A060528 -id:A060528 - cp's OEIS frontend

A060529 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of three complementary pairs of simple musical tones: 7/6 and 12/7, 6/5 and 5/3 and 7/5 and 10/7.

1, 2, 3, 4, 12, 14, 15, 18, 19, 23, 27, 45, 68, 72, 99, 171, 346, 445, 517, 616, 688, 787, 1133, 1304, 3912, 7136, 8440, 9744, 11048, 12352, 18355, 19659, 20963, 22267, 26795, 28099, 29403, 30707, 40451, 41755, 69854, 71158, 72462, 143620, 216082

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 12 2001

nonn

The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 216082. The self-accumulating nature of this sequence fails once, between the fourth and fifth terms. The sequence therefore does not meet the rigorous definition of 'impeccable' recurrence. The otherwise perfect recurrence in this sequence is of the type seen in sequences A054540, A060526 and A060527. The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

A054540, A060525, A060526, A060527, A060528, A001149, A000045.

A060233 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to six complementary pairs of ratios which generate simple musical tones (scale steps): 8/7 and 7/4, 6/5 and 5/3, 16/13 and 13/8, 5/4 and 8/5, 4/3 and 3/2 and 11/8 and 16/11.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 15, 19, 22, 24, 26, 31, 37, 41, 46, 50, 53, 72, 84, 87, 130, 137, 140, 171, 183, 217, 224, 270, 494, 764, 851, 1038, 1282, 1308, 1578, 2190, 2684, 3395, 4843, 5004, 5585, 6079, 8269, 14124, 14618, 17302, 20203, 22887, 31737

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 14 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to over 31737. The numerical value of each term represents a musical scale based on an equal division of the octave. 19, for example, signifies the scale which is formed by dividing the octave into 19 equal parts.

			6 = 4 + the previous term 2. Again, 48545 = 46625 + the previous terms (1578 + 270 + 72).

A054540, A060525, A060526, A060527, A060528, A060529, A001149, A000045.

Recurrence: The next term equals the current term plus one or more of the previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z), etc.

A061918 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 5/4 and 8/5 which generate two complementary tones of musical harmony, the Major 3rd (5/4) and the Minor 6th (8/5).

Original entry on oeis.org

1, 2, 3, 16, 19, 22, 25, 28, 59, 87, 146, 351, 497, 643, 2718, 3361, 4004, 8651, 12655, 21306, 55267, 76573, 97879, 489395, 1055363, 1153242, 1251121, 1349000, 1446879, 1544758, 1642637, 1740516, 1838395, 1936274, 5808822, 7647217

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 7647217. The numerical value of each term represents a musical scale based on an equal division of the octave. 19, for example, signifies the scale which is formed by dividing the octave into 19 equal parts. Among the terms listed, the self-accumulating nature (recurrence) in this sequence breaks down down five times, between the 3rd and 4th terms, between the 14th and 15th terms, between the 20th and 21st terms, between the 23rd and 24th terms and between the 24th and 25th terms. In later sequences, this pair of target ratios will appear in combination with other pairs of target ratios, resulting in new, different (and often recurrent), composite sequences. The examples of proper recurrence which do occur in this sequence are of the same type as is seen in sequences A054540, A060526, A060527, A060233.

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061416.

A254351 Numerators of increasingly better rational approximations to log(3)/log(2) with increasing denominators.

Original entry on oeis.org

2, 3, 5, 8, 11, 19, 46, 65, 84, 317, 401, 485, 569, 1054, 13133, 14187, 15241, 16295, 17349, 18403, 19457, 20511, 21565, 22619, 23673, 24727, 50508, 125743, 176251, 301994, 8632083, 8934077, 9236071, 9538065, 9840059, 10142053, 10444047, 10746041, 11048035

Offset: 1

K. G. Stier, Jan 29 2015

log(3)/log(2) = 1.5849625... (see A020857) is an irrational number. The fractions (2/1, 3/2, 5/3, 8/5, 11/7, 19/12, 46/29, 65/41, 84/53, 317/200, 401/253, 485/306, 569/359, 1054/665, ...) are a sequence of approximations to log(3)/log(2), where each is an improvement on its predecessors.

Numerators are shown here, the respective denominators are A060528 (and can also be found among the terms of A206788), both of which refer to equal divisions of the octave and good approximations to musical harmonics.

Cf. A060528 (denominators), A020857, A206788.

x:bfloat(log(3)/log(2)),fpprec:100, errold:2,for denominator:1 thru 10000 do (numerator:round(x*denominator), errnew:abs(x-numerator/denominator), if errnew < errold then (errold:errnew, print(numerator)));

A344443 Completely additive with a(2)=5; for odd prime p, a(p) = ceiling((a(p-1) + a(p+1))/2).

Original entry on oeis.org

0, 5, 8, 10, 12, 13, 14, 15, 16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 28, 29, 29, 29, 29, 29, 30, 29, 30, 30, 30, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31

Offset: 1

Peter Munn, May 19 2021

Monotonic until a(55) = 30 > 29 = a(56).
The length of a monotonic completely additive sequence using positive integers is limited by the size of the initial terms, and this constraint is related to the conflicts inherent in specifying musical scales. In either case, the route to a good compromise solution that avoids complexity can be seen as a search for rational approximations to ratios between logarithms of the first few prime numbers.
Here a(i)/a(j) is the intended approximation to log(i)/log(j), so by starting with a(2) = 5 we open the door to using 8/5 as an approximation to log(3)/log(2) = 1.58496... . This approximation underlies simple musical scales that divide an octave into 5, where a note of twice the frequency is 5 tones higher and a note of about 3 times the frequency is 8 tones higher. The most commonly used more complex musical scales divide an octave into 12, equivalent to starting a sequence like this with a(2) = 12 (see A344444).

			a(4) = a(2*2) = a(2) + a(2) from the definition of completely additive. So a(4) = 5 + 5 = 10.
3 is an odd prime number, so a(3) = ceiling((a(3-1) + a(3+1))/2). Using the values a(2) = 5 and a(4) = 10 that we already know, we get a(3) = ceiling((5 + 10)/2) = ceiling(7.5) = 8.
The sequence is defined as completely additive, so a(1) = 0, the identity element for addition. (To see this, note that "completely additive" implies a(2) = a(2*1) = a(2)+a(1), and solve the equation for a(1).)

Author?, Musical Temperaments and Ratios.
Encyclopedia of Mathematics, Additive arithmetic function.
Index entries for sequences related to music.

Equivalent sequence with a(2)=12, a(3)=19: A344444.
The first 11 terms match row 11 of A352957.
For other completely additive sequences see the references in A104244.
Cf. A020857, A060528, A119256.

a[1] = 0; a[n_] := a[n] = Plus @@ ((Last[#] * If[First[#] == 2, 5, Ceiling[(a[First[#] - 1] + a[First[#] + 1])/2]]) & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 27 2021 *)

A344443(n) = if(1==n,0, my(f=factor(n)); sum(k=1,#f~,f[k,2]*if(2==f[k,1],5,ceil((1/2)*(A344443(f[k,1]-1)+A344443(f[k,1]+1)))))); \\ Antti Karttunen, May 19 2021

a(n*k) = a(n) + a(k).

A344444 Completely additive with a(2) = 12, a(3) = 19; for prime p > 3, a(p) = ceiling((a(p-1) + a(p+1))/2).

Original entry on oeis.org

0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 57, 58, 59, 59, 60, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 66, 67, 67, 67, 68, 68, 68, 69, 69, 69, 70, 70, 70, 71, 71, 71, 72, 72, 72, 72, 73, 73, 73, 73, 74, 74

Offset: 1

Peter Munn, May 20 2021

Monotonic until a(143) = 87 > 86 = a(144).
The only infinite monotonic completely additive integer sequence is the all 0's sequence (cf. A000004). The challenge taken up here is to specify one that is monotonic for a modestly long number of terms, using a comparatively short prescriptive definition.
To start we specify values for a(2) and a(3) so that a(3)/a(2) approximates log(3)/log(2). 19/12 is a good approximation relative to the size of denominator. This reflects 2^19 = 524288 having a similar magnitude to 3^12 = 531441. Equivalently, we can say 3 is approximately the 19th power of the 12th root of 2. This approximation is used to construct musical scales. (See the Enevoldsen link, also A143800.) There is no better approximation with a denominator smaller than 29. [Revised by Peter Munn, Jun 14 2022]
To find a good specification to use for a(p) for larger primes, p, we are guided by knowing that if 2*a(n) < a(n-1) + a(n+1) then a completely additive sequence is not monotonic after a(n^2-1) because a(n^2) < a((n-1)*(n+1)) = a(n^2-1). Considering n = p, we see we want a(p) >= (a(p-1) + a(p+1))/2; but the same consideration for n = p-1 shows we don't want a(p) larger than necessary. These considerations lead towards the choice of "a(p) = ceiling((a(p-1) + a(p+1))/2)" for use in the definition.

			a(4) = a(2*2) = a(2) + a(2) from the definition of completely additive. So a(4) = 12 + 12 = 24. Similarly, a(6) = a(2*3) = a(2) + a(3) = 12 + 19 = 31.
5 is a prime number greater than 3, so a(5) = ceiling((a(5-1) + a(5+1))/2). Using the values a(4) = 24 and a(6) = 31 that we calculated earlier, we get a(5) = ceiling((24 + 31)/2) = ceiling(27.5) = 28.
The sequence is defined as completely additive, so a(1) = 0, the identity element for addition. (To see this, note that "completely additive" implies a(2) = a(2*1) = a(2)+a(1), and solve the equation for a(1).)

Encyclopedia of Mathematics, Additive arithmetic function
Keith Enevoldsen, Twelve-Tone Musical Scale
Index entries for sequences related to music

Equivalent sequence with a(2)=5, a(3)=8: A344443.
First 10 terms match A143800.
Cf. row 23 of A352957.
Cf. A000004, A020857, A060528, A061920.
For other completely additive sequences see the references in A104244.

a[1] = 0; a[n_] := a[n] = Plus @@ ((Last[#] * Which[First[#] == 2, 12, First[#] == 3, 19, First[#] > 3, Ceiling[(a[First[#] - 1] + a[First[#] + 1])/2]]) & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 27 2021 *)

a(n*k) = a(n) + a(k).

A061919 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 6/5 and 5/3 which generate two complementary musical harmonies, the Minor 3rd (6/5) and the Major 6th (5/3).

Original entry on oeis.org

1, 2, 3, 4, 11, 15, 19, 95, 232, 251, 270, 289, 308, 327, 346, 365, 384, 403, 422, 1285, 1707, 2129, 3836, 19180, 28981, 32817, 36653, 40489, 44325, 48161, 51997, 259985, 3591629, 3643626, 3695623, 3747620, 3799617, 3851614, 3903611, 3955608

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 3955608. The numerical value of each term represents a musical scale based on an equal division of the octave. 19, for example, signifies the scale formed by dividing the octave into 19 equal parts. Within the terms shown, the self-accumulating nature of this sequence breaks down five times, between the 4th and 5th terms, between the 7th and 8th terms, between the 8th and 9th terms, between the 23rd and 24th terms and between the 32nd and 33rd terms, but the sequence is of interest because it shows the terms generated when this pair of target ratios stands alone.

Later, in other sequences, this pair of target ratios will appear in combination with other pairs of target ratios, resulting in new, different (and often recurrent), composite sequences. The examples of proper recurrence which do occur in this sequence are of the same type which is seen in sequences A054540, A060526, A060527 and A060233.

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061416, A061918.

A061920 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the 7 pairs of complementary target ratios needed to express the 12 unsymmetrical steps of the untempered (Just Intonation) scale known as the Duodene: 3/2 and 4/3, 5/4 and 8/5, 6/5 and 5/3, 9/8 and 16/9, 10/9 and 9/5, 16/15 and 15/8 and 45/32 and 64/45.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 19, 22, 31, 34, 41, 53, 118, 171, 289, 323, 376, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, 12888, 16572, 20868, 25164, 44249, 48545, 52841, 57137, 69413, 73709, 78005, 151714, 229719, 307724, 537443, 714321

Offset: 1

Mark William Rankin (MarkRankin95511(AT)yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 714321. The numerical value of each term represents a musical scale based on an equal division of the octave. The term 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

			118 = 53 + [34 + 31]; Again, 69413 = 57137 + [4296 + 3684 + 2513 + 1783].

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061918, A061919.

Recurrence Rule: The next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z), etc.

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Original entry on oeis.org

0, 2, -1, 8, 1, -3, 26, 7, -1, -7, 80, 25, 5, -5, -15, 242, 79, 23, 1, -13, -31, 728, 241, 77, 19, -7, -29, -63, 2186, 727, 239, 73, 11, -23, -61, -127, 6560, 2185, 725, 235, 65, -5, -55, -125, -255, 19682, 6559, 2183, 721, 227, 49, -37, -119, -253, -511, 59048, 19681, 6557, 2179, 713, 211, 17, -101, -247, -509, -1023

Offset: 0

K. G. Stier, Jan 22 2015

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

			Table begins
   0    2   8  26  80..
  -1    1   7  25  79..
  -3   -1   5  23  73..
  -7   -5   1  19  65..
  -15 -13  -7  11  49..
  ..   ..  ..  ..  ..

K. G. Stier, Table of n, a(n) for n = 0..5150

Row 0 (=3^n-1) is A024023.
Row 1 (=3^n-2) is A058481.
Row 2 (=3^n-4) is A168611.
Column 0 (=1-2^n) is (-1)A000225.
Column 1 (=3-2^n) is (-1)A036563.
Column 2 (=9-2^n) is (-1)A185346.
Column 3 (=27-2^n) is (-1)A220087.
0,0-Diagonal (=3^n-2^n) is A001047.
1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.
0,1-Diagonal (=3^n-2^(n+1)) is A003063.
0,2-Diagonal (=3^n-2^(n+2)) is A214091.

Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)

for(i=0, 10, {
     for(j=0, i, print1((3^(i-j)-2^j),", "))
});

A061921 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the 11 pairs of target ratios needed to express the 22 steps of the theoretical Hindu scale known as the 22 Srutis: 45/32 and 64/45, 27/20 and 40/27, 4/3 and 3/2, 81/64 and 128/81, 5/4 and 8/5, 6/5 and 5/3, 32/27 and 27/16, 9/8 and 16/9, 10/9 and 9/5, 16/15 and 15/8, 256/243 and 243/128.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 30, 32, 34, 37, 39, 40, 41, 53, 118, 171, 323, 335, 376, 388, 441, 494, 506, 559, 612, 1171, 1783, 2513, 3072, 3125, 3684, 4296, 12276, 16572, 20868, 40565, 44861, 48545, 52841, 57137, 61433, 69413, 73709

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 73709. The numerical value of each term represents a musical scale based on an equal division of the octave. The term 32, for example, signifies the scale which is formed by dividing the octave into 32 equal parts.

			118 = 53 + [34 + 31]; Again, 229719 = 78005 + [73709 + 69413 + 4296 + 3684 + 612].

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061416, A061918, A061919, A061920.

Recurrence rule: The next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z), etc.

cp's OEIS Frontend

A060529 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of three complementary pairs of simple musical tones: 7/6 and 12/7, 6/5 and 5/3 and 7/5 and 10/7.

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A061918 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 5/4 and 8/5 which generate two complementary tones of musical harmony, the Major 3rd (5/4) and the Minor 6th (8/5).

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A254351 Numerators of increasingly better rational approximations to log(3)/log(2) with increasing denominators.

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A344443 Completely additive with a(2)=5; for odd prime p, a(p) = ceiling((a(p-1) + a(p+1))/2).

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A344444 Completely additive with a(2) = 12, a(3) = 19; for prime p > 3, a(p) = ceiling((a(p-1) + a(p+1))/2).

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A061919 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 6/5 and 5/3 which generate two complementary musical harmonies, the Minor 3rd (6/5) and the Major 6th (5/3).

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A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

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