptual Pitch Shift for Sounds with Similar Waveform Autocorrelation". Acoustics Research Letters Online. 3 (1): 1–6. doi:10.1121/1.1416671.
21. Burns, E.M.; Viemeister, N. F. (October 1976). "Nonspectral Pitch". Journal of the Acoustical Society of America. 60 (4): 863–69. Bibcode:1976ASAJ...60..863B. doi:10.1121/1.381166.
22. Fitzgerald, M. B.; Wright, B. (December 2005). "A Perceptual Learning Investigation of the Pitch Elicited by Amplitude-Modulated Noise". Journal of the Acoustical Society of America. 118 (6): 3794–3803. Bibcode:2005ASAJ..118.3794F. doi:10.1121/1.2074687. PMID   16419824.
23. Birger Kollmeier; Thomas Brand & B. Meyer (2008). "Perception of Speech and Sound". In Jacob Benesty; M. Mohan Sondhi & Yiteng Huang (eds.). Springer Handbook of Speech Processing. Springer. p. 65. ISBN   978-3-540-49125-5.
24. Levitin, Daniel (2007). This Is Your Brain on Music. New York: Penguin Group. p. 40. ISBN   978-0-452-28852-2. The one with the slowest vibration rate—the one lowest in pitch—is referred to as the fundamental frequency, and the others are collectively called overtones.
25. The Concise Grove Dictionary of Music: Hermann von Helmholtz, Oxford University Press (1994), Answers.com. Retrieved 3 August 2007.
26. Helmholtz, Hermann (1885). On the Sensations of Tone (English Translation). p. 15. ISBN   9781602066397.
27. Sachs, C. and Kunst, J. (1962). In The Wellsprings of Music, edited by J. Kunst. The Hague: Marinus Nijhoff. Cited in Burns (1999).
28. Malm, W.P. (1967). Music Cultures of the Pacific, the Near East, and Asia. Englewood Cliffs, NJ: Prentice-Hall. Cited in Burns (1999).
29. Burns, Edward M. (1999). "Intervals, Scales, and Tuning", The Psychology of Music, second edition. Deutsch, Diana, ed. San Diego: Academic Press. ISBN   0-12-213564-4.
• Moore, B.C. & Glasberg, B.R. (1986) "Thresholds for Hearing Mistuned Partials as Separate Tones in Harmonic Complexes". Journal of the Acoustical Society of America, 80, 479–83.
• Parncutt, R. (1989). Harmony: A Psychoacoustical Approach. Berlin: Springer-Verlag, 1989.
• Schneider, P.; Sluming, V.; Roberts, N.; Scherg, M.; Goebel, R.; Specht, H.-J.; Dosch, H.G.; Bleeck, S.; Stippich, C.; Rupp, A. (2005). "Structural and Functional Asymmetry of Lateral Heschl's Gyrus Reflects Pitch Perception Preference". Nat. Neurosci.[ full citation needed ] 8, 1241–47.
• Terhardt, E., Stoll, G. and Seewann, M. (1982). "Algorithm for Extraction of Pitch and Pitch Salience from Complex Tonal Signals". Journal of the Acoustical Society of America, 71, 679–88.