Musical sequence

[Henry Gould]

I am reminded, of course, of Euler's definition of sweetness W of a 
musical chord.
He called it the 'gradus suavitatis' and defined it as follows.
Let the chord be given by natural numbers a, b, c, d. Then

W(a,b,c,d) = g(lcm (a,b,c,d),gcd(a,b,c,d)), where the function g(n) i 
defined as
follows:  g(1) = 1;  g(p) =p for any prime p;   g(mn) = g(m) + g(n) -1 for
arbitrary natural numbers n.
Reference: Balthaser van der Pol, Music and elementary theory of numbers,
The Music Review, Vol.7, 1946, pp. 1-25. See his Selected Papers, 
Amsterdsam,
1960, Vol. 2, pp. 1101-1125.
An explicit formula is known for writing g(n ) in terms of the 
fsactorization  of n.

I wrote a paper on a related function 'Note on functions studied by 
LeVan, Chawla, and Euler', Journal of Natural Science and Mathematics, 
(Lahore, Pakistan). Vol. 12, No.2,  October, 1972, pp.421-430.  pp. 428 
and 429  I presented two tables of values of values that led to the two 
sequences

1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, . . .

and

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 17, 17, 26, 26, 37, 37, . . .

all of which are INTERESTING, being that they are related to musical 
chords that sound good! 

I have not checked to see if any of this in the OEIS.

Regards,

Henry W. Gould


Jon Wild wrote:

> On Wed, 18 Jan 2006, David Wilson wrote:
>
>> Is there a measure of goodness of r that would rate r = 2^(1/12) 
>> high? Presumably this rating would be based on the closeness of tone 
>> ratios to simple rationals which represent pleasing harmonies.  Given 
>> this rating, what would be the best r?  If we return to octave-based 
>> scales, could be construct a sequence of t with increasingly better 
>> ratings of 2^(1/t), which would represent increasingly good choices 
>> for number of tones in an octave?
>
>
> There are several candidates for such a sequence; they generally 
> include (5,7,12),19,31,53... (the closeness of 3^53 to a power of 2, 
> which causes 53-tone equal temperament to have an excellent perfect 
> fifth, was discovered by the Chinese more than 5000 years ago)
>
> As a couple of examples out of hundreds, you could look into Joseph 
> Yasser's book "A Theory of Evolving Tonality" for precisely what 
> you're talking about; Norman Carey & David Clampitt's article 
> "Well-formed Scales" in _Music Theory Spectrum_ for a more 
> mathematical treatment (and a slightly different series), or see the 
> yahoo discussion groups "tuning" and "tuning-math" for many, many more 
> details than you'd care to know about.
>
> Jon
>