By Ken Emerson

Throughout the past due Fifties and early Sixties, after the surprise of Elvis Presley and sooner than the Beatles spearheaded the British Invasion, fourteen proficient younger songwriters huddled in midtown Manhattan's mythical Brill development and a warren of workplaces a section farther uptown and composed essentially the most beguiling and enduring entries within the nice American Songbook. Always Magic within the Air is the 1st thorough heritage of those well known songwriters-tunesmiths who melded black, white, and Latino sounds, built-in audiences prior to the USA desegregated its faculties, and taken a brand new social attention to pop music.

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The number of A;-chord classes is the number of G-orbits which is the coefficient of zk in the cycle index for variables tk = 1 + zk Proposition 61 The cycle index of the cyclic group Cn is the polynomial d\n where if is the Euler totient function. Example. -, t12) = ^(t\2 + 4 + 2t\ + 2*2 + 2t\ + 4*12) Substituting tk by 1 -f zk, we get the following counting series P(c12,z12)(z) = l + 2 + 6*2 + 192 3 +43* 4 + 6625 + 80;z6 +66* 7 + 43* 8 + 19z9 + 6z10 + z11 + z12 Each coefficient of zk is the number of &-chord classes under the Cn group action.

Daniel Muzzulini [1995] has calculated all minimal cadential sets for different local compositions. Guerino Mazzola has shown that for the well-tempered tuning M = Z ^ , if K is the major scale, there are only five minimal cadential sets Jx = {IIJII}, J2 = {II, V}, J 3 = {IIIJV}, J4 = {IV, V}, J5 = {VII}. For the harmonic minor scale K = {0,2,3,5,7,8,11}, there are 21 minimal cadential sets. Every pair of two different degrees defines a minimal cadential set. Example 2. In the diatonic Pythagorean tuning, the frequency ratios are generated by pure fifths {C (1), D (9/8), E (81/64), F (4/3), G (3/2), A (27/16), B (243/128)}.

In the following table, the first column is the representative of the set, the second column gives the number of the class set under the action of the cyclic group, the third gives the class names according to Mazzola's classification table, the fourth column provides a representative and the last column is the symmetry group of the class. | Representative **0000000000 *0000*000000 *0*000000000 *00*00000000 *000*0000000 *00000*00000 tt C2-1 C2-5 C2-2 C2-3 C2-4 C2-6 Maz. 1 4 5 6 7 Sets 0,1 0, 5 0,2 0,3 0,4 0,6 Sym.

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