Re: Pitch learning (Complex Systems )


Subject: Re: Pitch learning
From:    Complex Systems  <large@xxxxxxxx>
Date:    Wed, 28 Feb 2007 19:18:52 -0500
List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>

--Apple-Mail-7--1019879744 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=WINDOWS-1252; delsp=yes; format=flowed Hi Susan, Your message leaves the impression that non-Western tuning systems are unrelated to the Western system and do not follow small-integer ratio principles. But that's not quite true. First, our modern Western tuning does not follow Pythagorean intervals, either. It uses equal temperament, as I'm sure you know. Second, the =20 tuning systems of the rest of the world are related in many ways to the Western tuning system and tend to use small integer ratios (with some exceptions, such as gamelan). =46rom something I'm currently working on: The oldest Western theory of musical consonance is due to Pythagoras: Musical consonance is determined by the ratios of small whole numbers. The principle was that intervals of small integer ratios produced =20 harmonies that were pleasing and mathematically pure. The Pythagorean tuning system is the oldest extant Western system devised explicitly according to this principle, and is thought to have been devised by Pythagoras =20 himself. Interestingly, the available evidence suggests that a scale similar =20 to this was already in use in the West before any mathematical principle was advanced to describe its structure. According to Iambiclus=92s Life of Pythagoras (Guthrie, 1987), Pythagoras did not invent the Pythagorean =20= scale, he discovered a principle to explain a scale that was already in use. =20= A related tuning system, Just Intonation (JI), originally proposed by Ptolemy =20 (Hunt, 1992), derives all interval ratios in relation to one single tonic, and =20 chooses the smallest possible integer ratios that divide the octave (approximately) =20 equally. Just intonation satisfies the principle of small integer ratios more =20 nearly than the Pythagorean, and it is sometimes referred to as the natural scale. The three largest non-Western tuning systems are Indian, Chinese and =20 Arab-Persian. Each of these has inclusive 12-tone scales whose frequency =20 relationships are similar to the Western chromatic scales. Two of these systems, the =20 Indian and the Arab-Persian, use more than 12 intervals per octave (Burns, 1999). =20 The musical systems of India are theoretically based on 22 intervals per octave. =20 However, the basic scale consists of 12 tones tuned according to a form of just =20 intonation. The remaining 10 tones are slight variations of certain intervals, =20 the exact frequencies of which depend upon the individual melodic framework (raga) being =20 played. The Arab-Persian system theoretically employs intervals that bisect the =20 distance between Western chromatic intervals. However, there is some controversy as to =20= the exact number of possible intervals and the actual intervals produced in =20 performance. Most sources list the small integer ratio tuning relationships. Ed On Feb 28, 2007, at 1:41 AM, Susan Allen wrote: > It is astonishing to me that all of you are talking about western =20 > scales and octaves! This is not the music of the world! This is =20 > colonial music, discovered in the West.... > The WORLD of music does not follow Pythagorean intervals! There =20 > are many more notes! > > FORGET perfect pitch - it only has to do with relative pitch on the =20= > piano keyboard - within the Western (colonial) paradigm! > > > Susan Allen PhD > > http://music.calarts.edu/~susie > > > > On Feb 27, 2007, at 10:03 PM, Annabel Cohen wrote: > >> Dear Martin and Stewart and others: >> >> I am willing to concede that sensitivity to overlapping harmonics may >> not be the basis of the musical and octave sensitivity of monkeys; >> what remains unclear to me is whether there is an "octave circular >> pitch processor" or rather than a "small-integer / periodicity- >> sensitive processor". >> >> If there is only an "octave circular pitch processing" to account for >> octave generalization, one would predict performance in monkeys on >> transpositions to the perfect fifth (ratio 3/2 =3D 7 semitones up) = to >> be as poor as performance on transposition to the tritone (half >> octave =3D 6 semitones). A study including the perfect fifth >> transposition has not been carried out to the best of my knowledge. >> If performance were superior for the perfect fifth, the "octave >> processor" theory would be incomplete. >> >> How also does one explain the monkey's superior performance on tonal >> as opposed to atonal melodies, when tonal melodies are characterized >> by tones related by small integer ratios (though typically not >> octaves) as compared to tone relations in atonal melodies. >> >> Annabel >> >> On 24 Feb 2007 at 0:43, Martin Braun wrote: >> >>> Dear Annabel, Stew, and others, >>> >>> Annabel Cohen wrote: >>> >>> "The evidence in this paper [ >>> http://web.telia.com/~u57011259/Wright.htm ] for octave >>> generalization for tonal melodies by rhesus monkeys is impressive, >>> however, whether this reflects something special about sensitivity >>> to the octave (chroma) rather than sensitivity to the overtone >>> series or periodicity is still not clear from this study." >>> >>> Sorry, it IS clear from this study. The authors reported that >>> generalization over the distance of two octaves is even stronger >>> than that over the distance of one octave. This finding definitely >>> rules out the possibility that the monkeys generalized according to >>> similarities in the sound spectrum (harmonics). The only remaining >>> possibility is that the monkeys, the same as humans, have an octave >>> circular pitch processing, which provides the basis for a chroma >>> percept. >>> >>> Martin >>> >>> -------------------------------------------------------------------- >>> - Martin Braun Neuroscience of Music S-671 95 Kl=E4ssbol Sweden web >>> site: http://w1.570.telia.com/~u57011259/index.htm >> >> >> ------- End of forwarded message -------Annabel J. Cohen, Ph. D. >> Department of Psychology >> University of Prince Edward Island >> Charlottetown, P.E.I. C1A 4P3 CANADA >> e:mail acohen@xxxxxxxx >> phone: (902) 628-4325 office; (902) 628-4331 lab >> fax: (902) 628-4359 >> www.upei.ca/~musicog >> www.upei.ca/~cmtc --Apple-Mail-7--1019879744 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=WINDOWS-1252 <HTML><BODY style=3D"word-wrap: break-word; -khtml-nbsp-mode: space; = -khtml-line-break: after-white-space; "><DIV>Hi Susan,</DIV><DIV><BR = class=3D"khtml-block-placeholder"></DIV><DIV>Your message leaves the = impression that non-Western tuning</DIV><DIV>systems are unrelated to = the Western system and do not</DIV><DIV>follow small-integer ratio = principles. But that's not quite true.</DIV>First, our modern Western = tuning does not follow Pythagorean intervals,=A0<DIV>either. It=A0uses = equal temperament, as I'm sure you know.=A0Second, the = tuning=A0<DIV>systems of the rest of the world are related in many = ways=A0<DIV>to the Western tuning system and tend to use small integer = ratios (with</DIV><DIV>some exceptions, such as gamelan). =46rom = something I'm currently</DIV><DIV>working = on:</DIV><DIV>=A0</DIV><DIV><DIV><DIV><DIV><DIV><DIV><DIV = class=3D"Bodynochange">The oldest Western theory of musical consonance = is due to Pythagoras:=A0</DIV><DIV class=3D"Bodynochange">Musical = consonance is determined by the ratios of small whole = numbers.=A0</DIV><DIV class=3D"Bodynochange">The principle was that = intervals of small integer ratios produced harmonies</DIV><DIV = class=3D"Bodynochange">=A0that were pleasing and mathematically pure. = The Pythagorean tuning=A0</DIV><DIV class=3D"Bodynochange">system is the = oldest extant Western system devised explicitly according=A0</DIV><DIV = class=3D"Bodynochange">to this principle, and is thought to have been = devised by Pythagoras himself.=A0<SPAN style=3D""></SPAN></DIV><DIV = class=3D"Bodynochange"><SPAN style=3D"">Interestingly, the available = evidence suggests that a scale similar to this=A0</SPAN></DIV><DIV = class=3D"Bodynochange"><SPAN style=3D"">was already in use in the West = before any mathematical principle was=A0</SPAN></DIV><DIV = class=3D"Bodynochange"><SPAN style=3D"">advanced to describe its = structure<SPAN class=3D"MsoFootnoteReference">.</SPAN> According to = Iambiclus=92s Life of=A0</SPAN></DIV><DIV class=3D"Bodynochange"><SPAN = style=3D"">Pythagoras </SPAN><SPAN style=3D"">(Guthrie, = 1987)</SPAN><SPAN style=3D"">, Pythagoras did not invent the Pythagorean = scale,=A0</SPAN></DIV><DIV class=3D"Bodynochange"><SPAN style=3D"">he = discovered a principle to explain a scale that was already in use. = </SPAN>A related=A0</DIV><DIV class=3D"Bodynochange">tuning system, Just = Intonation (JI), originally proposed by Ptolemy (Hunt, 1992),=A0</DIV><DIV= class=3D"Bodynochange">derives all interval ratios in relation to one = single tonic, and chooses the smallest=A0</DIV><DIV = class=3D"Bodynochange">possible integer ratios that divide the octave = (approximately) equally. Just=A0</DIV><DIV = class=3D"Bodynochange">intonation satisfies the principle of small = integer ratios more nearly than the=A0</DIV><DIV = class=3D"Bodynochange">Pythagorean, and it is sometimes referred to as = <I>the</I><SPAN style=3D""> natural scale</SPAN>.<O:P></O:P></DIV><DIV = class=3D"Bodynochange"><BR class=3D"khtml-block-placeholder"></DIV><DIV = class=3D"Bodynochange">The three largest non-Western tuning systems are = Indian, Chinese and Arab-Persian.=A0</DIV><DIV class=3D"Bodynochange">Each= of these has inclusive 12-tone scales whose frequency relationships = are=A0</DIV><DIV class=3D"Bodynochange">similar to the Western chromatic = scales. Two of these systems, the Indian and the=A0</DIV><DIV = class=3D"Bodynochange">Arab-Persian, use more than 12 intervals per = octave (Burns, 1999). The musical=A0</DIV><DIV = class=3D"Bodynochange">systems of India are theoretically based on 22 = intervals per octave. However, the=A0</DIV><DIV = class=3D"Bodynochange">basic scale consists of 12 tones tuned according = to a form of just intonation.=A0</DIV><DIV class=3D"Bodynochange">The = remaining 10 tones are slight variations of certain intervals, the exact = frequencies=A0</DIV><DIV class=3D"Bodynochange">of which depend upon the = individual melodic framework (raga) being played. The=A0</DIV><DIV = class=3D"Bodynochange">Arab-Persian system theoretically employs = intervals that bisect the distance between=A0</DIV><DIV = class=3D"Bodynochange">Western chromatic intervals. However, there is = some controversy as to the exact number=A0</DIV><DIV = class=3D"Bodynochange">of possible intervals and the actual intervals = produced in performance. Most sources</DIV><DIV = class=3D"Bodynochange">list the small integer ratio tuning = relationships.</DIV><DIV class=3D"Bodynochange"><BR = class=3D"khtml-block-placeholder"></DIV><DIV = class=3D"Bodynochange">Ed</DIV><DIV><BR = class=3D"khtml-block-placeholder"><DIV><DIV><DIV><DIV>On Feb 28, 2007, = at 1:41 AM, Susan Allen wrote:</DIV><BR = class=3D"Apple-interchange-newline"><BLOCKQUOTE type=3D"cite"><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">It is astonishing to me that all of you are talking = about western scales and<SPAN class=3D"Apple-converted-space">=A0 = </SPAN>octaves!<SPAN class=3D"Apple-converted-space">=A0 </SPAN>This is = not the music of the world!<SPAN class=3D"Apple-converted-space">=A0 = </SPAN>This is colonial music, discovered in the West....</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">The WORLD of music does not follow Pythagorean = intervals!<SPAN class=3D"Apple-converted-space">=A0 </SPAN>There are = many more notes!</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">FORGET perfect pitch - it only has to do with = relative pitch on the piano keyboard - within the Western (colonial) = paradigm!</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Susan = Allen PhD</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; "><A = href=3D"http://music.calarts.edu/~susie">http://music.calarts.edu/~susie</= A></DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: = 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">On Feb 27, = 2007, at 10:03 PM, Annabel Cohen wrote:</DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = min-height: 14px; "><BR></DIV> <BLOCKQUOTE type=3D"cite"><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">Dear Martin and Stewart and others:</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">I am = willing to concede that sensitivity to overlapping harmonics = may</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">not be the basis of the musical = and octave sensitivity of monkeys;</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">what remains = unclear to me is whether there is an "octave circular</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">pitch processor" or rather than a "small-integer / = periodicity-</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">sensitive processor".</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">If there = is only an "octave circular pitch processing" to account for</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">octave generalization, one would predict performance = in monkeys on</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">transpositions to the perfect = fifth (ratio 3/2<SPAN class=3D"Apple-converted-space">=A0 </SPAN>=3D 7 = semitones up)<SPAN class=3D"Apple-converted-space">=A0 = </SPAN>to</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">be as poor as performance on = transposition to the tritone (half</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">octave =3D 6 = semitones). A study including the perfect fifth</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">transposition has not been carried out to the best = of my knowledge.</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">If performance were superior for = the perfect fifth, the "octave</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">processor" = theory would be incomplete.</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: = 14px; "><BR></DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">How also does one explain the = monkey's superior performance on tonal</DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">as = opposed to atonal melodies, when tonal melodies are = characterized</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">by tones related by small = integer ratios (though typically not</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">octaves) as = compared to tone relations in atonal melodies.</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = ">Annabel</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">On 24 Feb 2007 at 0:43, Martin Braun = wrote:</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> = <BLOCKQUOTE type=3D"cite"><DIV style=3D"margin-top: 0px; margin-right: = 0px; margin-bottom: 0px; margin-left: 0px; ">Dear Annabel, Stew, and = others,</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">Annabel Cohen wrote:</DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">"The evidence = in this paper [</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; "><A = href=3D"http://web.telia.com/~u57011259/Wright.htm">http://web.telia.com/~= u57011259/Wright.htm</A> ] for octave</DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = ">generalization for tonal melodies by rhesus monkeys is = impressive,</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">however, whether this reflects = something special about sensitivity</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">to the octave = (chroma) rather than sensitivity to the overtone</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">series or periodicity is still not clear from this = study."</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">Sorry, it IS clear from this study. The authors = reported that</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">generalization over the distance = of two octaves is even stronger</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">than that = over the distance of one octave. This finding definitely</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">rules out the possibility that the monkeys = generalized according to</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">similarities = in the sound spectrum (harmonics). The only remaining</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">possibility is that the monkeys, the same as humans, = have an octave</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">circular pitch processing, which = provides the basis for a chroma</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = ">percept.</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">Martin</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: = 14px; "><BR></DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; = ">--------------------------------------------------------------------</DI= V><DIV style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">- Martin Braun Neuroscience of Music S-671 95 = Kl=E4ssbol Sweden web</DIV><DIV style=3D"margin-top: 0px; margin-right: = 0px; margin-bottom: 0px; margin-left: 0px; ">site: <A = href=3D"http://w1.570.telia.com/~u57011259/index.htm">http://w1.570.telia.= com/~u57011259/index.htm</A></DIV> </BLOCKQUOTE><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = min-height: 14px; "><BR></DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: = 14px; "><BR></DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">------- End of forwarded message = -------Annabel J. Cohen, Ph. D.</DIV><DIV style=3D"margin-top: 0px; = margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Department of = Psychology</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">University of Prince Edward = Island</DIV><DIV style=3D"margin-top: 0px; margin-right: 0px; = margin-bottom: 0px; margin-left: 0px; ">Charlottetown, P.E.I. C1A = 4P3<SPAN class=3D"Apple-converted-space">=A0 </SPAN>CANADA</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">e:mail <A = href=3D"mailto:acohen@xxxxxxxx">acohen@xxxxxxxx</A></DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">phone: (902) 628-4325<SPAN = class=3D"Apple-converted-space">=A0 </SPAN>office;<SPAN = class=3D"Apple-converted-space">=A0 </SPAN>(902) 628-4331<SPAN = class=3D"Apple-converted-space">=A0 </SPAN>lab</DIV><DIV = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; ">fax: (902) 628-4359</DIV><DIV style=3D"margin-top: = 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; = ">www.upei.ca/~musicog</DIV><DIV style=3D"margin-top: 0px; margin-right: = 0px; margin-bottom: 0px; margin-left: 0px; ">www.upei.ca/~cmtc</DIV> = </BLOCKQUOTE></BLOCKQUOTE></DIV><BR></DIV></DIV></DIV></DIV></DIV></DIV></= DIV></DIV></DIV></DIV></DIV></BODY></HTML>= --Apple-Mail-7--1019879744--


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