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== Theory and measurement == Music theorists have long speculated that blue notes are intervals of [[just intonation]]<ref name=":2">{{Cite book|last=Kubik|first=Gerhard, 1934-|title=Africa and the blues|date=1999|publisher=University Press of Mississippi|isbn=0-585-20318-0|location=Jackson, Miss.|oclc=44959610}}</ref><ref>{{Cite journal|last=Kubik|first=Gerhard|date=2005|title=The African Matrix in Jazz Harmonic Practices|journal=Black Music Research Journal|volume=25|issue=1/2|pages=167–222|jstor=30039290|issn=0276-3605}}</ref><ref name=":3">Kubik, G. (2008). Bourdon, blue notes, and pentatonicism in the blues: An Africanist perspective. In D. Evans (Ed.), ''Ramblin’ on my mind: New perspectives on the blues'' (pp. 11–48)''.'' Urbana, IL: University of Illinois Press.</ref><ref name=":4">{{Cite journal|last=Curry|first=Ben|date=2015|title=Blues music theory and the songs of Robert Johnson: ladder, level and chromatic cycle|url=https://www.cambridge.org/core/journals/popular-music/article/blues-music-theory-and-the-songs-of-robert-johnson-ladder-level-and-chromatic-cycle/2E4C3BEECE5EF371DBA79A32498BD745|journal=Popular Music|language=en|volume=34|issue=2|pages=245–273|doi=10.1017/S0261143015000276|s2cid=145765888 |issn=0261-1430|url-access=subscription}}</ref><ref name=":5">{{Cite journal|last=Curry|first=Ben|date=2017|title=Two Approaches to Tonal Space in the Music of Muddy Waters: Two Approaches to Tonal Space|journal=Music Analysis|language=en|volume=36|issue=1|pages=37–58|doi=10.1111/musa.12084|s2cid=126072443 |url=http://pure-oai.bham.ac.uk/ws/files/43012584/Curry_two_approaches.pdf}}</ref><ref name=":6">{{Cite book|last=Van der Merwe|first=Peter|title=Origins of the popular style : the antecedents of twentieth-century popular music|date=1989|publisher=Clarendon Press|isbn=0-19-316121-4|location=Oxford [Oxfordshire]|oclc=18071070}}</ref> not derived from European [[Equal temperament|12-tone equal temperament tuning]]. Just intonation musical intervals derive directly from the [[Harmonic series (music)|harmonic series]]. Humans naturally learn the harmonic series as infants. This is essential for many auditory activities such as understanding speech (see [[formant]]) and perceiving tonal music.<ref>{{Cite book|last=Helmholtz|first=Hermann von|title=On the sensations of tone as a physiological basis for the theory of music|others=Ellis, Alexander John, 1814-1890.|isbn=0-486-60753-4|edition=Second English edition, translated thoroughly revised and corrected, rendered conformal to the 4th (and last) German ed. of 1877, with numerous additional notes and a new additional appendix bringing down information to 1885, and especially adapted to the use of music students by Alexander J. Ellis|location=New York|oclc=385076}}</ref> In the harmonic series, overtones of a fundamental tonic tone occur as integer multiples of the tonic frequency. It is therefore convenient to express musical intervals in this system as integer ratios (e.g. {{Fraction|2|1}} = octave, {{frac|3|2}} = perfect fifth, etc.). The relationship between just and equal temperament tuning is conveniently expressed using the 12-tone equal temperament cents system. Just intonation is common in music of other cultures such as the [[17 equal temperament|17-tone Arabic scale]] and the [[22 equal temperament|22-tone Indian classical music scale]].<ref>{{Cite book|last=Danielou|first=Alain|title=Introduction to the Study of Musical Scales.|publisher=Oriental Book Reprint Corporation|year=1999|isbn=8170690986}}</ref> In African cultures, just intonation scales are the norm rather than the exception.<ref>{{Cite book|last=Kubik|first=Gerhard|title=Theory of African music|date=2010|publisher=The University of Chicago Press|isbn=978-0-226-45690-4|location=Chicago|oclc=457769452}}</ref> As the blues appears to have derived from ''a cappella'' field hollers of African slaves, it would be expected that its notes would be of just intonation origin closely related to the musical scales of western Africa.<ref name=":7">{{Cite book|last=Oliver|first=Paul|title=Savannah syncopators: African retentions in the blues|publisher=Studio Vista|year=1970|location=London}}</ref><ref name=":2" /><ref name=":3" /> The blue "lowered third" has been speculated to be from {{frac|7|6}} (267 [[Cent (music)|cents]])<ref name=":3" /><ref name=":4" /> to 350 cents<ref name=":6" /> above the [[Tonic (music)|tonic]] tone. It has recently been found empirically to center at {{frac|6|5}} (316 cents, a [[minor third]] in just intonation, or a slightly sharp minor third in equal temperament) based on cluster analysis of a large number of blue notes from early blues recordings.<ref name=":8">{{Cite journal|last=Cutting|first=Court B|date=2019-01-17|title=Microtonal Analysis of "Blue Notes" and the Blues Scale|url=http://emusicology.org/article/view/6316|journal=Empirical Musicology Review|volume=13|issue=1–2|pages=84–99|doi=10.18061/emr.v13i1-2.6316|issn=1559-5749|doi-access=free}}</ref> This note is commonly slurred with a [[major third]] justly tuned at {{frac|5|4}} (386 cents)<ref name=":8" /> in what Temperley et al.<ref>{{Cite journal|last=Temperley|first=David|date=2017|title=Mediant mixture and "blue notes" in rock: An exploratory study.|url=https://mtosmt.org/issues/mto.17.23.1/mto.17.23.1.temperley.html|journal=Music Theory Online|volume=23|doi=10.30535/mto.23.1.7|doi-access=free}}</ref> refer to as a "neutral third". This bending or glide between the two tones is an essential characteristic of the blues.<ref name=":0" /><ref name=":1" /><ref name=":3" /><ref name=":4" /><ref name=":5" /> The blue "lowered fifth" has been found to be quite separate from the perfect fifth and clusters with the perfect fourth with which it is commonly slurred. This "raised fourth" is most commonly expressed at {{frac|7|5}} (583 cents).<ref name=":8" /> The [[eleventh harmonic]] (i.e. {{frac|11|8}} or 551 cents) as put forward by Kubik<ref name=":3" /> and Curry<ref name=":4" /> is also possible as it is in the middle of the slur between the perfect fourth at {{frac|4|3}} and {{frac|7|5}}. The blue "lowered seventh" appears to have two common locations at {{frac|7|4}} (969 cents) and {{frac|9|5}} (1018 cents).<ref name=":8" /> Kubik<ref name=":3" /> and Curry<ref name=":4" /> proposed {{frac|7|4}} as it is commonly heard in the barbershop quartet [[harmonic seventh chord]].<ref name=":9">{{Cite book|last=Averill|first=Gage|url=http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780195116724.001.0001/acprof-9780195116724|title=Four Parts, No Waiting|date=2003-02-20|publisher=Oxford University Press|isbn=978-0-19-511672-4|doi=10.1093/acprof:oso/9780195116724.001.0001}}</ref> The barbershop quartet idiom also appears to have arisen from African American origins.<ref>{{Cite journal|last=Abbott|first=Lynn|date=1992|title="Play That Barber Shop Chord": A Case for the African-American Origin of Barbershop Harmony|journal=American Music|volume=10|issue=3|pages=289–325|doi=10.2307/3051597|jstor=3051597}}</ref><ref name=":9" /> It was a surprising finding that {{frac|9|5}} was a much more common tonal location although both were used in the blues, sometimes within the same song.<ref name=":8" /> It should not be surprising that blue notes are not represented accurately in the 12-tone equal temperament system, which is made up of a cycle of very slightly flattened perfect fifths (i.e. {{frac|3|2}}). The just intonation blue note intervals identified above all involve prime numbers not equally divisible by 2 or 3. Prime-number harmonics greater than 3 are all perceptually different from 12-tone equal temperament notes. The blues has likely evolved as a fusion of an African just intonation scale with European 12-tone musical instruments and harmony.<ref name=":7" /><ref name=":2" /> The result has been a uniquely American music which is still widely practiced in its original form and is at the foundation of another genre, [[Jazz|American jazz]].
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