Here are the frequencies of the 8 notes in the A major scale A (440Hz) to
octave A (880 Hz). We can compare just tuning to equal-tempered tuning,
with the frequency differences all in Hz :
N=.'A';'B';'C#';'D';'E';'F#';'G#';'A' NB. A-major scale notes
ju=._1 x:440 *1 9r8 5r4 4r3 3r2 5r3 15r8 2r1 NB. Just tuning for A-major
scale (Hz) (rational ratios)
eq=. 0 2 4 5 7 9 11 12 {440*2^12%~i.13 NB. Equal temperament tuning for
A-major scale (Hz)
Now show the frequencies of the A-major scale using just and equal
temperament frequencies as well as the difference in Hz for each note:
N,.{ju,.eq,.-/"1 ju,.eq
┌──┬────────────────────────┐
│A │440 440 0 │
├──┼────────────────────────┤
│B │495 493.883 1.1167 │
├──┼────────────────────────┤
│C#│550 554.365 _4.36526 │
├──┼────────────────────────┤
│D │586.667 587.33 _0.662869│
├──┼────────────────────────┤
│E │660 659.255 0.744886 │
├──┼────────────────────────┤
│F#│733.333 739.989 _6.65551│
├──┼────────────────────────┤
│G#│825 830.609 _5.6094 │
├──┼────────────────────────┤
│A │880 880 0 │
└──┴────────────────────────┘
Skip Cave
On Mon, Mar 30, 2020 at 11:54 AM Skip Cave <[email protected]> wrote:
> 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 <[email protected]>
> 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 <[email protected]> wrote:
>> >
>> >
>> >
>> > > On Mar 30, 2020, at 07:04, Raul Miller <[email protected]> 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
>>
>
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