# The Logarithmic Piano

///////

Related KMR pages:

///////

Other related sources of information:

Ackord på Musikipedia
Pianoaccord – ackordfinnare på Musikipedia.

///////

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.

///////

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:

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

///////

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 $\, 2^{ \, 7} = 128$.

///////

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

///////

Music and math: The genius of Beethoven