By Barrie Nettles, Richard Graf

ISBN-10: 389221056X

ISBN-13: 9783892210566

Jazz concord, as taught on the Berklee university of tune is predicated at the so known as Chord Scale thought. this system - extra built - is now to be had as a finished textbook for the 1st time. Emerged from perform and designed for useful use, it presents theoretical wisdom invaluable for improvisation, composing, and arranging. This entire textbook is a needs to for newcomers, intermediate or complex scholars. it really is relatively logically built, really easy to realize, and the quantity of precious details contained during this pages is particularly extraordinary. it's also a beneficial and arranged instrument that may reduction any musician drawn to realizing concord because it pertains to the complicated subtleties of jazz improvisation. The excessive calls for on brand new expert musician calls for a well-founded perception of the interior video game of track. this technique is trustworthy for reading diverse idioms of tonal tune: jazz, pop, rock, and different modern types, even the research of Western paintings track among 1650 and 1900 profits a brand new measurement. benefit from the trip during the international of chord scales and jazz concord

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1 CHAPTER 2. 27 Let (S,F,T) be a probabilistic metric space and T = TT, where TT( F, G)( x) = sup {T( F( u), G( v )) I u + v = x} for at-norm T. Then (S,F,T) is the so ca lied Menger space, which will be denoted by (S,F,T). 27 is a tri angle function. 3) for all p,q,r in Sand x,y real numbers. This inequality also implies (iv). Namely, taking x E [0, 00) we have for all u, v E [0, 00) such that u + v = x Hence Fp,r(x) 2: (TT(Fp,q, Fq,r ))(x). 3) in the way of the classical metric spaces that the third side in a tri angle depends on the other two sides in the sense that if the knowledge of two sides increases then also the knowledge of third side increases or that knowing the upper bounds of two sides we have an upper bound for the third side.

29). The above proposition holds also for A 2: 0 since in this case Tfw 2: TL. 26) for a special sequence (1 - qn)nEN for q E (0,1). 76 We say that for some q E (0,1) at-norm T is q-convergent if lim n~oo Since lim (1- qn) = 1 and n--+oo all t-norms from the family 00 T (1 . qi) = 1. \E(O,oo) are q-convergent for every q E (0, 1). The following example shows that not every strict t-norm is q-convergent. 77 Let T be the strict t-norm with an additive generator t( x) -log(~-x)' In this case the series smce co L t(l - qi) for any q E (0,1) is not convergent i=1 co .

This inequality also implies (iv). Namely, taking x E [0, 00) we have for all u, v E [0, 00) such that u + v = x Hence Fp,r(x) 2: (TT(Fp,q, Fq,r ))(x). 3) in the way of the classical metric spaces that the third side in a tri angle depends on the other two sides in the sense that if the knowledge of two sides increases then also the knowledge of third side increases or that knowing the upper bounds of two sides we have an upper bound for the third side. 2) is satisfied for 8 = c/2. q,r (~) 2 > 1 - ~2' then Fp,r(c) > TL (Fp,q > Fp,q (~) ,Fq,r (~)) G) + Fq,r (~) - 1 > 1- c.

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