Kyle Sager Music . com

SMALL FRACTIONS AND MUSIC

The Fractional Mapping Models affords a fresh perspective for exploring small-number fractions in music (all fractions), for examining how two notes in a musical dyad interact with one another.

This model does not displace traditional thought about how fractions influence intervals, but expands it.

For example, with Fractional Mapping, "just intonation intervals" are still "just intonation intervals" (dyads with notes at fundamental frequencies expressing perfect irreducible fractions, typically small-number fractions).

Suppose we have a just intonation dyad expressing the ratio 3:2 (the perfect 5th, the top note fundamental frequency exactly 3/2 times the bottom note frequency)  Now imagine we slowly glissando this upper note in either direction, up or down, away from 3:2.  Fractional mapping asks the question, "How do different fractions influence this interval as the movement occurs?  As the upper note begins receding from that special 3:2 ratio and as other fractions begin asserting more centrally (some fractions quickly, some more slowly), how do all the fractions simultaneously exert upon this dyad?

In other words, "How do fractions systemically interact with each other around a dyad?"

Fractional Mapping makes the intuitive leap to imagine several fractions might impose influence on a dyad simultaneously. Fractional mapping is about beginning to understand these "in-between spaces" more clearly.

SIMPLER THAN IT LOOKS - AN INTUITIVE MODEL - JUST ONE OCTAVE
On one level, the Fractional Mapping Model is neither more nor less than simply one octave. That's really all there is to it. The map comprises the fractional landscape the top note of a dyad travels across as it glissandos from unison to an octave from the left to the right.

Perhaps the easiest way to understand the Fractional Mapping Model is to imagine a flight instrument with a moving needle or pointer, a gauge. The top note of the dyad is the moving needle, pinned at the apex. This downward pointing needle sweeps across the gauge as the top note glissandos to-and-fro across an octave.

The fixed fractions on the face of the gauge are the relationships we will examine as the top note moves.  The fractions are the fixed landscape we hunt around for clues to the mysteries of musical consonance and dissonance.

That left margin of the gauge (the left side of the 'pyramid') represents the bottom note of the dyad. That bottom note, the left side of the gauge, never moves for purposes of this model. If we bring this needle to rest overtop of the left side of the gauge, on that left margin, then the two notes are the same note, two voices together express perfect unison. Whenever the needle reaches the far right margin, then the top note expresses an octave.


A DYNAMIC MODEL - 
USING SIMPLE TECHNOLOGY
THE SPREADSHEET
There is one nice feature with this approach: Modern computing technology affords an easy, widely accessible platform for jointly exploring of this perceptual domain. Spreadsheet software is available to virtually anyone with web access.  This web-site offers related spreadsheets in LibreOffice Calc, which is a free and opensource.

On the one hand, the spreadsheet solution makes examining many numbers under dynamic conditions simultaneously easy.

On the other hand, spreadsheets are clearly constructed around a visual presentation of rows and columns, straight up-and-down, straight across left-to-right. Also, spreadsheets are not particularly tuned to more subtle graphical presentation.


WORKAROUND HACK FOR SPREADSHEETS - GETTING TO THE PYRAMID
Rows and columns do not offer the best presentation for an intuitive fractional landscape model. They just don't.

A pyramidal structure, however, is much better for this conceptual model than visible rows and columns because a pyramid presents the space in a fashion highlighting the inherent symmetry of the fractional landscape while the logarithmic narrowing at higher frequencies can still be intuited. One way to achieve a pyramid with spreadsheet software is to arrange cells into 'larger cells' (perhaps 4 or 8).  Each 'large cell' represents one fraction.  In this way, rows can be staggered.


SMALL CHALLENGES
Some spreadsheet mechanisms like absolute cell references may become tedious under a pyramidal arrangement. Typical copy-cut-paste operations do not always yield hoped for results along diagonals, making alterations to this spreadsheet a bit time-consuming.

The spreadsheets provided by this web-site are intended to jump-start exploration, allowing visitors here to bypass much of the tedium. With that said, spreadsheets are a terrific place to begin the exploration...because spreadsheets are so ubiquitous today.


BETTER VISUALIZATION TOOLS POSSIBLE - BEYOND THE SPREADSHEET
Psychoacoustic and music researchers and programmers keen on this conceptual framework may be interested in the immediate potential for better graphical tools than those offered on this website.

One significant deficiency in the spreadsheet approach arises in the left-right distance of fractions from the top-note fundamental.  The spreadsheet pushes fraction cells together as though they arrive at fixed distances horizontally. They do not. And the differences can be substantial.

The inclusion of a cents-reference in the bottom of each fraction cell is intended to help ameliorate this deficiency for the time being. A graphical tool that actually positioned fractions at physically appropriated distances along a logarithmic arc would be better.


LIMITING THE DEPTH OF THE MAP, BUT ONLY AT FIRST
THE BIG PRIME CONUNDRUM
This special gauge, The Fractional Map, contains every single just interval possible within an octave. Every possible irreducible fraction between 1:1 and 2:1 is represented (in theory). Nothing is left off the table, no stone is left unturned as we begin exploring fractional relationships. We'll initially tend to chop off the map at the bottom with a denominator of 20 or 30 typically, but this is not required.

The imaginary gauge could be infinitely tall, with massive denominators in the many thousands. We certainly will eventually ask at least a few of those related "what if" questions too:

"But what if a denominator is the huge prime number 3,517 and the top note asserts the irreducible fraction 5,275 : 3,517? Then we have a "very large number" irreducible fraction, right? Will this dyad then be musically consonant because the numbers are just so darn big? Just as Galileo Galilei's father Vincenzo might have been inclined to suggest in Dialogue on Ancient and Modern Music?"

For now the answer is simply, "No, it will not. This very large-number fraction dyad will be extremely consonant in contrary to the small number rule." This interval is an almost perfect expression of 3:2, the just perfect 5th. This very large fraction is in fact 0.16% closer to a just perfect fifth than an equal temperament perfect fifth played on a piano. And anyone but the trained musician might be very hard-pressed to distinguish between any of these three intervals, the difference is so slight among all three.

And yet some educators today, even one or two at premier collegiate music institutions, propagate the ancient notion that "large fractions" like the "Devil's interval," the tritone - 45:32, make for consonance.

The mistake is common and very understandable. We will eventually explore why it persists. 

A TABLE FOR MUSICAL CONVERSATIONS
AND A BRIDGE TO BETTER UNDERSTANDING -
A search for new hypotheses for the ear and brain
Is it possible, though, for us all to begin tightening up our language surrounding musical consonance and dissonance to enable us to find better answers together? We need to begin taking a closer look at how fractions impact wave forms. That is what Fractional Mapping is all about.

At first we limit the size and scope of our map, but only at first. We take things step by step, examining to what degree the map confirms and/or challenges understanding as we begin understanding it. 

Fractional Mapping is about fine-tuning how we see fractions interplay.

It is about a journey to better hypotheses (or even possible validation or circumstantial corroboration of a few recent hypotheses), leading to better ear and brain research, and ultimately better answers...better explanations for why we hear what we hear.

Fractional Mapping will punch upward into the field of psychoacoustics, proposing a precise terrain for researchers to explore...
 

...and Fractional Mapping will punch downward into intermediate music training, perhaps suggesting to high school and early college teachers simple new lenses for exploring and comparing equal temperament intervals with their more intermediate music students.

The general thrust will never be to displace our most important theories, but to simply expand them....to fill in gaps and improve how we talk about music.