Perfect fifth

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Perfect fifth
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equal tempered and
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perfect fifth
Inverse perfect fourth
Name
Other namesdiapente
AbbreviationP5
Size
Semitones 7
Interval class 5
Just interval 3:2
Cents
Equal temperament 700
24 equal temperament 700
Just intonation 702
Examples of perfect fifth intervals Perfect5s.PNG
Examples of perfect fifth intervals

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

Music theory considers the practices and possibilities of music

Music theory is the study of the practices and possibilities of music. The Oxford Companion to Music describes three interrelated uses of the term "music theory":

The first is what is otherwise called 'rudiments', currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, and so on. [...] The second is the study of writings about music from ancient times onwards. [...] The third is an area of current musicological study that seeks to define processes and general principles in music — a sphere of research that can be distinguished from analysis in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built.

In music theory, an interval is the 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.

Pitch (music) perceptual property in music

Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, 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. Pitch can be determined only in sounds that have a frequency that is clear and stable enough to distinguish from noise. Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre.

Contents

In classical music from Western culture, a fifth is the interval from the first to the last of five consecutive notes in a diatonic scale. [1] The perfect fifth (often abbreviated P5) spans seven semitones, while the diminished fifth spans six and the augmented fifth spans eight semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. Loudspeaker.svg Play  

Classical music broad tradition of Western art music

Classical music is art music produced or rooted in the traditions of Western culture, including both liturgical (religious) and secular music. While a more precise term is also used to refer to the period from 1750 to 1820, this article is about the broad span of time from before the 6th century AD to the present day, which includes the Classical period and various other periods. The central norms of this tradition became codified between 1550 and 1900, which is known as the common-practice period. The major time divisions of Western art music are as follows:

Western culture Heritage of norms, customs, belief and political systems, and artifacts and technologies associated with Europe (both indigenous and foreign origin)

Western culture, sometimes equated with Western civilization, Occidental culture, the Western world, Western society, and European civilization, is a term used very broadly to refer to a heritage of social norms, ethical values, traditional customs, belief systems, political systems and specific artifacts and technologies that have some origin or association with Europe. The term also applies beyond Europe to countries and cultures whose histories are strongly connected to Europe by immigration, colonization, or influence. For example, Western culture includes countries in the Americas and Australasia, whose language and demographic ethnicity majorities are European. The development of western culture has been strongly influenced by Christianity.

Musical note sign used in musical notation, a pitched sound

In music, a note is the pitch and duration of a sound, and also its representation in musical notation. A note can also represent a pitch class. Notes are the building blocks of much written music: discretizations of musical phenomena that facilitate performance, comprehension, and analysis.

The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note.

Harmonic series (music) sequence of sounds where the base frequency of each sound is an integer multiple of the lowest base frequency

A harmonic series is the sequence of sounds—pure tones, represented by sinusoidal waves—in which the frequency of each sound is an integer multiple of the fundamental, the lowest frequency.

Dominant (music) fifth scale degree of the diatonic scale, between the subdominant and the submediant

In music, the dominant is the fifth scale degree of the diatonic scale, called "dominant" because it is next in importance to the tonic, and a dominant chord is any chord built upon that pitch, using the notes of the same diatonic scale. The dominant is sung as so in solfege. The dominant function has the role of creating instability that requires the tonic for resolution.

In very much conventionally tonal music, harmonic analysis will reveal a broad prevalence of the primary harmonies: tonic, dominant, and subdominant, and especially the first two of these.

The scheme I-x-V-I symbolizes, though naturally in a very summarizing way, the harmonic course of any composition of the Classical period. This x, usually appearing as a progression of chords, as a whole series, constitutes, as it were, the actual "music" within the scheme, which through the annexed formula V-I, is made into a unit, a group, or even a whole piece.

Tonic (music) tonal center of a diatonic scale

In music, the tonic is the first scale degree of a diatonic scale and the tonal center or final resolution tone that is commonly used in the final cadence in tonal classical music, popular music and traditional music. The triad formed on the tonic note, the tonic chord, is thus the most significant chord in these styles of music. More generally, the tonic is the pitch upon which all other pitches of a piece are hierarchically referenced. Scales are named after their tonics, thus the tonic of the scale of C is the note C.

In very much conventionally tonal music, harmonic analysis will reveal a broad prevalence of the primary harmonies: tonic, dominant, and subdominant, and especially the first two of these.

The perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. [2] Its inversion is the perfect fourth. The octave of the fifth is the twelfth.

Consonance and dissonance categorizations of simultaneous or successive sounds

In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Consonance is associated with sweetness, pleasantness, and acceptability; dissonance is associated with harshness, unpleasantness, or unacceptability.

Unison musical parts sounding at the same pitch

In music, unison is two or more musical parts sounding the same pitch or at an octave interval, usually at the same time.

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.

A perfect fifth is at the start of "Twinkle, Twinkle, Little Star"; the pitch of the first "twinkle" is the root note and pitch of the second "twinkle" is a perfect fifth above it.

Twinkle, Twinkle, Little Star childrens song

"Twinkle, Twinkle, Little Star" is a popular English lullaby. The lyrics are from an early-19th-century English poem by Jane Taylor, "The Star". The poem, which is in couplet form, was first published in 1806 in Rhymes for the Nursery, a collection of poems by Taylor and her sister Ann. It is sung to the tune of the French melody Ah! vous dirai-je, maman, which was published in 1761 and later arranged by several composers including Mozart with Twelve Variations on "Ah vous dirai-je, Maman". The English lyrics have five stanzas, although only the first is widely known. It has a Roud Folk Song Index number of 7666. This song is usually performed in the key of C major.

Alternative definitions

The term perfect identifies the perfect fifth as belonging to the group of perfect intervals (including the unison, perfect fourth and octave), so called because of their simple pitch relationships and their high degree of consonance. [3] When an instrument with only twelve notes to an octave (such as the piano) is tuned using Pythagorean tuning, one of the twelve fifths (the wolf fifth) sounds severely discordant and can hardly be qualified as "perfect", if this term is interpreted as "highly consonant". However, when using correct enharmonic spelling, the wolf fifth in Pythagorean tuning or meantone temperament is actually not a perfect fifth but a diminished sixth (for instance G–E).

Perfect fourth musical interval

A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth is the fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished and augmented fourths span the same number of staff positions, but consist of a different number of semitones.

Pythagorean tuning

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ~702 cents wide.

Wolf interval particularly dissonant musical interval spanning seven semitones present in most meantone tuning systems

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including most meantone temperaments.

Perfect intervals are also defined as those natural intervals whose inversions are also perfect, where natural, as opposed to altered, designates those intervals between a base note and another note in the major diatonic scale starting at that base note (for example, the intervals from C to C, D, E, F, G, A, B, C, with no sharps or flats); this definition leads to the perfect intervals being only the unison, fourth, fifth, and octave, without appealing to degrees of consonance. [4]

The term perfect has also been used as a synonym of just , to distinguish intervals tuned to ratios of small integers from those that are "tempered" or "imperfect" in various other tuning systems, such as equal temperament. [5] [6] The perfect unison has a pitch ratio 1:1, the perfect octave 2:1, the perfect fourth 4:3, and the perfect fifth 3:2.

Within this definition, other intervals may also be called perfect, for example a perfect third (5:4) [7] or a perfect major sixth (5:3). [8]

Other qualities

In addition to perfect, there are two other kinds, or qualities, of fifths: the diminished fifth, which is one chromatic semitone smaller, and the augmented fifth, which is one chromatic semitone larger. In terms of semitones, these are equivalent to the tritone (or augmented fourth), and the minor sixth, respectively.

Pitch ratio

Just perfect fifth on D
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. The perfect fifth above D (A+, 27/16) is a syntonic comma (81/80 or 21.5 cents) higher than the just major sixth above middle C: (A♮, 5/3). Just perfect fifth on D.png
Just perfect fifth on D Loudspeaker.svg Play  . The perfect fifth above D (A+, 27/16) is a syntonic comma (81/80 or 21.5 cents) higher than the just major sixth above middle C: (A, 5/3).
Just perfect fifth below A
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. The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second above middle C: (D♮, 9/8). Just perfect fifth below A.png
Just perfect fifth below A Loudspeaker.svg Play  . The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second above middle C: (D, 9/8).

The justly tuned pitch ratio of a perfect fifth is 3:2 (also known, in early music theory, as a hemiola ), [10] [11] [12] meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. The just perfect fifth can be heard when a violin is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune.

Keyboard instruments such as the piano normally use an equal-tempered version of the perfect fifth, enabling the instrument to play in all keys. In 12-tone equal temperament, the frequencies of the tempered perfect fifth are in the ratio or approximately 1.498307. An equally tempered perfect fifth, defined as 700 cents, is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents.

Kepler explored musical tuning in terms of integer ratios, and defined a "lower imperfect fifth" as a 40:27 pitch ratio, and a "greater imperfect fifth" as a 243:160 pitch ratio. [13] His lower perfect fifth ratio of 1.48148 (680 cents) is much more "imperfect" than the equal temperament tuning (700 cents) of 1.4983 (relative to the ideal 1.50). Helmholtz uses the ratio 301:200 (708 cents) as an example of an imperfect fifth; he contrasts the ratio of a fifth in equal temperament (700 cents) with a "perfect fifth" (3:2), and discusses the audibility of the beats that result from such an "imperfect" tuning. [14]

Use in harmony

W. E. Heathcote describes the octave as representing the prime unity within the triad, a higher unity produced from the successive process: "first Octave, then Fifth, then Third, which is the union of the two former". [15] Hermann von Helmholtz argues that some intervals, namely the perfect fourth, fifth, and octave, "are found in all the musical scales known", though the editor of the English translation of his book notes the fourth and fifth may be interchangeable or indeterminate. [16]

The perfect fifth is a basic element in the construction of major and minor triads, and their extensions. Because these chords occur frequently in much music, the perfect fifth occurs just as often. However, since many instruments contain a perfect fifth as an overtone, it is not unusual to omit the fifth of a chord (especially in root position).

The perfect fifth is also present in seventh chords as well as "tall tertian" harmonies (harmonies consisting of more than four tones stacked in thirds above the root). The presence of a perfect fifth can in fact soften the dissonant intervals of these chords, as in the major seventh chord in which the dissonance of a major seventh is softened by the presence of two perfect fifths.

Chords can also be built by stacking fifths, yielding quintal harmonies. Such harmonies are present in more modern music, such as the music of Paul Hindemith. This harmony also appears in Stravinsky's The Rite of Spring in the "Dance of the Adolescents" where four C trumpets, a piccolo trumpet, and one horn play a five-tone B-flat quintal chord. [17]

Bare fifth, open fifth, or empty fifth

A bare fifth, open fifth or empty fifth is a chord containing only a perfect fifth with no third. The closing chords of Pérotin's Viderunt Omnes and Sederunt Principes, Guillaume de Machaut's Messe de Nostre Dame , the Kyrie in Mozart's Requiem , and the first movement of Bruckner's Ninth Symphony are all examples of pieces ending on an open fifth. These chords are common in Medieval music, sacred harp singing, and throughout rock music. In hard rock, metal, and punk music, overdriven or distorted electric guitar can make thirds sound muddy while the bare fifths remain crisp. In addition, fast chord-based passages are made easier to play by combining the four most common guitar hand shapes into one. Rock musicians refer to them as power chords . Power chords often include octave doubling (i.e., their bass note is doubled one octave higher, e.g. F3-C4-F4).

E5 power chord in eighth notes
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E5 power chord in eighth notes Loudspeaker.svg play  

An empty fifth is sometimes used in traditional music, e.g., in Asian music and in some Andean music genres of pre-Columbian origin, such as k'antu and sikuri . The same melody is being led by parallel fifths and octaves during all the piece. Examples: Loudspeaker.svg Play k'antu  , Loudspeaker.svg Play pacha siku  .

Western composers may use the interval to give a passage an exotic flavor. [18] Empty fifths are also sometimes used to give a cadence an ambiguous quality, as the bare fifth does not indicate a major or minor tonality.

Use in tuning and tonal systems

The just perfect fifth, together with the octave, forms the basis of Pythagorean tuning. A slightly narrowed perfect fifth is likewise the basis for meantone tuning.[ citation needed ]

The circle of fifths is a model of pitch space for the chromatic scale (chromatic circle), which considers nearness as the number of perfect fifths required to get from one note to another, rather than chromatic adjacency.

See also

Related Research Articles

Equal temperament The musical tuning system where the ratio between successive notes is constant

An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In other words, the ratios of the frequencies of any adjacent pair of notes is the same, and, as pitch is perceived roughly as the logarithm of frequency, equal perceived "distance" from every note to its nearest neighbor.

In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. A tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half.

Enharmonic (in modern musical notation and tuning) note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in twelve-tone equal temperament, the notes C and D are enharmonic notes. Namely, they are the same key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony and chord progressions. Arbitrary amounts of accidentals can produce further enharmonic equivalents, such as B, although these are much rarer and have less practical use.

Diesis

In classical music from Western culture, a diesis is either an accidental, or a very small musical interval, usually defined as the difference between an octave and three justly tuned major thirds, equal to 128:125 or about 41.06 cents. In 12-tone equal temperament three major thirds in a row equal an octave, but three justly-tuned major thirds fall quite a bit narrow of an octave, and the diesis describes the amount by which they are short. For instance, an octave (2:1) spans from C to C', and three justly tuned major thirds (5:4) span from C to B. The difference between C-C' (2:1) and C-B (125:64) is the diesis (128:125). Notice that this coincides with the interval between B and C', also called a diminished second.

Semitone musical interval

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.

Major third musical interval

In classical music from Western culture, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the minor third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two: the major third spans four semitones, the minor third three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, and there are three staff positions from C to E. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones.

Minor third musical interval

In the music theory of Western culture, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A, and (coincidentally) there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.

Major sixth musical interval

In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions, and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two. The major sixth spans nine semitones. Its smaller counterpart, the minor sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. Diminished and augmented sixths span the same number of note letter names and staff positions, but consist of a different number of semitones.

Minor sixth musical interval

In classical music from Western culture, a sixth is a musical interval encompassing six staff positions, and the minor sixth is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two: the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a minor sixth, as the note F lies eight semitones above A, and there are six staff positions from A to F. Diminished and augmented sixths span the same number of staff positions, but consist of a different number of semitones.

Minor chord chord having a root, a minor third, and a perfect fifth; e.g.  A–C–E or C–E♭–G

In music theory, a minor chord is a chord having a root, a minor third, and a perfect fifth. When a chord has these three notes alone, it is called a minor triad. Some minor triads with additional notes, such as the minor seventh chord, may also be called minor chords.

Comma (music) small musical interval, the difference between two tunings of the same note

In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81/80 are considered the same note because the 12 note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

Quarter-comma meantone, or 14-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80), with respect to its just intonation used in Pythagorean tuning ; the result is . The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

Music and mathematics

Music theory has no axiomatic foundation in modern mathematics, 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.

Interval ratio ratio of the frequencies of the pitches in a musical interval

In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2, 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12. If the A above middle C is 440 Hz, the perfect fifth above it would be E, at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.

References

  1. Don Michael Randel (2003), "Interval", Harvard Dictionary of Music, fourth edition (Cambridge, MA: Harvard University Press): p. 413.
  2. William Smith and Samuel Cheetham (1875). A Dictionary of Christian Antiquities. London: John Murray. p. 550.
  3. Walter Piston and Mark DeVoto (1987), Harmony, 5th ed. (New York: W. W. Norton), p. 15. ISBN   0-393-95480-3. Octaves, perfect intervals, thirds, and sixths are classified as being "consonant intervals", but thirds and sixths are qualified as "imperfect consonances".
  4. Kenneth McPherson Bradley (1908). Harmony and Analysis. C. F. Summy Co. p. 17.
  5. Charles Knight (1843). Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge. Society for the Diffusion of Useful Knowledge. p. 356.
  6. John Stillwell (2006). Yearning for the Impossible. A K Peters, Ltd. p. 21. ISBN   1-56881-254-X.
  7. Llewelyn Southworth Lloyd (1970). Music and Sound. Ayer Publishing. p. 27. ISBN   0-8369-5188-3.
  8. John Broadhouse (1892). Musical Acoustics. W. Reeves. p. 277.
  9. 1 2 Fonville, John. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.109, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106-137.
  10. Willi Apel (1969). Harvard Dictionary of Music (second ed.). Harvard University Press. p. 382. ISBN   978-0-674-37501-7.[ full citation needed ]
  11. Don Michael Randel (ed.) (1986), New Harvard Dictionary of Music, [third edition] (Cambridge, MA: Belknap Press of Harvard University Press, 1986), p. 376.[ full citation needed ]
  12. Don Michael Randel (ed.) (2003), "Hemiola", Harvard Dictionary of Music, fourth edition (Cambridge, MA: Harvard University Press): p. 389.
  13. Johannes Kepler (2004). Stephen W. Hawking, ed. Harmonies of the World. Running Press. p. 22. ISBN   0-7624-2018-9.
  14. Hermann von Helmholtz (1912). On the Sensations of Tone as a Physiological Basis for the Theory of Music. Longmans, Green. pp. 199, 313.
  15. W. E. Heathcote (1888), "Introductory Essay"", in Moritz Hauptmann, The Nature of Harmony and Metre , translated and edited by W. E. Heathcote (London: Swan Sonnenschein & Co.), p.xx.
  16. Hermann von Helmholtz (1912). On the Sensations of Tone as a Physiological Basis for the Theory of Music. p. 253.
  17. Piston and DeVoto (1987), pp. 503–505.
  18. Scott Miller, "Inside The King and I", New Line Theatre, accessed December 28, 2012