Draft:Normalized Tonality

In music theory, normalized tonality is a contemporary approach of very recent development that essentially consists of organizing musical scales as derivations of the major scale (a.k.a. diatonic) in such a way that every mode of every derived scale is matched to the derivator, the major, based on the mathematical principle of lydloc delta. This approach to the study of scales enormously simplifies the process of learning them theoretically as well as memorizing their fingerings on instruments; it also facilitates operations such as scale substitution and harmonization.

INTRODUCTION
Throughout music history scales have been studied and used in a totally dissociated manner comparable to speaking and writing without understanding that, for example, to go, goes, going, and went are simply the same verb, an infinitive form and its different conjugations, related by a common property, the same basic semantic meaning. Under normalized tonality, the heptatonic major scale is the one infinitive and all others are its “conjugations”, related by a common property, the lydloc delta. These “conjugations” are technically called derivees and are a pivotal concept that serves as the ground for derivation paths, derivation hierarchies, scale yield groups, tritone-glide levels, and other theoretical elaborations and their practical applications. The lyloc delta is comparable to a compass that always indicates the direction back to the derivator major scale, and in turn is based on the lydloc interval, a mathematical property of heptatonic scales[1]. Seven-tone pitch-class sets are the center of traditional Western music, the Carnatic melakartas, the Hindustani thaats, the Arabic maqam system, the Turkish makamlar, Chinese scale system, and virtually all music theories[2]; and they are also the center of normalized tonality, but the derivation system extends to pitch-class sets of other cardinalities, such as pentatonic, hexatonic, octatonic, etc.

GENERAL OVERVIEW AND BASIC PRINCIPLES
To understand normalized tonality, first, the lydloc interval and the lydloc delta have to be introduced as they are the backbone of this music theory system.

THE LYDLOC INTERVAL
It is a tritone formed by the two notes that absorb the error produced when trying to close the octave after a cycle of seven perfect fifths in order to produce a heptatonic scale, since it takes actually a cycle of 12 perfect fifths to reach the starting note and close the octave (with a small error called comma[3]). In fact, the last of the seven fifths has to be shortened by a semitone, causing the resulting heptatonic pitch-class set, the major scale, to be left with six perfect fifths plus one diminished fifth. As a clarifying example, in the key of C major (C-D-E-F-G-A-B) the two notes forming the lydloc interval are F and B. All others will have a perfect fifth and will be in turn, prefect fifths to another note, but F will be nobody’s perfect fifth and B receives a diminished fifth, not a perfect one. In the key of C major of the above example the cycle of seven perfect fifths strats on F, the fourth diatonic degree of C, and it goes F-C-G-D-A-E-B; after B, the next perfect fifth is F#, but the cycle can be closed short by lowering the F# by a semitone, making it a closing F, the starting note, at the expense of a diminished fifth.

The complete cycle of 12 perfect fifths, F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#, closes without error (except for the comma mentioned earlier), with the 13th note, E#, being enharmonically F, the starting one, but producing the dodecatonic chromatic scale, not a heptatonic[4].

This pair of scale degrees that is “perfect-fifth-defective”, is what in normalized tonality is called the Lydloc interval, the notes that form it, F and B in the C Major example, are called the lyd note and the loc note respectively. Their denominations come from the first three letters of the names of the modes of which they are root, the Lydian (IV) and Locrian (VII) modes respectively.

THE LYDLOC DELTA
When a third note is added to the lydloc interval in such a way that the lyd note becomes a perfect forth to it and the loc note becomes its major seventh, we obtain the lydloc delta[1]; delta meaning triangle as it is the shape of the Greek letter of the same name. In the C major example, the note C completes the delta C-F-B or I-IV-VII.

This lydloc delta is the mathematical principle behind normalized tonality. It is not possible to put seven pitches together in the 12-tone-equal-temperament scheme of Western music without forming one delta[1][5]; in other words, a heptatonic scale/pitch-class set does not exist that doesn’t contain it. It would be mathematically impossible. Normalized tonality music theory takes advantage of this relevant fact to put the major scale, the center of Western music’s tonal system, at the center of the normalization system and relate all other mathematically possible heptatonic scales to it in such way that the omnipresent lydloc delta serves as the invariable reference across scales.

NOMENCLATURE: The words “lyd” and “loc” are the names of scale degrees, while “lydloc”, one word, is an adjective. Besides the lydloc interval and lydloc delta that have been already explained above, there are lydloc chord substitutions, lydloc negative harmony, lydloc pentatonic flips, and other lydloc-labeled concepts which are beyond the scope of this article.

THE NORMALIZATION PROCESS
What follow are practical examples using a clock-face chromatic circle to represent the heptatonic scales to graphically explain the normalization process. Graphic 1 illustrates the major scale with the root of the ionian mode at the top (the 12 o’clock on the clock face). The lydloc delta, formed by the scale degrees I, IV, and VII is shown as a triangle connecting those three degrees.

When the III degree of the major/diatonic scale is lowered by a semitone (bIII) [graphic 2], the jazz melodic minor scale[6] is obtained. If instead, the VI degree is lowered by a semitone (bVI), the harmonic major scale is generated[7] [graphic 3]. If the second degree of the newly obtained harmonic major is lowered by a semitone (bII), the resulting scale is the double harmonic major, which possesses a flat second (bII) and a flat sixth (bVI)[8] [graphic 4]. It can be seen that the lydloc delta remains invariable and all the scales obtained by moving pitches around have a perfect match mode to mode with the original scale, the major, the mode I of the major scale becomes the mode I of the melodic minor, the harmonic major, and the double harmonic major with the remaining modes matching as well. The equivalent of the ionian mode in each new scale, which carries over the roman numeral I, is called the ionian match. The process is called normalized derivation. As stated in the introduction, if the major scale is symbolically regarded as an infinitive verb, the derivee scales are its conjugations.

The examples that follow are of capital importance to deeply understand the meaning of the word “normalized” in the name normalized tonality.

It is commonly said that by raising the VII degree (#VII) of the natural minor scale -the aeolian mode- the harmonic minor is obtained[9]; using as an example the key of A minor (A-B-C-D-E-F-G) the G would become G#. This is perfectly correct, but not the way normalized tonality works, otherwise there would be no normalization. In normalized tonality it is always the ionian mode, the major scale itself, what is used as a reference. In the clock face of graphic 5 the note to be sharpened to obtain the harmonic minor scale is regarded as the degree V of the major scale, not the degree VII of its relative minor, the natural minor. In our example keys, C major and A minor, the note is still the same, a G sharpened to G#, but in this way the one to one relation between modes is preserved. In the same way that the natural minor or aeolian mode is considered the mode VI of the major scale, the harmonic minor is considered a VI mode of a major scale with a sharp fifth degree (#V). This major scale mode with a sharp fifth, which could be called ionian #5, is the ionian match or mode I of the harmonic minor block. The word block in normalized tonality identifies the scale as a group or system as a whole that includes all modes, without referencing a specific mode and, as seen above, the name of the example scale group as a whole has not changed, it is still the harmonic minor block, only that the specific mode of the same name is now the normalized mode VI.

This solves innumerous problems that traditional theory had not been able to remedy when relating Western scales to one another or to their Indian, Persian, Arabic and Turkish counterparts. For instance, the double harmonic major scale of our early example contains a mode called double harmonic minor or hungarian minor, often regarded as mode I, to which the double harmonic major would become mode V. Additionally, they are also called gypsy minor and major respectively, and are the simhendramadhyamam (#57) and mayamalavagowla (#15) Carnatic melakartas as well as todi and bhairav Hindustani thaats. Furthermore, the same scale block contains another mode known as rasikapriya (#72) melakarta. And that is for only one scale block of seven modes. Without normalization it would be nearly impossible to relate scales and modes to one another. There are 66 possible heptatonic scale blocks which, with seven modes each, total 462 heptatonic modes. And they have pentatonic complements plus hexatonic/octatonic derivations. By keeping track of the invariable lydloc delta and its associated ionian mode, all of them can be normalized and related to each other; the neapolitan minor becomes, for instance, mode III of the neapolitan minor scale block, while the neapolitan major remains the mode I of its own block. The enigmatic gets normalized to mode II, the persian scale to mode VII and so on.

The normalization system opens the doors to an immense body of theoretical principles and their practical applications for composers that were never possible before under the schemes of traditional music theory. These principles, in addition, find direct practical application for instrumentalists as they can be applied to the visualization of scale fingerings on all instruments, such as those of the guitar, violin, and keyboard families greatly facilitating memorization and execution.