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

24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 64, 72, 80, 90, 96, 108, 120, 128, 144, 160, 180, 192, 216, 240, 256, 288, 320, 360, 384, 432, 480, 512, 576, 640, 720, 768, 864, 960, 1024, 1152, 1280, 1440, 1536, 1728, 1920, 2048, 2304, 2560, 2880

Offset

0,1

Comments

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

Links

Table of n, a(n) for n=0..48.

Wikipedia, Ptolemy's intense diatonic scale.

Wikipedia, Five-limit tuning. Diatonic scale.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2).

Formula

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

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

Example

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.

Mathematica

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 *)
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 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 2}, {24, 27, 30, 32, 36, 40, 45}, 50] (* Harvey P. Dale, May 23 2016 *)

Crossrefs

Cf. A071831, A071832, subset of A051037.

Sequence in context: A279427 A116203 A345499 * A064159 A141632 A308603

Adjacent sequences: A071830 A071831 A071832 * A071834 A071835 A071836

Keyword

nonn,frac,easy,nice

Author

N. J. A. Sloane, Jun 10 2002

Extensions

More terms from Kerri Sullivan (ksulliva(AT)ashland.edu), Oct 31 2005

Name made more specific by Jon E. Schoenfield, Sep 12 2022

Status

approved