A BIT O' BLARNEY

              Copyright (c) 2002, 2010 by George D. Secor


*** Foreword ***

I wrote this short story for inclusion in my book-in-progress as
the first Saint Paddy's day of the new millenium (2001) was fast
approaching, and I thought it would be fitting to throw in a bit
o' blarney to introduce a rather unorthodox approach to just
intonation.  I first had the idea for the following story some 25
years before but did not have occasion to develop it until
recently.  It is a tall tale of an encounter with some not-very-
tall characters who, according to legend, are inhabitants of the
Emerald Isle.


*** The Story ***

There once was a lad by the name of Justin, who loved to spend
time in the woods not far from his home.  In the course of his
frequent walks, he would sometimes stop to lie down, rest, and
think about all sorts of things, ranging from profound
philosophical questions to the most trivial matters.

Resting one day in a secluded spot, he noticed an odd little man
wandering toward him, apparently unaware of his presence.  As the
little man came closer, Justin realized that this was no ordinary
person, for he was much smaller than any human alive.  Justin had
heard tales of leprechauns inhabiting these woods, but he had
never before seen one.  As the little man passed by, Justin
lunged forward and grabbed hold of him.  Being quite tenacious
(so much so that it was practically his middle name), Justin was
able to prevent the leprechaun from breaking free of his grasp.

In exchange for freedom, the leprechaun agreed to grant him one
wish.  Being a just man, particularly in matters of musical
intonation, and an ardent xenharmonist besides, Justin expressed
his desire for a musical instrument that would be in just
intonation, yet be capable of free modulation, yet not have not
much more than 20 or so tones in the octave, not an unreasonable
request for a magical leprechaun, he thought.  The leprechaun
became increasingly bewildered as Justin launched into a
theoretical explanation in an attempt to clarify his objective.

After a few minutes of listening to this, the leprechaun
interrupted, "You couldn't wish for something simple, like a pot
o' gold, now could you?  I will try and persuade my mentor to
come and hear your request.  We are both musical, but he has a
way with numbers and would be better able than I to understand
what it is that you desire.  And if it is not in our power to
grant that, then you may ask another wish in its place."  The
leprechaun then swore an oath, promising to return shortly. 
Seeing that there was no reasonable alternative, Justin decided
to trust the leprechaun and agreed to this proposal.

In a short time the leprechaun returned as promised, accompanied
by another leprechaun, somewhat older and wiser in appearance,
and after having been properly introduced, Justin outlined the
requirements for his dream instrument.  As he spoke, the older
leprechaun asked questions and made mental notes, as Justin gave
the acoustical and historical background for his request,
clarifying and adding to the requirements as he went along.  At
one point the older leprechaun questioned him about the concept
of unequal temperaments that were almost equal, thereby
permitting free modulation, but which could have intervals
slightly different in size in different keys; Justin explained
that, while this was acceptable in principle, the result was a
temperament, whereas the first requirement in his list was that
all of the tones in the system be in just intonation.  At another
point, he asked Justin to clarify the concept of consistency, in
which it was required that intervals of the same size span an
equal number of degrees in the tonal system.  "So if, for
example, you are going to have a certain interval in one key, you
must have it in all the other keys to satisfy the requirement of
free modulation, and so on with all the other intervals?", the
older leprechaun asked.  "That's correct," Justin answered.  The
older leprechaun went on, "But if all the intervals of the same
number of degrees are the same size, then you have an equally
divided octave, thus a temperament.  But you don't want any of
the intervals to be tempered, so some of the intervals of the
same number of degrees will have to be different.  On the one
hand, they must be the same, only different, but on the other
hand, they must be different, only the same."  "Right again!",
exclaimed Justin, delighted that he had gotten his message
across.

When asked how the instrument might produce its sound, Justin
described the construction of tubulongs, tubular metal pipes
tuned by cutting and filing them to the proper lengths, supported
by a soft material at the nodes, and sounded by striking with a
mallet.  If these were made from a lightweight metal such as
aluminum, he would be able to place them in a sack and carry them
home without much difficulty.  "There are those who cannot carry
a tune," Justin joked, "but I will need to carry an entire tonal
system.  So you think you can grant me my wish?"

"There are certain wishes that are beyond our ability to bring
about, such as world peace," the older leprechaun explained, "but
your request is not of the same kind as that.  It will take a bit
of time to see if it is possible.  If we work through the night,
we should know by the end of the day tomorrow.  We will return
here by that time with your answer."

That night Justin found it difficult to put the anticipation of
the fulfillment of his wish out of his head, and he got virtually
no sleep.  Still awake at daybreak, he did not want to risk
sleeping through the day, so he arose, had a hearty breakfast,
and after locating a large sack in which to carry the tubulongs,
made his way through the woods to the place where he had seen the
leprechauns the day before.  When he arrived he did not see
anyone else there, so he sat down to wait.  Drowsiness began to
get the better of him, and he eventually lay back and, after a
short time, fell asleep.

He was awakened by a few repeated tugs at the sleeve of his
jacket and opened his eyes to see the two leprechauns standing
before him, empty-handed, illuminated by the rays of the setting
sun.  "You were unsuccessful," he offered in answer to a question
that apparently did not need to be asked.  "Nay!  On the
contrary," the older leprechaun responded with enthusiasm, "only
there are such a multitude of ways to build an instrument such as
you wished, that it was not a simple matter to decide which.  We
would have had to build them all and let you choose one, but for
that there was not time, so we built none.  But no matter; you
could build any of them yourself, once I tell you how.  All of
the requirements of your wish can be met in almost any manner of
instrument in just about any number of tones in the octave
without violating any of the laws of acoustics, once you know
how.  We leprechauns have a reputation for being clever, and we
have learned that if you don't see the solution to the problem by
looking at it one way, maybe you should turn it on its head and
look at it another way.  

"Here is the solution:  In your history of tuning, free
modulation was the first priority and intonation was put second. 
To make sure that you got free modulation, you divided the octave
equally or nearly so with the minimum number of tones that would
reasonably approximate the just intervals you wanted.  But if you
make just intonation the first priority and put free modulation
second, you could then decide how many tones in the octave you
want and use a set of just tones to approximate that equal
division of the octave.  In different keys the just intervals
that will represent a certain number of degrees of the division
of the octave could be somewhat different in size.  For example,
so many degrees could be 7:9 in one key, but 11:14 or 10:13 in
another key; each of these could approximate the same tempered
interval, but they are all just intervals, so nothing is
tempered.  You would have the ability to modulate to any key,
since each key would have a just interval for each of the scale
degrees, but the intervals in each key would not be exactly the
same as in all of the others.  Like numbers of degrees would be
similar in sound, but different, yet all just and none tempered."

"But how do I keep the numbers in the ratios from getting too
large?" asked Justin.

"By making all of the tones multiples of a very low tone," the
older leprechaun answered.

"Like harmonics of a fundamental," Justin chimed in.

"Indeed!  And the best part about the whole thing is that the
lower the number of tones in the octave, the easier it is to keep
the numbers in the ratios low.  Is it not good to have these two
things helping one another instead of tugging in opposite
directions?"

Justin puzzled over this for a couple of moments, trying to
decide whether this was really going to work or if he had been
tricked.  His silence provided the leprechauns with their cue. 
"It is getting dark now, and we must be getting on.  I trust you
will enjoy the many new instruments that you can build, using
this idea.  Farewell and tootle-oo!"  With that the two
leprechauns disappeared into the shadows.

Justin realized that it was getting too dark to get home safely,
so he covered himself with the sack to keep warm and remained
there the entire night, pondering this new approach to tonality
that the leprechauns had given him before falling asleep.  And in
his dreams he heard a paradoxically strange, yet familiar sort of
music emanating from the swirl of thoughts that had been stirred
up in his head.

When he awoke at daybreak, he began to wonder, "Did that really
happen, or was it all a dream?"  And the more time that passed,
the less sure he was of the answer.  But it mattered very little,
for his wish had been granted, and he went on to build many new
instruments and to make an abundance of strange and wonderful
music.  And so ends the tale of Justin Tenacious, the man who got
his just intonation from the leprechauns.


*** Afterword ***

It was not until after I had written the story of Justin and the
leprechauns that I remembered an incident from my own past that
took place under somewhat similar circumstances.  I had been
studying microtonality for only a couple of years and was
relaxing in my bedroom listening to music from a classical FM
radio station.  I fell asleep for about an hour or two, and the
next thing I remembered was hearing some strange-sounding music
as I passed through a dreamlike state on my way back to
consciousness.  Only when the music ended and the radio announcer
identified the composer that I became fully awake with the
startling realization that drowsiness had just deprived me of my
first chance to hear some of Harry Partch's music played from an
out-of-print Gate Five recording (Partch's private label).  If
Justin thought that he might have been tricked, I definitely felt
that I had been cheated, with no one to blame but myself.  Ten
years passed before I was able to obtain a recording of Partch's
music, and until that time, like Justin, I had only the vague
memory of music that I had heard in a dream.

Oh, lest I forget, you'll probably want to know about some of the 
quasi-equal rational tunings that Justin came up with!  May you 
enjoy exploring this generous baker's dozen:

5-EDO approximations
12:14:16:18:21
13:15:17:20:23
26:30:34:39:45

7-EDO approximations
18:20:22:24:27:30:33
17:19:21:23:25:28:31

9-EDO approximations
24:26:28:30:33:36:39:42:45
22:24:26:28:30:32:35:38:41
23:25:27:29:31:34:37:40:43

10-EDO approximations
24:26:28:30:32:34:36:39:42:45
26:28:30:32:34:37:40:43:46:49
25:27:29:31:33:35:38:41:44:47

12-EDO approximation
30:32:34:36:38:40:42:45:48:51:54:57

14-EDO approximation
36:38:40:42:44:46:48:51:54:57:60:63:66:69