A piano has approximately seven octave patterns. For notation of notes I'll use {A, B, C, D, E, F, G} and the so called accidental signs (b for flat) and (# for sharp). The repeating octave pattern can be visualized as

           C#  Eb      F#  Ab  Bb
           |-| |-|     |-| |-| |-|         Black Keys
          C   D   E   F   G   A   B        White Keys

As far as a modern keyboard with well tempering goes:

           C# = Db,   D# = Eb,  F# = Gb,  G# = Ab,  A# = Bb
           B  = Cb,   E  = Fb,  C  = B#,  F  = E#,

   The "ascending chromatic sequence" is

      ...  C C# D Eb E F F# G Ab A Bb B (C)  ...

   where the ratios of frequencies: (C# / C), (D / C#), (Eb / D), ...

   are all equal to the 12th root of 2.

   This defines "well tempering".

Return to Home Page
Return to Music Page
Circle of Fifths
Medieval Modes
Greek Modes
Traditional Harmony
Patterns, Transformations and Groups in Musical Composition
Evolving Dodecaphony
Mathematical Groups
Pitch Sets in Composition

Email me, Bill Hammel at

The URL for this document is:
Created: September 1997
Last Updated: May 28, 2000
Last Updated: March 1, 2011