April 2, 2019
1: “in tune” or “out of tune”:
The tonally trained ear expects to hear things in a certain way, and clings to that way in spite of gradually mounting evidence that what they are hearing is not tonal but microtonal.
I’ve demonstrated this with a sound experiment in which a major triad (such as C-E-G) is gradually transformed into a minor triad (C-Eb-G) followed by a reverse direction. The root note and fifth remain constant; the third is gradually lowered over the course of a certain duration until it has fallen a half step, at which point the third starts rising in pitch, at the same rate as it the pitch was lowered, until it is back to its usual position as the third of a major triad.
In this particular experiment the third is lowered (and later raised) at the rate of one hundredth of a semi-tone (a “cent”) every third of a second.
For many people, no change is noticed for a while. Their ear continues to hear, or cling to hearing, a major triad – one albeit that is “out of tune” but still clearly intended to be a major triad. The ear does not accept that it is perceiving a microtonal tonal triad that is neither major nor minor. The microtonal change is considered an imperfection in the intonation. There is no recognition of the triad as being of a new aesthetic species – neither major or minor.
Then a sudden switch occurs. At a certain point in the migration downwards of the third of the triad, most interestingly a point that is closer to the eventual minor chord and further from the initial major chord, the sound, almost instantly, changes in the listener’s ear from being heard as an out of tune major triad to an out of tune minor triad.
This “inaccurate” minor triad persists until the third is close to its final value, at which point the sensation the minor chord at last is getting more and more “in tune”, until at the end it sounds very in tune.
The most interesting part of this sound experiment is that when the third starts traveling in the other direction, the location of the point where the ear ceases to hear the triad as an out of tune minor chord and flips over to hearing it as an out of tune major chord, does not occur at the same point as the similar position during the first phrase of the experiment. This change in perception now occurs when the triad is closer to the final major chord and further from the minor triad.
2. A self-similar “fractal” chord:
I’ve made many experiments in discovering types of microtonal chords that have a distinct effect that is aesthetically interesting.
My aim was to create a microtonal analogy to a ‘self similar’ fractal design. The results I got were extremely beautiful, and unlike in case number one, above, could not under any circumstances be heard as an ‘out of tune’ version of a more tonal chord.
I started with an arbitrary selection of a lowest and a highest pitch. I then inserting a medium pitch that had the effect of dividing the overall range into two parts that bore a certain ratio (in my first experiment this ratio in pitch was 2 : 3). I continued to divide up each of the smaller pitch intervals by the same ratio. What started as just two pitches, became successively, hree pitches, five pitches, nine pitches, seventeen pitches, etc..*
*In computing the frequencies of the microtones I started with the unit of the “cent” (one hundredth of a half step) and then converted each cents value to a frequency.
3. Overtone series:
When an ear perceives a single tone or pitch from an orchestral instrument or the voice, an analysis of the sound vibration shows that there are actually a ‘chord’ of different pitches executing separate vibrations. These additional tones are known as overtones. If one could separate one overtone from the rest we would hear a sound at a different pitch from the one the ear first perceives.
One of the properties of the overtones is that they are the most spread out in pitch near the ‘fundamental frequency’ (the pitch that the ear perceives) and clump closer and closer together as they continue upwards in pitch.
There is a piece by Stockhausen called “Stimmung”* which has a group of singers each singling one of the upper overtones of a constant fundamental frequency. A ‘range’ of overtones is chosen by the composer. The fundamental is never sung, but a consecutive group of overtones is used. A variable in this selection is what should the lowest pitched overtone in the overtone series that should be sounded and which is the highest pitched one, and how many overtones does that ‘interval’ contains.
In its application to microtonal music, such a group of overtones, can be used as a ‘scale’ of available pitches out of which the notes of the piece are formed. However, one property of a scale is that it repeats over and over, usually at the octave. We can make a series of overtones do this by taking just one octave of the overtone series and transposing its pitches up and down various numbers of octaves so as to form a continuous scale from bass to treble.
For most orchestral instruments the overtones are linear in frequency. The first overtone is twice the fundamental frequency, the second overtone is three times the fundamental frequency, etc.. But the more three dimensional the instrument is the more it deviates from this simple linear pattern. A bell, for instance, whose vibrating mechanism does not approximate a one dimensional line, has a different arrangement of overtones.
And if a four-dimensional creature were to suspend a four-dimensional bell from a string, and then set it into vibration, there would be an less linear overtone series. It does not matter that we cannot construct such an instrument, for mathematics enables to predict what the overtones would be, and they can be reproduced exactly on an electronic synthesizer that is set up for microtones. So we can form scales out of the overtone series for n-dimensional objects (where n goes beyond three).
Here is a list of sample possible constants for generating an ‘altered’ linear overtone series on a particular note:
In the following n is a whole number, and ff the fundamental frequency:
(pi) x (n) x (ff)
(e) x (n) x (ff)
Here are some other possibilities of generating a linear overtone series not based on multiplying the fundamental frequency by whole numbers.
2 to the 1/2 power
pi times e
sine of an angle
* There was work by Maurice Béjart’s modern ballet company which was set to the music of Stimmung which was sung on stage at Carnegie by the “Swingle Singers”. Each singer intoned the pitch corresponding to one of the linear overtones of a single fundamental pitch. So that the tones were not too widely separated pitch-wise, they used a part of the overtone series where there were approximately as many overtones within the scope of one octave as there are notes forming one octave of a more familiar scale. For instance the following numbered overtones, in the fourth octave about the fundamental, span an octave and divide that octave into 8 parts: 7 8 9 10 11 12 13 14 15
4. Graphing a mathematical function:
There is an aesthetic fallacy in trying to find a means of translation between something spatial (as a graph) and something temporal (as music). However, if one is willing to experiment, one could try to derive the notes of microtonal chord from the y-values of some function f(x). Each next note in the chord would be f(x) for each whole number value of x. What would a parabola sound like? A hyperbola? A sine way? We don’t know until we ‘hear it’. We may stumble on a function whose sound as a chord is pleasing and unique aesthetically.
5. Expanding or compressing a tonal piece around a constant center of pitch:
This is more productive of interesting sounding tone groups. Bach Chorales lend themselves nicely to this procedure.
Take each chord, translate it into cents, and then either increase or decrease each pitch in the chord relative to some stable frequency that is either be one of the pitches of the original chord or a pitch that is chosen randomly but which remains throughout the chorale as the center of expansion and/or contraction. Or, another way would be to use the notes in one of the four voices as the “stable” pitch (even though it may change from beat to beat) and contract or expand the pitches of the other three voices relative to it.
For those of you who dabble in microtones would you let me know what methods you use or whether any of the methods described above have proven useful. Thanks, Joe