A061918 -id:A061918 - cp's OEIS frontend

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).

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.

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.

A328883 Denominators of the best rational approximations of log(6/5)/log(2).

Original entry on oeis.org

1, 2, 3, 4, 11, 15, 19, 232, 251, 270, 289, 308, 327, 346, 365, 384, 403, 422, 1285, 1707, 2129, 3836, 28981, 32817, 36653, 40489, 44325, 48161, 51997, 3591629, 3643626, 3695623, 3747620, 3799617, 3851614, 3903611, 3955608, 4007605, 4059602, 4111599, 4163596, 4215593

Offset: 1

Daniel Hoyt, Oct 29 2019

A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the minor third, 6/5 and its complement the major sixth, 5/3.
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.
The 19 equal temperament, first proposed and used by Guillaume Costeley in the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal 12-semitone equal temperament at the cost of a flatter fifth.

Wikipedia, Guillaume Costeley
Index entries for sequences based on music

Cf. A054540, A060525, A060526, A060527, A060528, A061918, A254351.

import decimal
from math import floor
from decimal import Decimal as D
from collections import namedtuple
def continued_fraction(x, k):
    cf = []
    q = floor(x)
    cf.append(q)
    x = x - q
    i = 0
    while x != 0 and i < k:
        q = floor(1 / x)
        if q > k:
            break
        cf.append(q)
        x = 1 / x - q
        i += 1
    return cf
def best_rational_approximation(clist, app):
    hn0, kn0 = 0, 1
    hn1, kn1 = 1, 0
    ran, rad = 0, 0
    conlist, finallist = [], []
    fraction = namedtuple("fraction", "ratio, denom")
    for n in clist:
        for i in range(1, n + 1):
            ran = hn0 + (i * hn1)
            rad = kn0 + (i * kn1)
            try:
                if D.copy_abs(app-D(ran)/D(rad)) < D.copy_abs(app-D(hn1)/D(kn1)):
                    conlist.append(fraction(f'{ran}/{rad}', rad))
            except:
                pass
        hn2 = (n * hn1) + hn0
        kn2 = (n * kn1) + kn0
        conlist.append(fraction(f'{hn2}/{kn2}', kn2))
        hn0, kn0 = hn1, kn1
        hn1, kn1 = hn2, kn2
    #Change x.denom to x.ratio for the full ratio as a string
    finallist = [ x.denom for x in sorted(conlist, key=lambda i: i.denom) ]
    return list(dict.fromkeys(finallist))
if _name_ == "_main_":
    prec = 200
    decimal.getcontext().prec = prec
    value = D(6/5).ln()/D(2).ln()
    vc = continued_fraction(value, prec)
    print(', '.join([str(x) for x in best_rational_approximation(vc, value)]))

cp's OEIS Frontend

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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A328883 Denominators of the best rational approximations of log(6/5)/log(2).

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Python