Steve Burian wrote:
RE: Metronome "Scale"
Take the first 17 numbers:
40 42 44 46 48 50 52 54 56 58 60 63 66 69 72 76 80
Notice that they count by twos from 40 to 60. Then by threes. Since 60-80 can't
be divided evenly by threes he (Maelzel?) fudged by counting fours at the end.
"fudged" might not capture the likely reason for the shift in
increment. Note the 1st marking that begins each sequence of constant
increment: 40, 60, 72. Divide the increment by the leading tempo marking:
2/40 = .05 = 5%
3/60 = .05 = 5%
4/72 = .055 = 5.5%
and the sequence continues:
6/120 = .05 = 5%
8/144 = .055 = 5.5%
It makes sense to keep the *percent* change of tempo relatively
constant, and change of increment is how to do it; our sensibilities
are more sensitive to fraction of change or relative change, rather than
absolute change. For example, a change of 10 beats per minute in the
area of 60 bpm is much more noticeable than a change of 10 bpm in the
area of 200 bpm.
The percent change between successive metronome markings varies from
about 3.5% to 5.5% throughout the sequence, or an average change of
about 4% between markings. If this is the underlying principle of the
variation of increment, it is consistent with Hans' notion of selecting
'nice' numbers with which to divide the number 60; 60 being not only the
connection between minutes and seconds, but also has more factors than
any number smaller than itself: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,
60, and so is more capable of making nice fractions of a minute.
Here is a "logorhythmic" scale I found on the web:
40.00 41.77 43.62 45.55 47.57 49.67 51.87 54.17
56.57 59.07 61.69 64.42 67.27 70.25 73.36 76.61 80.00
But forget that crazy stuff.
which is of equal interest as this sequence that I found on the web:
14, 18, 23, 28, 34, 42, 50, 59, 66, 72, 86, 96, 103, 110, 116, 125, 137,
145, 157, 168, 181, 191, 200, 207, 215, 225, 231, 238, 242
and which is related to the metronome sequence, and part of which an
ancient calculus prof wrote on the blackboard and made the class
struggle with for a very long time. Interested persons read no farther
until you give up.
.
.
.
.
.
This is the sequence of street stops on the IRT subway line in New York
City. And there is a relation to the metronome. Well, the subway is a
'metro'.
By the way, for a metronome set at 66 bpm on a moving train to sound
like 63 bpm, the train has to travel away from the observer at about 35
mph. This is also of equal interest.
David G
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