May the 4ths be with you

[Richard Moody]

The beat rates of contiguous 3rds if tuned in ET beat at the ET
ratio, which is NOT 5/4.  Take the beat rate of C--E  and then
E--G#   and you will see the ratio is not the EXACT ratio of 5/4.
Actually the ratio of beat rates of  two contiguous 5/4 3rds are
zero because those two thirds 3rds have no beat.   Dr Sanderson
must have been giving a generalized explanation rather than a
mathematically correct one.   The proof of the ratios of 3rds in
ET is that ratio cubed, (or ^3)  equals 2.     1.25992105  cubed
equals 2

The ratio of the frequencies of ET 3rds is 1.25992105  ,  the
ratio of their beat frequencies is 1.25992105  .    How can it be
otherwise? -----ric





----- Original Message -----
From: Richard Brekne <Richard.Brekne@grieg.uib.no>
To: PTG <pianotech@ptg.org>
Sent: Saturday, August 17, 2002 10:47 AM
Subject: May the 4ths be with you


> List.
>
> The following is an except from appendix F of the SAT
> manual. It gives an explanation by Dr Sanderson > "Two
contiguous musical intervals are intervals that touch
> each other, in other words, share the note in the middle.
> Tests that use contiguous intervals are easy to learn and
> use, and tell the tuner explicitly which notes are at fault
> and what to do to correct them.
> Contiguous major thirds will beat in the ratio of four to
> five because the major third itself consists of two notes
> whose frequencies are in the ratio of four to five.
> Displacing any interval up the keyboard will speed it up
> theoretically in the ratio of the frequencies of the two
> root notes involved.  Therefore two contiguous major thirds
> should beat in the ratio of four to five, two contiguous
> minor thirds in the ratio of five to six.Similarly, two
> contiguous fourths should beat in the ratio of three to four
> and two contiguous fifths in the
> ratio of two to three.  However, on the piano this
> theoretical relationship holds well only for the major and
> minor  thirds.    The  fourths  and  fifths  are  so
> strongly  affected  by  inharmonicity  that  these
> contiguous intervals beat at almost the same speeds"
>
>
> Cheers !
>
> Ricb