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].

To find the recipe for making a physical model

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.

|Graph of this pattern and comparison to integer frequency ratios| |Chapter One| |Chapter Three|

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

To download more technical information go to LucyTuning homepage