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A133269
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Fractal sequence consisting of quadruples, each of which contains the tones comprising a major 7th chord (i.e., root, major third, fifth, and major 7th), with the tones in an octave assigned to the numbers 1..12, and with the n-th quadruple using a(n) as its root.
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4
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1, 5, 8, 12, 5, 9, 12, 4, 8, 12, 3, 7, 12, 4, 7, 11, 5, 9, 12, 4, 9, 1, 4, 8, 12, 4, 7, 11, 4, 8, 11, 3, 8, 12, 3, 7, 12, 4, 7, 11, 3, 7, 10, 2, 7, 11, 2, 6, 12, 4, 7, 11, 4, 8, 11, 3, 7, 11, 2, 6, 11, 3, 6, 10, 5, 9, 12, 4, 9, 1, 4, 8, 12, 4, 7, 11, 4, 8, 11, 3, 9, 1, 4, 8, 1, 5, 8, 12, 4, 8, 11, 3, 8
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OFFSET
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1,2
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COMMENTS
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A major 7th chord consists of its root tone and the tones that lie 4, 7, and 11 semitones (a.k.a. half tones or half steps) above the root.
Starting from root tones at C, C#, D, D#, E, F, etc., wrapping around the tones higher than an octave (i.e., 13->1, 14->2, etc.), and assigning numbers 1 to 12 to the half tones within the octave, the major 7th chords (4 tones each) are represented as 1-> {1, 5, 8, 12}, 2-> {2, 6, 9, 1}, 3-> {3, 7, 10, 2}, 4-> {4, 8, 11, 3}, 5-> {5, 9, 12, 4}, 6-> {6, 10, 1, 5}, 7-> {7, 11, 2, 6}, 8-> {8, 12, 3, 7}, 9-> {9, 1, 4, 8}, 10-> {10, 2, 5, 9}, 11-> {11, 3, 6, 10}, 12-> {12, 4, 7, 11}.
The sequence starts with the chord at root 1, which defines a(1)..a(4), and then appends iteratively in groups of 4 the chords at base a(k), k = 2, 3, 4, ... to the sequence.
Note that construction of a chord in terms of the intervals from its root to each of the chord's constituent notes is simpler if we instead number the notes in an octave from 0 through 11, rather than from 1 through 12 (and a sequence built this way would also have a simpler formula, not requiring us to add 1 after taking modulo-12 residues). (See the link to the Wikipedia article, which says that a major 7th chord "can be represented by the integer notation {0, 4, 7, 11}.) - Jon E. Schoenfield, Aug 27 2017
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LINKS
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FORMULA
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a(1) = 1, a(2) = 5, a(3) = 8, a(4) = 12;
for k > 1,
a(4k - 3) = a(k);
a(4k - 2) = (a(k) + 3) mod 12 + 1;
a(4k - 1) = (a(k) + 6) mod 12 + 1;
a(4k) = (a(k) + 10) mod 12 + 1. (End)
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EXAMPLE
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If we assign the number 1 to the note C in some octave (e.g., middle C) and assign the numbers 2, 3, 4, ..., to the notes reached as we move up the scale by half tones, then the C notes at successive octaves will be assigned the numbers 1, 13, 25, ... (see the illustration below, which is intended to represent a portion of a standard musical keyboard).
| |C| |D| | |F| |G| |A| | |C| |D| | |F| |G| |A| |
| |#| |#| | |#| |#| |#| | |#| |#| | |#| |#| |#| |
| |/| |/| | |/| |/| |/| | |/| |/| | |/| |/| |/| |
| |D| |E| | |G| |A| |B| | |D| |E| | |G| |A| |B| |
| |b| |b| | |b| |b| |b| | |b| |b| | |b| |b| |b| |
| | | | | | | | | | | | | | |
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| C | D | E | F | G | A | B | C | D | E | F | G | A | B | C
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
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Then the sequence consists of quadruples constructed as follows:
- the values of this sequence's first quadruple, a(1)..a(4) (i.e., 1, 5, 8, and 12) give a C major 7th chord (the tones C, E, G, B);
- the sequence's 2nd quadruple, a(5)..a(8), starts from a(5) = a(2) = 5 and thus gives an E major 7th chord (i.e., 5, 9, 12, 16, but the 16 is wrapped down an octave to 16-12 = 4);
- the sequence's 3rd quadruple, a(9)..a(12), starts from a(9) = a(3) = 8 and thus gives a G major 7th chord (8, 12, 15->3, 19->7);
- the sequence's 4th quadruple, a(13)..a(16), starts from a(13) = a(4) = 12, giving a B major 7th chord (12, 16->4, 19->7, 23->11);
- in general, the sequence's k-th quadruple, a(4k-3)..a(4k), starts with a(4k-3) = a(k), and its three remaining terms are a(k)+4, a(k)+7, and a(k)+11 (with any that exceed 12 decreased by 12, i.e., by one octave).
Thus, the sequence is fractal: keeping a(1) and every 4th term after it, and discarding the rest, yields the original sequence. (End)
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MATHEMATICA
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Clear[s, p] s[i_] = {i, If[i + 4 > 12, i - 8, i + 4], If[i + 7 > 12, i - 5, i + 7], If[i + 11 > 12, i - 1, i + 11]}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]]; p[4]
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CROSSREFS
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See A133270 for a companion sequence which provides more explanation.
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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EXTENSIONS
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Comments inserted for clarification - The Assoc. Eds. of the OEIS - Aug 29 2010
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STATUS
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approved
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