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

Author

N. J. A. Sloane, Jun 10 2002

Keywords

• nonn
• frac
• easy
• nice

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

From Federico Provvedi, Apr 19 2024: (Start)

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

Sum_{k=1..n} i(k) = Sum_{k=1..n} j(k), for n=4 a solution is:

1 + 4/3 + 5/3 + 15/8 = 9/8 + 5/4 + 3/2 + 2 ,

I + IV + VI + VII = II + III + V + VIII,

1 + 4 + 6 + 7 = 2 + 3 + 5 + 8 ,

a(0) + a(3) + a(5) + a(6) = a(1) + a(2) + a(4) + a(7). (End)

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

Examples

			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.
		

Links

• 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).

Crossrefs

Cf. A071831, A071832, subset of A051037, A010774.

Programs

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 *)

Python

def a(n): return [24, 27, 30, 32, 36, 40, 45][n % 7] << (n // 7) # Peter Luschny, Aug 22 2024

Formula

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

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

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

a(n) = 2^floor(n/7) * round(24 * 2^(floor( (12*(n mod 7)+5)/7) / 12)). - Robert B Fowler, Aug 22 2024

Extensions

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

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