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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.
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