The Logarithmic Piano

This page is a sub-page of our page on Music.

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Related KMR pages:

Pythagorean Intervals
Uniformisation of Intervals
Solfege: An Abstract Key System
Instantiations of Solfege in different Keys
Abstract and Concrete Chord Circles
Interactions of Quarts and Quints
Chord Ladder
Chord-Set Inclusion Graph
Transposition of Key = Shift of Basis in Music
Bengt Ulin om Musik (in Swedish)

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Other related sources of information:

Ackord på Musikipedia
Pianoaccord – ackordfinnare på Musikipedia.

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Underlying frequencies of the notes:

Pythagorean tuning: \, q_{uart} \, * \, q_{uint} = \frac{4}{3} \, * \, \frac{3}{2} = \frac{2}{1} = o_{ctave} \, .

Equally Tempered tuning: \, q_{uart} \, * \, q_{uint} = 2^{\, \frac{5}{12}} \, * \, 2^{ \, \frac{7}{12}} = 2^{\, (\frac{5}{12} + \frac{7}{12})} = 2^{\, \frac{12}{12}} = 2^{\, 1} = 2 \, d_o = o_{ctave} \, .

NOTE: Counting the steps between adjacent notes on the piano gives a logarithmic behaviour independently of tuning.

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Multiplication of musical intervals:

Every square of a note is equal to \, 1 \, :

\, d^2_o = r^2_e = m^2_i = f^2_a = s^2_o = l^2_a = t^2_i = 1 \,

since such a square represents an interval (= a frequency multiplier) of size \, 1 = 2^{ \, 0} ,
which does not change the pitch of the tone. Hence we have:

\, (d_o r_e) (r_e m_i) = d_o r_e r_e m_i = d_o m_i \,
\, (d_o m_i) (m_i f_a) = d_o m_i m_i f_a = d_o f_a \,
\, (d_o f_a) (f_a s_o) = d_o f_a f_a s_o = d_o s_o \,
\, (d_o s_o) (s_o l_a) = d_o s_o s_o l_a = d_o l_a \,
\, (d_o l_a) (l_a t_i) = d_o l_a l_a t_i = d_o t_i \,

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On an equally tempered 12-tone scale, such as the one employed by the piano,
we therefore have:

\, d_o m_i = d_o r_e r_e m_i = (d_o r_e) (r_e m_i) = 2^{ \, \log(d_o r_e)} \, 2^{ \, \log(r_e m_i)} = 2^{ \, \log(d_o r_e) \, + \, \log(r_e m_i) } = 2^{ \, \frac{2}{12} \, + \, \frac{2}{12}} = 2^{ \, \frac{4}{12}} = 2^{ \, \log(d_o m_i)} = m_{ajorThird} \,

\, d_o f_a = d_o m_i \textcolor{red} {m_i f_a} = (d_o m_i) ( \textcolor{red} {m_i f_a} ) = 2^{ \, \log(d_o m_i)} \, 2^{ \, \log( \textcolor{red} {m_i f_a} ) } = 2^{ \, \log(d_o m_i) \, + \, \log( \textcolor{red} {m_i f_a} ) } = 2^{ \, \frac{4}{12} \, + \, \textcolor{red} {\frac{1}{12}}} = 2^{ \, \frac{5}{12}} = 2^{ \, \log(d_o f_a)} = q_{uart} \,

\, d_o s_o = d_o f_a f_a s_o = (d_o f_a) (f_a s_o) = 2^{ \, \log(d_o f_a)} \, 2^{ \, \log(f_a s_o)} = 2^{ \, \log(d_o f_a) \, + \, \log(f_a s_o) } = 2^{ \, \frac{5}{12} \, + \, \frac{2}{12} } = 2^{ \, \frac{7}{12} } = 2^{ \, \log(d_o s_o)} = q_{uint} \,

Hence we have \, \log m_{ajorThird} \, = \, \frac{4}{12} \, , \log q_{uart} \, = \, \frac{5}{12} \, , \, \log q_{uint} \, = \, \frac{7}{12} \, ,

and therefore, for example:

\, q_{uart} * q_{uint} = 2^{ \, \log q_{uart} \, + \, \log q_{uint} } = 2^{ \, \frac{5}{12} \, + \, \frac{7}{12} } = 2^{ \, \frac{12}{12}} = 2^{ \, 1} = 2 = o_{ctave} \, ,

which is the musicological term that denotes this interval.

For example, when we say that we are “raising a note by an octave,” we mean that we are producing a new note that has twice the pitch (= frequency) of the original one. Together these two notes form the interval of an octave. Since a normal piano spans a width of seven octaves, the ratio between the pitches of the highest and the lowest tone (normally two A:s), is \, 2^{ \, 7} = 128 .

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The tune “The Entertainer” (by Scott Joplin) represented in “piano coordinates”:

The Entertainer - represented in sheet-music for piano

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Music and math: The genius of Beethoven
Natalya St. Clair on YouTube:

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Why Not Admit There is a Problem With Math and Music?
(Dan Formosa at TEDxDrexelU):

From the description on YouTube: Dan Formosa’s work covers many areas of design. A unifying theme is the thought that great design first requires understanding people. To accomplish that, design needs to embrace complexity. To that end, Dan’s TEDxDrexelU talk will focus on math, music, design, and how it all seems to have gone terribly wrong. In 1980 Dan helped establish Smart Design to explore ways design can positively affect peoples lives. He frequently lectures on biomechanics, information design, behavior, gender and social responsibility. He also recently helped create the new Masters in Branding program at the School of Visual Arts in New York City.

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