# Ergotonic

## History

The concept of "ergotonic" scale theory, proposed by Bostjan Zupancic, is in development. The idea came about from broadening the concepts behind the diatonic scale to come up with musical constraints that were free enough to encompass some of microtonality, but still constrained enough to allow for deeper discussion.

## Etymology

From the Greek "ergon," meaning function and "tonic," meaning tones. Ergotonic theory deals with examining scales and developing new scale ideas based on the functions of the tones contained in the scale.

## Definitions

An ergotonic scale[idiosyncratic term] is defined by it's "reference," "basis" ("recursion" and "number of degrees"), and the "character" of each degree.

Reference: The root note, i.e. tonic, of the scale. Depending on how generally you define this, it could be as simple as "C" or as specific as "440.0 Hz." This is the starting pitch of the scale.

Recursion: The stopping point of the scale. An ergotonic scale must have both a reference and a recursion to act as "bookends" for the starting and stopping points of the scale. In general, the recursion tone could be anything, but most often, it will be the octave (second harmonic) of the reference. For example, a C scale will generally end on C an octave higher than the tonic. A Bohlen-Pierce scale, for another example, might have a recursion at the perfect twelfth (third harmonic) instead.

Degree: The notes of the scale. In many contexts, this might be the same concept as an "interval."

Number of degrees: How many tones are in the scale, from the reference up to the recursion (exclusive). For example, a pentatonic scale has five degrees.

Basis: The basis of the scale consists of two parts: 1. the "recursion" and 2. the "number of degrees."

Character: This describes the tonality of the degrees of the scale, depending on the allowed tonalities.

Tonality: The types of character available for a degree of an ergotonic scale. These are limited only by what we can define. Here are some examples (tonality is not limited to only these, but these are common):

• Perfect - Degrees consisting of intervals of sufficiently simple enough ratios that they do not necessitate major and minor varieties. For example, perfect unison 1:1, the perfect octave 2:1, the perfect fifth 3:2, and the perfect fourth 4:3.
• Major - Degrees, where two or more consonant options exist, consisting of sharper ratio of the options (higher in pitch). These are colloquially described as cheerful pitches. There is understanding that these pitches are more strongly associated with the harmonic series. As an example, the Major Diatonic Scale consists of only perfect and major degrees.
• Minor - Degrees, where two or more consonant options exist, consisting of the flatter of the options (lower in pitch). These are colloquially described as moody pitches. There is understanding that these pitches are more strongly associated with the subharmonic series. As an example, the classical mode called the "Phrygian Mode" consists of only perfect and minor intervals.
• Diminished - Degrees flatter than the most consonant option(s). The tones were classically regarded as depicting evil or danger.
• Augmented - Degrees sharper than the most consonant option(s). The tones were classically regarded as depicting surprise or amazement.
• Neutral - Degrees where there are two or more consonant options, consisting of a tone that lies between the two most consonant options. Neutral degrees are not typically found in western music theory, but are common in other traditions, in experimental music, and even in infantsong (vocalizations of human infants).

Western Standard Basis (WSB): The set of scales that recur at the octave and have seven degrees. These scales are limited to the perfect, major, minor, diminished, and augmented tonalities listed above, with or without further constraints. This includes the so-called "diatonic" scale. Many scales in western music fit into this basis.

Generalized Standard Basis (GSB): The set of scales that recur at the octave and have seven degrees, but with generalized tonality at each degree. Western Standard Basis is a subset of Generalized Standard Basis.

Step - The difference in relative frequency between two consecutive degrees in a scale. For example, the distance between the tonic and the major second in the major diatonic scale is called a "whole step." A whole step is a type of step. The same relative difference can be applied to the perfect fourth, for example, to get to the perfect fifth. Ergotonic scales can, and usually do, have constraints on steps as well.

Backstep - A backstep is a step where the higher degree is flatter than the lower degree. For example, if the augmented third is defined by the ratio of 675:512 (approximately 478 cents) to the tonic, and the diminished fourth is defined by the ratio of 32:25 (approximately 427 cents) to the tonic, then the fourth degree is flatter than the third degree, so a backstep is necessary to go from the third to the fourth.

## Example 1 - The Major Diatonic/Ionian Scale

The Major Diatonic Scale, also called Ionian Mode or Ionian Scale, is the most familiar musical scale in western music theory. It is recursive at the octave (2:1), has seven degrees, and fits nicely in WSB. It can be spelled out with only perfect and major intervals. Let's say, for this example, that we use a reference of "C" or "do," and talk briefly about how this scale is spelled:

Using C: C D E F G A B C

Using do: do re mi fa sol la ti do

Using intervals(P for "perfect", M for "major," and m for "minor"): P1 M2 M3 P4 P5 M6 M7 P1

If you play a song using this scale, it would be said that the song is in the "key of C major." The notes can be selected from the palette of options listed above in any order, at the composer's discretion. If a composer or performer decides to play any tones not contained in that template, for example B-flat (Bb), a number of music theory justifications can be made, but they are outside of the scope of this theory.

## Example 2 - The Harmonic Minor Scale

The Harmonic Minor Scale was a favourite of 18th and 19th century western music composers (for example, Mozart).

The intervals given in the scale are: P1 M2 m3 P4 P5 m6 M7 P1.

In comparison with the Major Diatonic Scale, the third degree and sixth degrees are both flatter.

Again, in the key of C, this would be spelled: C D Eb F G Ab B C.

Again, this is a WSB scale (seven degrees, repeats after the octave).

## Example 3 - The Melodic Minor Scale

The Melodic Minor Scale, often utilized by Baroque composers, like J.S. Bach, has a unique quirk about it. If you are unfamiliar with this scale, please read the link for better understanding.

The scale is defined in two ways: first, ascending, by the degrees P1 M2 m3 P4 P5 M6 M7 P1, and second, descending, by the degrees P1 m7 m6 P5 P4 m3 M2 P1.

In the key of C: up - C D Eb F G A B C and down - C Bb Ab G F Eb B C

There are seven degrees, although two of those seven come in two different types, depending on the context of ascending or descending. And the basis is the octave once again.

## Applying Constraints

After determining recursion (whether the scale encompasses one octave or something else), the number of degrees, and the constraints on tonalities, there will be a specific number of possible combinations available to make scales.

There may be additional constraints applied. For example: if you wish to use the Western Standard Basis, you may wish to exclude the possibility of combinations of degrees that result in a backstep in a scale. That is, you might disregard any scale that descends in frequency between two degrees as the scale is played ascending.

Of course, being a generalized theory, no additional constraints are truly necessary, although they will make it easier to construct a less abstract foundation.

## With respect to Bozuji Tuning

Bozuji Tuning is an approach which applies the constraints of seven degrees with an octave recursion, and the basis that only the following tonalities are allowed:

Table 1: Scale Degree/Interval Framework for Bozuji Approach
Degree Type Allowed tonailities Formal accidentals Informal accidentals
1 reference P
2 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #
3 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #
4 perfect d, P, A ♭, ♯ b, #
5 perfect d, P, A ♭, ♯ b, #
6 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #
7 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #

This is the Western Standard Basis, formalized, in the broadest sense.

## Applications

This approach can be applied to any musical system with discrete and constrained intervals placed into scales that follow a repeating pattern.

One example where the theory does not apply, is when the number of degrees changes depending on some variable. For example, a scale based on the harmonic series, which has a different number of degrees per octave, is not ergotonic.

The theory can work to build a framework around just intonation scales. or work backward from scales into a basis and then into a tuning from there.