A little musico-statistical analysis of 20er music

Hi -- Yesterday, I asked myself (why? holidays? too many other
important things to do?) how many different chords are used in 20er
arrangements.  I immediately set out to answer this eminently important
question; and then proceeded to write a scientific article on my
enterprise.  Here it is.  :-) 

 Introduction
 - - - - - - 

(Let's keep this short.)  How many different chords are actually used
in 20er arrangements?  How many can be used?

 Previous work
 - - - - - - -

I don't know of any.  But I remember foggily that someone did a
statistical analysis of note and chord frequencies in Bach fugues
(I think Douglas Hofstadter's "Goedel, Escher, Bach" contains something
about this [1]).  I leave it as an exercise to the reader to search the
MMD archives, the various journals of various societies, and the millions
of scientific journals out there (but if you actually do search around
and, as it may happen, do find anything, let me know, please!).

 Theoretical foundation
 - - - - - - - - - - - 

The standard 20er-scale has been in use for at least 200 years
(see [2]).  In the key of B-flat (which I use in all my arrangements),
it contains the following 20 notes:

f bb c' d' eb' e' f' g' a' bb' c" d" eb" e" f" g" a" bb" c"' d"'

where a' is assumed to be the note with 440Hz (or thereabouts).

The number of possible chords for the 20er with various numbers of
notes can easily by computed:

 1 note      20 (not really "chords", I admit)
 2 notes    190 = 20! / (2! 18!) = 20*19/2*1
 3 notes   1140 = 20! / (3! 17!)
 4 notes   4845 = 10! / (4! 16!)
 5 notes  15504
 6 notes  38760
 7 notes  77520
 8 notes 125970
 9 notes 167960
10 notes 184756
etc.

 [ The '!' symbol ('exclamation point') represents the math operation
 [ called factorial.  '20!' is read '20 factorial'.  -- Robbie

Obviously, all the (i+k)-note chords contain all the i-note chords for
any k > 0.  So we can now define two sorts of "occurrences" of a chord
in an arrangement:

Definition: "Strict Occurrence" -- A chord consisting of notes n1, ...,
nk occurs strictly in the arrangement if exactly the notes n1...nk
sound at the same time; and no other note.

Definition: "(General) Occurrence" -- A chord consisting of notes n1,
..., nk occurs (generally) in the arrangement if the notes n1...nk
sound at the same time; and possibly other notes.  The attribute
"General" will, in general, be omitted.

 Data and Measuring equipment
 - - - - - - - - - - - - - - 

As a data base, I had only my own arrangements -- a trifle more than
100 from the last 5 years.

The punching program I wrote can also "punch" to a text file, where it
writes a nice little "o" for each hole - so I ran the program over the
100+ arrangements mentioned above.  After 15 minutes, I had a file with
156MB and a few million o's in it.

With the friendly help of a good text editor (UltraEdit), I managed to
format that file in various ways which allowed me to arrive at the
first counts.

 Strict occurrences
 - - - - - - - - - 

Strict occurrences were relatively easy to count.  Here are the
aggregated results:

# of notes/different chords occurring strictly
  1         20
  2        189
  3        846
  4       1621
  5       1383
  6        699
  7        211
  8         45
  9          5
 10          1

>>10          0


Hence, 5020 different chords are used in the data base.

Comparing this with the list above, we can conclude the following
facts:

1. All notes are used alone somewhere (ha!).

2. There is a two-note chord that does not occur strictly.  Further
research revealed that this is the chord consisting of eb"+e" (however,
eb'+e' does occur strictly somewhere).

3. For 3 or more notes, there is a significant number of chords that do
not occur strictly.

Because strict occurrence is relatively uninteresting (as even adding a
note occurring in the chord again one octave higher or lower makes the
chord "strictly" a different one), I stopped researching here.

 General occurrences
 - - - - - - - - - -

Statistics for general occurrences are harder to find - I had to write
a little program (about 30 lines not counting the data arrays).  From
this, I found the following:

1. All notes occur somewhere (Ha!!)

2. All pairs of notes occur somewhere (also the pair eb"+e" not
occurring strictly occurs as part of some chord somewhere)

3. From the 1140 possible 3-note chords, only 38 do not occur.  That's
a mere 3.33333% (by coincidence, 38 also divides 1140 evenly ...  just
as a side-note) and definitely the most interesting result.  I have
attached a GIF image which shows all these chords.

4. From the 4845 possible 4-note chords, 1378 do not occur -- 28.44%
(and no nice division this time).

5. From the 15504 possible 4-note chords, 11147 do not occur.

Here I stopped counting, because it got boring.

 Further work
 - - - - - - 

(a) "-statistical" -- The definition "a chord is everything which
sounds at the same time" is quite technical and not really musical.
One should try the same playing around ...  ahem: research with a
narrower definition.  Ideas coming to mind:

* Notes playing together longer than some fixed time.  The minimal
sounding time for a note of a keyless 20er is somewhere around 50
milliseconds (3mm hole at 60mm/s and the assumption that the time from
the start of the hole to the pipe's sounding is roughly the same as the
time from end-of-hole to sound-stopping).  So useful "chord times"
would start at 100 ms and above.

* Notes playing together at certain times ("beats") in a measure.  Very
hard to define and argue.  Sounds like real work.

Did anyone try to find out when a group of notes sounding at the same
time is perceived as a "chord," i.e., as some "larger sounding entity"?

(b) "Musico-"  -- Of course, I'll now try to sneak in the 38 missing
3-note-chords (and the one strictly missing note pair) into some
arrangements of mine ...  watch out  ;-)  !!

 Thanks
 ------

My computer and all the programs on it don't get any thanks -- it's
their duty to provide support for such research.  But I have to thank
(or at least apologize to) my wife Astrid who did not complain when
I went to bed yesterday somewhat later because of all of this.


 References
 ----------

[1] Douglas Hofstadter, "Goedel, Escher, Bach".  Written in the '80s
and very good!

[2] ...  well, this whole thing is not _that_ important.  So I'll leave
out any references further on.  But feel free to solve the exercise in
section "Previous work"!

Harald Mueller