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[[Image:Perfect fifth on C.png|thumb|right|Perfect fifth {{audio|Perfect fifth on C.mid|Play}} equal tempered and {{audio|Just perfect fifth on C.mid|Play}} just.]]
 
{{Infobox Interval|
      main_interval_name = perfect fifth|
      inverse = [[perfect fourth]]|
      complement = [[perfect fourth]]|
      other_names = diapente|
      abbreviation = P5 |
      semitones = 7 |
      interval_class = 5 |
      just_interval = 3:2|
      cents_equal_temperament = 700|
      cents_24T_equal_temperament = 700|
      cents_just_intonation = 702
}}
[[Image:Perfect5s.PNG|right|thumb|Examples of perfect fifth intervals]]
In [[classical music]] from [[Western culture]], a '''fifth''' is a [[Interval (music)|musical interval]] encompassing five [[staff position]]s (see [[Interval (music)#Number|Interval number]] for more details), and the '''perfect fifth''' (often abbreviated '''P5''') is a fifth spanning seven [[semitone]]s, or in [[Meantone temperament|meantone]], four [[diatonic]] semitones and three [[chromatic (music)|chromatic]] semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five staff positions from C to G. [[Diminished fifth|Diminished]] and [[augmented fifth]]s span the same number of staff positions, but consist of a different number of semitones (six and eight, respectively).
 
The perfect fifth may be derived from the [[Harmonic series (music)|harmonic series]] as the interval between the second and third harmonics. In a [[diatonic scale]], the [[dominant (music)|dominant]] note is a perfect fifth above the [[tonic (music)|tonic]] note.
 
The perfect fifth is more [[consonance and dissonance|consonant]], or stable, than any other interval except the [[unison]] and the [[octave]]. It occurs above the [[root (chord)|root]] of all [[Major chord|major]] and [[Minor chord|minor]] chords (triads) and their [[extended chords|extensions]]. Until the late 19th century, it was often referred to by one of its Greek names, '''''diapente'''''.<ref>{{cite book | title = A Dictionary of Christian Antiquities | author = William Smith and Samuel Cheetham | publisher = London: John Murray | year = 1875 | page = 550 | url = http://books.google.com/books?id=1LIPFk6oFVkC&pg=PA550&dq=diatessaron+diapason+diapente+fourth+fifth }}</ref> Its [[inversion (music)|inversion]] is the [[perfect fourth]].  The octave of the fifth is the twelfth.
 
==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 and dissonance|consonance]].<ref>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".</ref> Note that this interpretation of the term is not in all contexts compatible with the definition of ''perfect fifth'' given in the introduction. In fact, 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 interval|wolf fifth]]) sounds severely dissonant 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{{Music|sharp}}–E{{Music|flat}}).
 
Perfect intervals are also defined as those natural intervals whose [[inversion (music)|inversions]] are also perfect, where natural, as opposed to altered, designates those intervals between a base note and the major diatonic scale starting at that 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]], [[perfect fourth|fourth]], fifth, and [[octave]], without appealing to degrees of consonance.<ref>{{cite book | title = Harmony and Analysis | author =
Kenneth McPherson Bradley | publisher = C. F. Summy Co. | year =  1908 | page = 17 | url = http://books.google.com/books?id=QsAPAAAAYAAJ&pg=PA16&dq=intitle:Harmony+perfect-interval#PPA17,M1 }}</ref>
 
The term ''perfect'' has also been used as a synonym of ''[[just interval|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]].<ref>{{cite book | title = Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge | author = Charles Knight | publisher = Society for the Diffusion of Useful Knowledge | year = 1843 | page = 356 | url = http://books.google.com/books?id=muBPAAAAMAAJ&pg=PA356&dq=%22perfect+fifth%22+%22imperfect+fifth%22+tempered }}</ref><ref>{{cite book | title = Yearning for the Impossible | author = John Stillwell | publisher = A K Peters, Ltd. | year = 2006 | isbn = 1-56881-254-X | page = 21 | url = http://books.google.com/books?id=YonsAAWIA7sC&pg=PA21&dq=%22perfect+fifth%22+%22imperfect+fifth%22+tempered }}</ref> The perfect unison has a [[interval ratio|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)<ref>{{cite book | title = Music and Sound | author = Llewelyn Southworth Lloyd | publisher = Ayer Publishing | year = 1970 | isbn = 0-8369-5188-3 | page = 27 | url = http://books.google.com/books?id=LxTwmfDvTr4C&pg=PA27&dq=%22perfect+third%22++%22perfect+major%22 }}</ref> or a perfect [[major sixth]] (5:3).<ref>{{cite book | title = Musical Acoustics | author = John Broadhouse | publisher = W. Reeves | year = 1892 | page = 277 | url = http://books.google.com/books?id=l9c5AAAAIAAJ&pg=PA277&dq=%22perfect+major+sixth%22+ratio }}</ref>
 
==Other qualities of fifth==
 
In addition to perfect, there are two other kinds, or qualities, of fifths: the [[diminished fifth]], which is one [[semitone|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.
 
==The pitch ratio of a fifth==
[[Image:Just perfect fifth on D.png|thumb|right|Just perfect fifth on D {{audio|Just perfect fifth on D.mid|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]] (A{{music|natural}}, 5/3).<ref name="Fonville">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.</ref>]]
[[Image:Just perfect fifth below A.png|thumb|right|Just perfect fifth below A {{audio|Just perfect fifth below A.mid|Play}}. The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second (D{{music|natural}}, 9/8).<ref name="Fonville"/>]]
 
The justly intoned [[interval ratio|pitch ratio]] of a perfect fifth is 3:2 (also known, in early music theory, as a ''[[hemiola]]''<ref>
{{cite book
| title = Harvard dictionary of music
| edition = 2nd
| author = Willi Apel
| publisher = Harvard University Press
| year = 1969
| isbn = 978-0-674-37501-7
| page = 382
| url = http://books.google.com/books?id=TMdf1SioFk4C&pg=PA382
}}</ref><ref>Don Michael Randel (ed.), ''New Harvard Dictionary of Music'' (Cambridge, MA: Belknap Press of Harvard University Press, 1986), p.&nbsp;376.</ref>), meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. In the [[Cent (music)|cent]] system of pitch measurement, the 3:2 ratio corresponds to approximately 702 cents, or 2% of a semitone wider than seven semitones. 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. Just perfect fifths are employed in [[just intonation]]. The 3:2 just perfect fifth arises in the C [[major scale]] between C and G.<ref>Paul, Oscar (1885). ''[http://books.google.com/books?id=4WEJAQAAMAAJ&dq=musical+interval+%22pythagorean+major+third%22&source=gbs_navlinks_s A manual of harmony for use in music-schools and seminaries and for self-instruction]'', p.165. Theodore Baker, trans. G. Schirmer.</ref> {{audio|Just perfect fifth on C.mid|Play}}
 
[[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.<ref>{{cite book | title = Harmonies of the World | author = Johannes Kepler | editor = Stephen W. Hawking | publisher =  Running Press| year = 2004 | isbn = 0-7624-2018-9 | page = 22 | url = http://books.google.com/books?id=br1sKvZPIKQC&pg=PA22&dq=%22perfect+fifth%22+%22imperfect+fifth%22 }}</ref> His lower perfect fifth ratio of 1.4815 (680 cents) is much more "imperfect" than the equal temperament tuning (700 cents) of 1.498 (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 [[beat (acoustics)|beats]] that result from such an "imperfect" tuning.<ref>{{cite book | title = On the Sensations of Tone as a Physiological Basis for the Theory of Music | author =  Hermann von Helmholtz | publisher = Longmans, Green | year = 1912 | url = http://books.google.com/books?id=po6fAAAAMAAJ&dq=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor%3AHelmholtz%20tempered&pg=PA199#v=onepage&q=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor:Helmholtz%20tempered&f=false | pages = 199, 313}}</ref>
 
In keyboard instruments such as the [[piano]], a slightly different version of the perfect fifth is normally used: in accordance with the principle of [[equal temperament]], the perfect fifth is slightly narrowed to exactly 700 cents (seven [[semitones]]). (The narrowing is necessary to enable the instrument to play in all keys.) Many people can hear the slight deviation from the idealized perfect fifth when they play the interval on a piano.
 
==Use in harmony==
 
[[Moritz Hauptmann]] describes the octave as a higher unity appearing as such within the triad, produced from the prime unity of first the octave, then fifth, then third, "which is the union of the former."<ref>[http://books.google.com/books?id=a8o5AAAAIAAJ&pg=PR20&dq=%22first+octave%22+%22then+fifth%22&hl=en&ei=1d4qTbyMBsGnnQf81bXmAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA#v=onepage&q=%22first%20octave%22%20%22then%20fifth%22&f=false Hauptmann, Moritz (1888). ''The nature of harmony and metre, Volume 1888, Part 1'', p.xx.]</ref> 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.<ref>{{cite book | title = On the Sensations of Tone as a Physiological Basis for the Theory of Music | author =  Hermann von Helmholtz | year = 1912 | url = http://books.google.com/books?id=po6fAAAAMAAJ&dq=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor%3AHelmholtz%20tempered&pg=PA199#v=onepage&q=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor:Helmholtz%20tempered&f=false | pages = 253}}</ref>
 
The perfect fifth is a basic element in the construction of major and minor [[triad (music)|triad]]s, and their [[extended chords|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 [[Consonance and dissonance|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.
 
One can also build chords 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 [[Trumpet]]s, a [[Piccolo Trumpet]], and one [[Horn (instrument)|Horn]] play a five-tone B-flat quintal chord.<ref>Piston and DeVoto (1987), p. 503–505.</ref>
 
==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 chord of the [[Kyrie]] in [[Wolfgang Amadeus Mozart|Mozart]]'s [[Requiem (Mozart)|Requiem]] and of the first movement of [[Anton Bruckner|Bruckner]]'s [[Symphony No. 9 (Bruckner)|Ninth Symphony]] are both examples of pieces ending on an empty fifth. These "chords" are common in [[Sacred Harp]] singing and throughout [[rock music]]. In [[hard rock]], [[heavy metal music|metal]], and [[punk music]], overdriven or distorted guitar can make thirds sound muddy while the '''bare fifth''' remains 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 chord]]s'' and often include octave doubling (i.e., their bass note is doubled one octave higher, e.g. F3-C4-F4).
 
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. Hear examples: {{audio|Kantu.mid|Play K'antu}}, {{audio|Antara.mid|Play Pacha Siku}}.
 
Western composers may use the interval to give a passage an exotic flavor.<ref>Scott Miller, "[http://www.newlinetheatre.com/kingandichapter.html Inside ''The King and I'']", ''New Line Theatre'', accessed December 28, 2012</ref>
 
==Use in tuning and tonal systems==
A perfect fifth in [[just intonation]], a '''just fifth''', corresponds to a frequency ratio of 3:2, while in 12-tone [[equal temperament|equally-tempered]] fifth, the frequencies are in the ratio <math>(\sqrt [12]{2})^7</math>, approximately 1.498307.
Measured against an equally-tempered scale, where each semitone spans 100 [[cent (music)|cents]], a just fifth is approximately 701.955001 cents, about two cents larger than the equally tempered fifth.
 
The just perfect fifth, together with the [[octave]], forms the basis of [[Pythagorean tuning]]. A flattened perfect fifth is likewise the basis for [[meantone temperament|meantone]] tuning.
 
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.
 
==References==
{{Reflist}}
 
==See also==
 
* [[All fifths]]
* [[Circle of fifths]]
 
{{Intervals}}
 
{{DEFAULTSORT:Perfect Fifth}}
[[Category:Fifths]]
[[Category:Perfect intervals]]
[[Category:3-limit tuning and intervals]]

Revision as of 12:44, 27 June 2013

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Template:Infobox Interval

Examples of perfect fifth intervals

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions (see Interval number for more details), and the perfect fifth (often abbreviated P5) is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five staff positions from C to G. Diminished and augmented fifths span the same number of staff positions, but consist of a different number of semitones (six and eight, respectively).

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.

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.[1] Its inversion is the perfect fourth. The octave of the fifth is the twelfth.

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.[2] Note that this interpretation of the term is not in all contexts compatible with the definition of perfect fifth given in the introduction. In fact, 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 dissonant 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 GTemplate:Music–ETemplate:Music).

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 the major diatonic scale starting at that 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.[3]

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.[4][5] 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)[6] or a perfect major sixth (5:3).[7]

Other qualities of fifth

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.

The pitch ratio of a fifth

Just perfect fifth on D My name: Lindsey Gavin
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Look into my weblog :: http://www.hostgator1centcoupon.info/. The perfect fifth above D (A+, 27/16) is a syntonic comma (81/80 or 21.5 cents) higher than the just major sixth (ATemplate:Music, 5/3).[8]
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Look into my weblog :: http://www.hostgator1centcoupon.info/. The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second (DTemplate:Music, 9/8).[8]

The justly intoned pitch ratio of a perfect fifth is 3:2 (also known, in early music theory, as a hemiola[9][10]), meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. In the cent system of pitch measurement, the 3:2 ratio corresponds to approximately 702 cents, or 2% of a semitone wider than seven semitones. 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. Just perfect fifths are employed in just intonation. The 3:2 just perfect fifth arises in the C major scale between C and G.[11] My name: Lindsey Gavin
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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.[12] His lower perfect fifth ratio of 1.4815 (680 cents) is much more "imperfect" than the equal temperament tuning (700 cents) of 1.498 (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.[13]

In keyboard instruments such as the piano, a slightly different version of the perfect fifth is normally used: in accordance with the principle of equal temperament, the perfect fifth is slightly narrowed to exactly 700 cents (seven semitones). (The narrowing is necessary to enable the instrument to play in all keys.) Many people can hear the slight deviation from the idealized perfect fifth when they play the interval on a piano.

Use in harmony

Moritz Hauptmann describes the octave as a higher unity appearing as such within the triad, produced from the prime unity of first the octave, then fifth, then third, "which is the union of the former."[14] 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.[15]

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.

One can also build chords 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.[16]

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 chord of the Kyrie in Mozart's Requiem and of the first movement of Bruckner's Ninth Symphony are both examples of pieces ending on an empty fifth. These "chords" are common in Sacred Harp singing and throughout rock music. In hard rock, metal, and punk music, overdriven or distorted guitar can make thirds sound muddy while the bare fifth remains 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 and often include octave doubling (i.e., their bass note is doubled one octave higher, e.g. F3-C4-F4).

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. Hear examples: My name: Lindsey Gavin
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Western composers may use the interval to give a passage an exotic flavor.[17]

Use in tuning and tonal systems

A perfect fifth in just intonation, a just fifth, corresponds to a frequency ratio of 3:2, while in 12-tone equally-tempered fifth, the frequencies are in the ratio , approximately 1.498307. Measured against an equally-tempered scale, where each semitone spans 100 cents, a just fifth is approximately 701.955001 cents, about two cents larger than the equally tempered fifth.

The just perfect fifth, together with the octave, forms the basis of Pythagorean tuning. A flattened perfect fifth is likewise the basis for meantone tuning.

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.

References

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See also

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  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. ↑ 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".
  3. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  5. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  6. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  7. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  8. ↑ 8.0 8.1 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.
  9. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  10. ↑ Don Michael Randel (ed.), New Harvard Dictionary of Music (Cambridge, MA: Belknap Press of Harvard University Press, 1986), p. 376.
  11. ↑ Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction, p.165. Theodore Baker, trans. G. Schirmer.
  12. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  13. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  14. ↑ Hauptmann, Moritz (1888). The nature of harmony and metre, Volume 1888, Part 1, p.xx.
  15. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  16. ↑ Piston and DeVoto (1987), p. 503–505.
  17. ↑ Scott Miller, "Inside The King and I", New Line Theatre, accessed December 28, 2012
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