A361378 Number of musical scales in n tone equal temperament respecting the property that alternate notes are 3 or 4 semitones apart.

0, 1, 2, 3, 3, 3, 8, 8, 12, 16, 25, 33, 45, 66, 91, 128, 177, 252, 351, 491, 689, 966, 1354, 1894, 2658, 3723, 5217, 7309, 10244, 14355, 20112, 28185, 39494, 55343, 77547, 108667, 152272, 213372, 298992, 418968, 587089, 822665, 1152777, 1615350

Offset

1,3

Comments

If you take any three consecutive notes in the scales counted by a(n) (with cyclic identification) then the distance between the first and third is either 3 or 4 semitones. a(n) is also the number of subsets of Z/nZ that 1) contain 0; 2) contain no subset of the form {x,x+1,x+2}; 3) have no superset satisfying property 2).

Links

Table of n, a(n) for n=1..44.

Martin Epstein, mastodon.xyz thread, Feb 23 2023.

Donovan Young, The number of scales with the natural thirds property.

Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1).

Formula

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6).

G.f.: x^2*(1+2*x+2*x^2-3*x^4)/(1-x^2-x^3-x^4+x^6).

Example

For n=4 there are four notes, call them 0, 1, 2, and 3. The scales are 01, 02, and 03 and so a(4)=3.

Mathematica

LinearRecurrence[{0, 1, 1, 1, 0, -1}, {0, 1, 2, 3, 3, 3}, 100]

Crossrefs

Sequence in context: A088041 A334579 A199457 * A347208 A210483 A022296

Adjacent sequences: A361375 A361376 A361377 * A361379 A361380 A361381

Keyword

nonn,easy

Author

Donovan Young, Mar 09 2023

Status

approved