One of the advantages of computer generated music, is that one can generate intervals using just interval tuning (rational ratios), rather than the equal temperament approximations used on most keyboard instruments today. Rational ratio or just intervals are the most pleasing to the ear.
Equal temperament tuning is used on typical keyboard instruments such as a piano, so dissonance (rational ratio error) can be minimized when playing in multiple keys. Equal temperament tuning divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2 (12√2 ≈ 1.05946). So intervals can be close to rational in all keys, but not exact. Just tuning ratios: NameCDEFGABC Ratio from C 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1 Twelve-tone equal temperament <https://en.wikipedia.org/wiki/File:Monochord_ET.png> <https://en.wikipedia.org/wiki/File:Monochord_ET.png> One octave of 12-tet on a monochord In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone <https://en.wikipedia.org/wiki/Semitone>, i.e. the frequency ratio <https://en.wikipedia.org/wiki/Interval_ratio> of the interval between two adjacent notes, is the twelfth root of two <https://en.wikipedia.org/wiki/Twelfth_root_of_two>: {\displaystyle {\sqrt[{12}]{2}}=2^{\frac {1}{12}}\approx 1.059463}[image: {\displaystyle {\sqrt[{12}]{2}}=2^{\frac {1}{12}}\approx 1.059463}] This is equivalent to: {\displaystyle e^{{\frac {1}{12}}\ln 2}\approx 1.059463}[image: {\displaystyle e^{{\frac {1}{12}}\ln 2}\approx 1.059463}] This interval is divided into 100 cents <https://en.wikipedia.org/wiki/Cent_(music)>. Calculating absolute frequencies[edit <https://en.wikipedia.org/w/index.php?title=Equal_temperament&action=edit§ion=12> ] See also: Piano key frequencies <https://en.wikipedia.org/wiki/Piano_key_frequencies> To find the frequency, *Pn*, of a note in 12-TET, the following definition may be used: {\displaystyle P_{n}=P_{a}\left({\sqrt[{12}]{2}}\right)^{(n-a)}}[image: {\displaystyle P_{n}=P_{a}\left({\sqrt[{12}]{2}}\right)^{(n-a)}}] Skip Skip Cave Cave Consulting LLC On Mon, Mar 30, 2020 at 10:56 AM Raul Miller <rauldmil...@gmail.com> wrote: > I wish I did... but I was on a different page (where the price has now > dropped to $814): > > https://www.amazon.com/Cybernetic-Music-Jaxitron/dp/0830608567 > > FYI, > > -- > Raul > > On Mon, Mar 30, 2020 at 10:45 AM Mike Powell <mdpow...@gmail.com> wrote: > > > > > > > > > On Mar 30, 2020, at 07:04, Raul Miller <rauldmil...@gmail.com> wrote: > > > > > > Hmm... > > > > > > Currently only $854 for a paperback copy of Cybernetic Music from > > > Amazon ($34 for hardcover, though (and $14 for a paperback copy from > > > Albris -- plus $4.80 for tax and shipping...)). > > > > I think you meant $8.45 for the Amazon paperback. > > See https://www.amazon.com/Cybernetic-music-Jaxitron/dp/0830618562 > > > > Mike > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
