Music

C. Wayne Buhrman says:

 Middle C     261.63 Hz
        D     293.66 Hz
        E     329.63 Hz
        F     349.23 Hz
        G     391.99 Hz
        A     440.00 Hz
        B     493.88 Hz
        C     523.25 Hz

Harley L. Miller hmiller@sound.net says:

The theorem underlying the modern scale was set forth by Pythagoras, a Greek philosopher/mathemetician in the 6th century B.C. He discovered that the major consonances (sounds that harmonize well) were the octave, perfect fifth and perfect fourth. Perfect fourth and fifth being the notes four and five up from the fundamental. The length of the strings relate to the fundamental in the ratios of 1:2 (octave), 3:2 (fifth) and 4:3 (fourth).

The perfect scale was found to not be workable when used in a number of different keys, and J.S. Bach demonstrated the versatility of the equal-tempered scale when he wrote 'The Well-Tempered Clavier', a series of pieces written in a succession of keys.

Dave Johnson says:

The current scale is the equal tempered scale (which just means that the intervals between all adjacent notes are equal). Originally the notes were based on the "ideal" frequencies exhibited by vibrating strings or columns of air - it came from physics - but the modern scale has been shifted around slightly to even up the intervals. In fact, it was during Bach's time that the changeover occurred: Bach was one of the main proponents of using equal temperament, but many people at the time thought it was an abomination. In fact, Bach wrote the Well-Tempered Clavier in part to demonstrate how useful equal temperament could be: you can play a piece in any key on the same instrument without re-tuning. In "true" temperament, instruments had to be tuned to a particular key, so you coouldn't, for instance, change keys in the middle of a piece.

Bob McClure says

Ideally, the ratio between the frequency of two adjacent notes should be 2**-12 (twelth root of 2). The problem is that there is an ever so slight difference between 2**-12 (going up) and 1/2**-12 (going down). Equal tempering splits the difference. It takes a very good ear to hear the difference in most music.

Darrel Johansen says:

For an excellent book on this topic see "On the Sensations of Tone" by Herman Helmoholtz. Equal temperament is possible on the piano, and is exactly reproducible on a keyboard synthesizer, but most piano tuners "fine tune" the equal temperament and end up with something a little in-between, especially for the higher notes on the piano. When playing a wind instrument or singing, without piano accompaniment, often the tuning will vary from equal temperament because the harmonies will be much more satisfying (again, see Helmholtz).

Other temperaments are heard not just in avant garde music, but in other cultures including native American. Middle-Eastern music divides the tones up even more divisions. That allows closer harmonies to the "ideal" physically derived ratios than equal temperament, as well as providing nuances that are not notated (which doesn't mean they aren't used, consciously or not) in Western music.

Harold Hallikainen harold@hallikainen.com says:

Herman Von Helmholtz published what we know as Thevenin's Theorem in 1853, while Thevenin did not publish it until 1883.

Also:

• PWM@
• Digital to Analog@
• Speakers
• Digital Signal Processing
• Audio I/O
• Pure Data Visual programming language often used in electronic music.

See also:

• https://z.musictools.live/ A synthesizer visual programming language in a browser window. Inspired by Pure Data and others.+
• Books
  • On the Sensations of Tone by Herman Helmoholtz
• http://www.hallikainen.com/rw/theory/theory5.html.
• http://gigue.peabody.jhu.edu/~philho/sc/chapter1-2.html
• http://hyperphysics.phy-astr.gsu.edu/hbase/waves/cavity.html#c4
• http://www.tubetrap.com/articles/db1191.htm room acoustics
• http://www.campanellaacoustics.com/faq.htm. Acoustics FAQ

Edirol UA-4fx USB Audio/MIDI IO device +

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