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Re: Musical notes...

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Show of hands, please.  How many of you view rec.puzzles in a
fixed-width font?  IIRC back in the 20th century, this newsgroup
had an over-riding default to fixed-width (Courier) because many
presentations relied on that.

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Music scales lead to interesting exercises in simple arithmetic.
Nothing new in this post, but it might be a starting point for those who've
never thought about the arithmetic of music intervals.

In the following diagram white keys on the piano are shown as AAA, BBB, etc.
Black keys are shown as ,,, or mmm.  The keys C major and A minor use
only white keys, but for ease of comparison we look at the key of C minor
where three black keys (E♭, A♭ and B♭) are used.

     CCC  ,,,  DDD  mmm  EEE  FFF  ,,,  GGG  mmm  AAA  mmm  BBB  CCC
      24  ...   27  28.8  30   32  ...   36  38.4  40  43.2  45   48
                        "3rd" "4th"    "5th"     "6th"     "7th"

Starting with C = 24 arbitrarily, the other white keys are simple integers:
Initially 24, 27, 30.  The numbers need to be sort of logarithmic,
so the whole step difference increases from 3 to 4: 32, 36, 40.
(The step from 30 to 32 is just a half step: There's no black key in between.)
Finally, the difference is 5 (40, 45) with the final 45, 48 another
half-step.

If you ever forget the simple rational numbers which define the Just Scale,
the above explanation for (24, 27, 30, 32, 36, 40, 45, 48) should
obviate the need for Google.  Unfortunately if you tune your piano
to produce the Just Scale for C Major, it won't be Just in any other key.

The black keys shown above for C Minor have ratios to C=24 that may look
uninviting (28.8/24 "minor 3rd", 38.4/24 "minor 6th", 43.2/24 "minor 7th")
but multiply throughout by 25/24 and get nice numbers: 6/5, 8/5, 9/5.
(In "principle."  Your instrument is very unlikely to be tuned this way.
6/5, 14/9, 16/9 is another solution for the black keys in C minor.)

The scales evolved, and some cultures used quite different scales.
Etc., etc.

James

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Thread

Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 08:20 +0000
  Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 12:58 +0000
    Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 18:06 +0000
      Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-28 10:15 +0000
  Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-27 16:25 +0000
    Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 17:10 +0000
  Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-28 12:30 +0000
    Re: Musical notes... HenHanna@NewsGrouper <user4055@newsgrouper.org.invalid> - 2026-05-28 18:21 +0000
    Re: Musical notes... msb@vex.net (Mark Brader) - 2026-05-28 18:50 +0000
    Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-29 17:19 +0000
    Re: Musical notes... Charlie Roberts <croberts@gmail.com> - 2026-05-29 15:48 -0400
      Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-29 20:59 +0000
        Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:07 +0000
          Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:25 +0000
            Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:33 +0000
            Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-30 12:56 +0000
    Re: Musical notes... Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-05-29 22:38 +0100
      Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-30 06:38 +0000
    Re: Musical notes... Phil Carmody <pc+usenet@asdf.org> - 2026-06-05 16:41 +0300
      Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-06-05 14:54 +0000

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