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



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



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





7-EDO approximations




9-EDO approximations





10-EDO approximations





12-EDO approximation



14-EDO approximation





main page