LucyScaleDevelopments presents extracts from:

Pitch, Pi, and Other Musical Paradoxes (a practical guide to natural microtonality)

Chapter Two

FOURTHS AND FIFTHS: CIRCLES, SPIRALS, OR CYLINDERS?

Twelve tone Equal Temperament (12tET)

In twelve note equal temperament, as the octave is divided into twelve equal intervals, it is possible to construct a circle of twelve notes in intervals of fourths of (500 cents) in a clockwise direction and of fifths (700 cents) in the opposite direction. Moving twelve steps in either direction will arrive back at the starting note in the next octave. Spiral of fourths and fifths.

This principle of cumulative fifths is also used to arrive at the pitches of the Pythagorean tuning, which like many of the fractional scales, uses the ratio of 3:2 for the fifth = ratio of 1.5.. The Pythagorean tuning having completed this circle arrives at a ratio of 531441:262144 = 2.0272865 and hence a spiral (see Chapter One of Pitch, Pi,....)

LucyTuning spiral

In LucyTuning, a sharp and its adjacent flat are not assumed to be the same pitch, instead of a circle of fourths and fifths; there is an expanding spiral of fourths (504.51 cents) and a contracting spiral of fifths (695.49 cents).

Twelve LucyTuned fourth intervals; 1.33832012 = 33.015632 instead of (32.00000 for 5 octaves) or 1.333333312 = 31.569292 (12* 504.51) = 6054.12 cents

compared to 1200 ^ 5 = 6000 cents i.e. 6054.12 - 6000 = 54.12 cents sharper, for each fourths revolution of the spiral. and for twelve fifth intervals (1.49441112 = 124.06254 instead of (128.0000 for seven octaves) or 1.5^12 = 129.74634 (12* 695.49) = 8345.88 cents compared to 1200 ^ 7 = 8400 cents i.e. 8400 - 8345.88 = 54.12 cents flatter, for each fifths revolution of the spiral. [This is the bbIInd interval 2s-L].

As this scale has an infinite number of intervals, the sharpened notes become closer to the adjacent flattened notes as the number of intervals per octave increases. By increasing the number of notes per octave, eventually adjacent pitches become too close for the human ear to distinguish between them. Any interval may therefore be described in musical terms of single or multiple sharps or flats, as shown below.

The significance of the columns is as follows:

The leftmost five columns are for the cycle of fifths.

Name. This is the name of the note starting from A. and follows the sequence A E B F# C# G# D# after which the sequence repeats with one extra sharp A# E# B# F## C## G## D## and continues the next step with A## etc.

Position in scale. This column shows the position in the A Major scale. Remember A Major has three sharps. The scale positions are expressed in Roman numerals. A=I B=II C#=III D=IV E=V F#=VI G#=VII. As with the note names the pattern is again repeated after seven steps and for the fifths is I V II VI III VII #IV followed by I# V# II# etc.

Cents from A. This column shows the interval upwards from A to the nearest named note, expressed in cents (1200 cents = one octave).

Large and small intervals (L&s). This column shows the number of Large and small intervals from which this interval is also derived. The values are always multiple addition and subtraction of whole Large and small intervals. The sequence of the pattern in this column (for fifths) is continued additions of 3L+s. So that for the second step the value is (3L+s)*2 = (6L+2s), but since this now takes us above the first octave and into the second it has been reduced by (5L+2s) to give a value of less than 1200 cents, and therefore (6L+2s)-(5L+2s)= L, which is less than one octave above our starting point. To find the value for any step of fifths or {fourths} multiply the step number by (3L+s) or {2L+s} and subtract the nearest number of whole octaves (5L+2s) below. The result is your remainder and the value for this step in the first octave.

Hertz This is the frequency of the named note in the octave between A2=110 Hz. and A3=220 Hz.

Step number. Surprisingly, this is exactly what it says; the number of steps in fifths or fourths from the starting point of A2=110Hz, 0 cents, as the tonic (I). The rightmost five columns are the equivalent columns for fourths and are the mirror image of the columns explained above. The fourth interval is (2L+s), and the note name, and scale position sequences are the exact reverse of those for the fifths. You will notice that for each step the fourths columns added to the fifths columns exactly equals one octave.

Table of first 43 Note Names, Hertz and Cents for cumulative fifths and fourths (A=110 Hz.)

This table shows the result of cumulative fourth and cumulative fifth intervals. Large (L) and small (s)intervals are shown from A at 110 Hertz. The cent values are in relation to A and the frequencies assume A=110 Hz.

 Note name Position cents from A L & s from A Hertz Steps Note name Position cents from A L & s from A Hertz A I ------- ----- 110.000 00 A I ------- ----- 110.000 E V 0695.493 3L+s 164.385 01 D IV 0504.507 2L+s 147.215 B II 0190.986 L 122.830 02 G bVII 1009.014 4L+s 197.021 F# VI 0886.479 4L+s 183.558 03 C bIII 0313.521 L+s 131.838 C# III 0381.972 2L 137.156 04 F bVI 0818.028 3L+2s 176.442 G# VII 1077.465 5L+s 204.967 05 Bb bII 0122.535 s 118.068 D# #IV 0572.958 3L 153.153 06 Eb bV 0627.042 2L+2s 158.013 A# #I 0068.451 L-s 114.436 07 Ab bVIII 1131.549 4L+3s 211.471 E# #V 0763.944 4L 171.015 08 Db bIV 0436.056 L+2s 141.508 B# #II 0259.438 2L-s 127.784 09 Gb bbVII 0940.563 3L+3s 189.383 Fx #VI 0954.931 5L 190.961 10 Cb bbIII 0245.070 2s 126.727 Cx #III 0450.424 3L-s 142.687 11 Fb bbVI 0749.577 2L+3s 169.602 Gx #VII 1145.917 6L 213.234 12 Bbb bbII 0054.084 2s-L 113.491 Dx xIV 0641.410 4L-s 159.329 13 Ebb bbV 0558.591 L+3s 151.887 Ax xI 0136.903 2L-2s 119.052 14 Abb bbVIII 1063.098 3L+4s 203.273 Ex xV 0832.396 5L-s 177.912 15 Dbb bbIV 0367.605 3s 136.022 Bx xII 0327.889 3L-2s 132.937 16 Gbb 3bVII 0872.112 2L+4s 182.041 F3# xVI 1023.382 6L-s 198.663 17 Cbb 3bIII 0176.619 3s-L 121.815 C3# xIII 0518.875 4L-2s 148.442 18 Fbb 3bVI 0681.126 L+4s 163.027 G3# xVII 0014.368 2L-3s 110.917 19 B3b 3bII 1185.633 3L+5s 218.182 D3# 3#IV 0709.861 5L-2s 165.755 20 E3b 3bV 0490.140 4s 145.999 A3# 3#I 0205.354 3L-3s 123.853 21 A3b 3bVIII 0994.647 2L+5s 195.393

This progression continues infinitely. Pi is both an irrational and a transcendental number. Each further step will produce two new and unique values and hence more intervals and positions. Notice that the patterns of note names and note positions repeat themselves every seven steps adding an extra # through the fifths and b through the fourths.

For each step the cents, intervals, and note positions from the fifths added to the corresponding cents, intervals and note names from the fourths result in an exact octave.

Eg. after nine steps 763.94 + 436.06 = 1200.00 cents; #V + bIV = VIII and 4L + (L+2s) = 5L +2s.

You may have noticed that the intervals which are closest on the spiral of fourths and fifths tend to sound more consonant.
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