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A071833 Frequency ratios for notes of C-major scale starting at c = 24 and using Ptolemy's intense diatonic scale.

%I A071833 #101 Aug 24 2024 05:41:02
%S A071833 24,27,30,32,36,40,45,48,54,60,64,72,80,90,96,108,120,128,144,160,180,
%T A071833 192,216,240,256,288,320,360,384,432,480,512,576,640,720,768,864,960,
%U A071833 1024,1152,1280,1440,1536,1728,1920,2048,2304,2560,2880
%N A071833 Frequency ratios for notes of C-major scale starting at c = 24 and using Ptolemy's intense diatonic scale.
%C A071833 All terms are 5-smooth numbers due to the 5-limit-tuning of the natural major scale, where all the ratios prime factors are all less than or equal to 5. - _Federico Provvedi_, Sep 09 2022
%C A071833 From _Federico Provvedi_, Apr 19 2024: (Start)
%C A071833 This natural scale has interesting musical and mathematical Diophantine relations between the sum of distinct interval ratios a(n)/a(0) and their own indices: with indices i(k) != j(k), Sum_{k=1..n} a(i(k)) = Sum_{k=1..n} a(j(k)) and
%C A071833   Sum_{k=1..n} i(k)  = Sum_{k=1..n} j(k), for n=4 a solution is:
%C A071833      1 + 4/3 + 5/3 + 15/8  = 9/8 + 5/4 + 3/2 +   2 ,
%C A071833      I +  IV +  VI +  VII  =  II + III +  V  + VIII,
%C A071833      1 +   4 +   6 +    7  =   2 +  3  +  5  +   8 ,
%C A071833   a(0) + a(3) + a(5) + a(6) = a(1) + a(2) + a(4) + a(7). (End)
%C A071833 In the terminology of classical music theory, a(0) to a(7) are the frequencies of the diatonic C-major scale (C,D,E,F,G,A,B,C) as tuned in "Just Intonation", starting with frequency C=24=a(0). On keyboard instruments, these are the "white notes". Each higher octave of 8 notes doubles the frequencies of the prior octave, hence, a(n+7) = 2*a(n). The a(n) frequencies of Just Intonation are uniquely determined by requiring that the notes in each of the three principal major triads, namely, the tonic triad (C:E:G), the dominant triad (G:B:D), and the subdominant triad (F:A:C), all have frequencies with exact ratios of 4:5:6. The base frequency of C=24=a(0) is the lowest frequency of C for which all a(n) are integers. (In actual practice, keyboard notes are usually tuned to non-integer frequencies, are based on a "middle C" frequency around 261.62 Hz, and have irrational frequency ratios due to "equal temperament" - see A010774.) - _Robert B Fowler_, Aug 21 2024
%H A071833 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ptolemy%27s_intense_diatonic_scale">Ptolemy's intense diatonic scale</a>.
%H A071833 Wikipedia, <a href="https://en.wikipedia.org/wiki/Five-limit_tuning#Diatonic_scale">Five-limit tuning. Diatonic scale</a>.
%H A071833 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,2).
%F A071833 a(n) = 2^floor(n/7) * (3*(91 + (-1)^(n mod 7)) + 42*(n mod 7) + 8*sqrt(3) * sin(Pi*(1+(n mod 7))/3))/12. - _Federico Provvedi_, Aug 28 2012
%F A071833 G.f.: -(45*x^6 + 40*x^5 + 36*x^4 + 32*x^3 + 30*x^2 + 27*x + 24) / (2*x^7 - 1). - _Colin Barker_, Feb 14 2014
%F A071833 a(b(n)) - a(b(n)+1) - a(b(n)+2) + a(b(n)+3) - a(b(n)+4) + a(b(n)+5) + a(b(n)+6) - a(b(n)+7) = 0, where b(n) = A047274(n). - _Federico Provvedi_, Apr 19 2024
%F A071833 a(n) = 2^floor(n/7) * round(24 * 2^(floor( (12*(n mod 7)+5)/7) / 12)). - _Robert B Fowler_, Aug 22 2024
%e A071833 The ratios are 24 times 1 (c), 9/8 (d), 5/4 (e), 4/3 (f), 3/2 (g), 5/3 (a), 15/8 (b), followed by these 7 numbers multiplied by successive powers of 2.
%t A071833 Table[ 2^Floor[n/7] ( 3*(91 + (-1)^Mod[n, 7] ) + 42 Mod[n, 7] + 8 Sqrt[3] Sin[Pi(1 + Mod[n, 7])/3] ) / 12,  {n, 0, 70}] (* _Federico Provvedi_, Aug 28 2012 *)
%t A071833 3*2^(3+Floor[#/7])*Rationalize[2^((-1+Floor[12(1+Mod[#,7])/7])/12),2^-6]&/@Range[0,70] (* _Federico Provvedi_, Oct 13 2013 *)
%t A071833 LinearRecurrence[{0,0,0,0,0,0,2},{24,27,30,32,36,40,45},50] (* _Harvey P. Dale_, May 23 2016 *)
%o A071833 (Python)
%o A071833 def a(n): return [24, 27, 30, 32, 36, 40, 45][n % 7] << (n // 7) # _Peter Luschny_, Aug 22 2024
%Y A071833 Cf. A071831, A071832, subset of A051037, A010774.
%K A071833 nonn,frac,easy,nice
%O A071833 0,1
%A A071833 _N. J. A. Sloane_, Jun 10 2002
%E A071833 More terms from Kerri Sullivan (ksulliva(AT)ashland.edu), Oct 31 2005
%E A071833 Name made more specific by _Jon E. Schoenfield_, Sep 12 2022