Fig. 2. A sequence of thirteen consecutive 3:2 fifths.
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Fig. 1 shows a sequence of thirteen consecutive 3:2 perfect fifths beginning on C2. Notice how quickly the sequence rises in pitch. If we continue the sequence beyond 13 pitches, it is interesting to note that this sequence will never generate a pitch that is octave-related (i.e., a pitch whoses frequency is 2x, 4x, 8x, 16x, and so on) to C2. The pitch sequence in Fig. 1 can be expressed using pitch notation as:

Notice that after thirteen consecutive 3:2 perfect fifths we reach the pitch B#8, not C9. If the frequeny of C2 is 65.41 Hz., the frequency of C9 would be 65.41 Hz. * (2:1)⁷ = 65.41 Hz. * 128 = 8,372.48 Hz. On the other hand, the frequency of the B#8 in Fig. 1 would be calculated in the following manner: 65.41 Hz. * (3:2)¹² = 65.41 Hz. * 129.75 = 8,486.95 Hz.--a difference of 114.47 Hz. This discrepency in pitch, or comma as it is known in tuning theory, is called the Pythagorean comma because of its associaton with Pythagoran tuning. The size of the Pythagorean comma may perhaps best be expressed as the interval frequency ratio or in cents as shown below:

Mathematically speaking, these two pitches, B#8 and C9, are obviously not the same. In the sense that it never produces an octave-related pitch, the pitch sequence shown in (1) may be regarded as an infinite sequence, because it never generates an octave-related pitch. The elipses symbol "..." in Fig 1 and (1) is used to indicate that the sequence continues forever. Correspondingly, a descending sequence of consecutive 3:2 perfect fifths never produces an ocatve-related pitch. Sequence (1) can be collapsed under octave equivalence into the sequence of pitch-class symbols known as the line of fifths which is shown in Fig. 2.

Note that the sequence shown in Fig. 2 can be produced by extending sequence (1) in both directions (up/down in fifths) and by dropping the octave numbers. Furthermore, note that our modern pitch notation system uses only seven basic pitch-class name symbols

These seven symbols corrospond to the seven tones of the major diatonic scale built on C. When an eighth pitch symbol is required in the upward direction, a sharp symbol modifier (#) is concatenated to the end of the symbol (e.g., F, C, G, D, A, E, B, F#, ...), and so on, for the next six fifths. In the downward direction, a flat (b) is used as a modifier. Once seven sharps or flats have been used, new modifiers are required called the double-sharp (##) and double-flat (bb), respectively.

The line of fifths is perhaps most useful as a way of concepualizing of tonal distances. For example, C is one fifth away from G, but two fifths away from D, and three fifths away from A. Thus, from the point of view of fifth relations, the key of G (G:) is more closely related to C:, than D: is to C:, and so on. Additionally, it is interesting to note that any set of seven consecutive pitch classes in the line of fifths produces a diatonic scale, while any set of five consecutive pitch classes produces a major pentatonic scale.