User:Aura/Aura's Diatonic Scales

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Introduction

Just about every non-microtonal musician these days is acquainted with the 12edo diatonic scale. Some may also know the diatonic modes- Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian. However in higher EDOs, as well as in Just Intonation, things are a little different. To start with, there are two well-known tunings for diatonic scales: the Pythagorean Diatonic Scale, and Ptolemy's Intense Diatonic Scale, but the Pythagorean Diatonic Scale is ill-suited for modulation, while the selections of notes in Ptolemy's Intense Diatonic Scale is ill-suited for building chords and chord progressions using just the notes in those scales. In both cases, it has been one sort of wolf-fifth or other that has caused problems due to the wolf fifth having a sound other than that of the 3/2 Perfect 5th. For the longest time, these wolf fifths were considered unusable, which is part of what led to the dominance of 12edo in the first place. However, I've found that the 40/27 grave fifth and intervals that approximate it are in fact usable in chord progressions- assuming that they are positioned correctly within a given just diatonic scale, and I have adopted my own version of a Just Diatonic Scale accordingly. Furthermore, I've found that in retuning the traditional diatonic modes in accordance with their differences from standard Major and Minor, the traditional diatonic modes cease to qualify as modes of the same diatonic scale, and instead give rise to several different diatonic scales of their own, complete with their own modes. As a final note, I prefer all the notes of my scales to be related to the tonic by ratios which have a power of 2- for reasons which I'll explain below- in either the numerator or the denominator, and this leads not only to 5/3 getting replaced by 27/16 as the ideal ratio for the Major Sixth scale degree above the tonic- a change also made in light of my findings on the ideal location for the grave fifth and intervals that approximate it- but also to the substitution of 77/64 for 6/5 for the Minor Third scale degree in cases where I'm not restricted to using just the 3-limit and the 5-limit for defining notes in the diatonic scale. I shall use this page for detailing my findings, as well as to document the modes of the now-separated diatonic scales.

Definitions of Scale Degree Names

In order to foster the understanding of some of the terms used for the scale degrees used on this page- to say nothing of their associated diatonic functions- it is necessary for me to define them, though these definitions will likely be amended in the future.

Tonic - This is the note that serves as the tonal center, and thus, the main resolution tone, and is the note for which scales are named (e.g. the key of C major is so-named because in this scale, C serves as the Tonic). For more information about the general functionalities and properties of the Tonic, see Wikipedia's article [1].

Reverse Lead - This is my term for a note that occurs at intervals between 160/153 and 14/13 away from the Tonic, and, which serves as a leading tone in the direction opposite that of the scale's direction of construction- which, in most modern music, is from the Bass upwards. This means that in most music, the Reverse Lead occurs as a lowered second scale degree.

Supertonic - This is the note that occurs roughly at intervals between 567/512 and 256/225 above the Tonic as the second scale degree, with 9/8 proving to be the most harmonically stable ratio between the Supertonic and the Tonic, and 10/9 often occurring as a common but less stable alternative, thus, in diatonic scales, the Supertonic generally occurs within 5 cents of either 10/9 or 9/8. For more information about the general functionalities and properties of the Supertonic, see Wikipedia's article [2], but do note that this article does not distinguish between a Supertonic and a Reverse Lead.

Mediant - This is the note that occurs roughly at intervals between 75/64 and 32/25 away from the Tonic in the scale's direction of construction. This is the first of the two scale degrees with the most possibilities for realization, though in true diatonic scales, it is generally within 20 cents of either 6/5 or 5/4. As only notes at intervals with powers of 2 in either the numerator or the denominator are harmonically or subharmonically connected with the Tonic and 6/5 fails to meet this critera, I often replace the traditional 6/5 Minor 3rd with the 77/64 Minor Third. For more information about the general functionalities and properties of the Mediant, see Wikipedia's article [3].

Serviant - This is my term for the note that occurs roughly at an interval of 4/3 away from the Tonic in the scale's direction of construction. Although this is commonly called the "Subdominant" in traditional music theory, the problems with that term are two-fold. Firstly, not all possible "Subdominant" harmonies have the same harmonic properties relative to the Tonic, as there is an extremely close connection between the Tonic and the the 4/3 Perfect 4th, and this is not the case for other intervals between 22/17 and 7/5, which might otherwise be called "Subdominants". Secondly, in music built from the Treble downwards, the notes with these sorts of functions are actually located above the Dominant. Like with notes at other intervals between 32/25 and 7/5 away from the Tonic- the Serviant tends to resolve towards the Dominant, or, as is the case in Locrian, a certain type of Antitonic. As for the notes at other intervals between 22/17 and 7/5, I would divide them into two classes depending on what side of the 4/3 Perfect 4th they fall on, however, aside from 27/20, none of these other intervals occur in diatonic scales.

Antitonic - This is my general term for notes that occur around half an octave away from the Tonic- specifically the region extending from 7/5 to 10/7- on account of harmonies built on notes in this area tending to oppose that of the Tonic. The exact outcome of this opposition depends on the exact distance of the Antitonic from the Tonic. If the Antitonic is less than half an octave away from the Tonic, it tends to cause the Dominant to become a new Tonic unless followed up by a different note- one that is usually a Major 7th away from the Tonic. Because of this tendency to "kiss up to" and tonicize the Dominant, I call any type Antitonic less than half an octave away from the Tonic a "Sycophant". Conversely, if the Antitonic is more than half an octave away from the Tonic, it tends to contrast with the Tonic in a manner somewhat akin to that of a Dominant, but by sheer brute force and contrary harmonic nature- e.g. if the Tonic harmony is Minor in nature, the Antitonic harmony will be Major- or more rarely, Supermajor- in nature. Furthermore, in scales such as the Locrian scale, any type of Serviant harmony tends to resolve towards either this type of Antitonic, or some other type of substitute for a Dominant. Because of these Dominant-esque tendencies, I call any type of Antitonic more than half an octave away from the tonic a "Tyrant".

Dominant - As per the name, and as noted in the relevant Wikipedia article [4], this note is the second most important after the Tonic, though in contrast to what is stated about the Dominant in the article, I would add several caveats. Firstly, I would prefer to restrict the term "Dominant" to where it only refers to the note that occurs roughly at an interval of 3/2 away from the Tonic in the the scale's direction of construction, not only because other intervals between 10/7 and 17/11 away from the Tonic in the scale's direction of construction have the tendency to create tension which requires the Tonic to resolve, but also because the 3/2 Fifth is by far the best choice for this sort of functionality on account of the the extremely close harmonic connection between the Tonic and the 3/2 Perfect 5th. Secondly, I would also add the caveat that the level of importance typically associated with the Dominant goes instead to the Tyrant Antitonic in those cases where one occurs on the 5th scale degree instead of a 3/2 Perfect 5th. As for the notes at other intervals between 10/7 and 25/16, I would divide them into two classes depending on which side of the 3/2 Perfect Fifth they fall on, however, aside from 40/27, none of these other intervals occur in diatonic scales.

Contramediant - This is my term for the note that occurs roughly at intervals between 25/16 and 128/75 away from the Tonic in the scale's direction of construction. The Contramediant is the second of two scale degrees with the most possibilities for the realization, however, in true diatonic scales of the variety I'm defining here- it is generally within 20 cents of either 8/5 or 27/16. While the 5/3 Major 6th is the traditional Contramediant, it- like the 6/5 Minor 3rd- is harmonically disconnected from the Tonic due to not occurring as an interval distance from the Tonic in the Tonic's own harmonic series, or even the Tonic's own subharmonic series. At the same time, since a note forming a 5/3 ratio with the Tonic occurs very early in the Serviant's harmonic series, this sort of configuration can very easily result in the tonicization of the Serviant- this is one of the key weaknesses of Ptolemy's Intense Diatonic Scale. However, replacing this interval with the nearby 128/77 for the Major 6th is not a good option outside of accidentals, as the note at 128/77 away from the Tonic forms a wolf fifth with a Supertonic at 9/8 away from the Tonic. This is not a good position for a wolf fifth, for if the Keenanisma is tempered out, the the resulting Perfect 5th between the Contramediant and the Mediant can still lend itself to the tonicization of the Contramediant itself where one might otherwise want a clear sense that "we're not done yet"- a further weakness of Ptolemy's Intense Diatonic Scale, and a persistent weakness of the traditional 12edo major scale. In contrast, if the Contramediant is set at the Tonic's 27th harmonic, a grave fifth is then positioned between the Contramediant and the Mediant, and the slightly-off sound of the resulting minor triad provides a more clear indication that one should expect a follow up- this is a deceptive cadence at its finest. Furthermore, setting the Contramediant at the Tonic's 27th harmonic alters the character of the Serviant chord to be less consonant, and thus allows the Serviant to unambiguously perform its harmonic functions relative to both the Tonic and the Dominant.

Subtonic - This is the note that occurs roughly at intervals between 225/128 and 1024/567 above the Tonic as the seventh scale degree, with 16/9 proving to be the most harmonically stable ratio between the Supertonic and the Tonic, and 9/5 often occurring as a common but less stable alternative, thus, in diatonic scales, the Subtonic generally occurs within 5 cents of either 16/9 or 9/5. For more information about the general functionalities and properties of the Subtonic, see Wikipedia's article [5].

Lead - This is a note that occurs at intervals between 153/80 and 13/7 away from the Tonic, which serves as a leading tone in the scale's direction of construction. For more information about the general functionalities and properties of the Lead, see [6], but do note that this article does refers to what I call a "Reverse Lead" by the term "Upper Leading-Tone".

Diatonic Scales

All of the chief diatonic scales listed here are named for their most useful mode. Yes, I've actually used all of these particular scales as the the basis for tonality, and yes, all of the most useful modes of all of these diatonic scales are actually capable of circle progressions using just the notes in the scale, although this often requires using tricks such as having a Suspended-4 chord on the Dominant. Phrygian, Lydian and Locrian require additional tricks, which I'll mention under their respective scales.

Ionian

My preferred version of this scale differs from Ptolemy's Intense Diatonic Scale only by having 27/16 as the ratio between the Tonic and the Submediant. Therefore, it consists of notes related to the Tonic by the following ratios:

As for the other modes of this scale, I give them the following names:

  • Ionic Dorian
  • Ionic Phrygian
  • Ionic Lydian
  • Ionic Mixolydian
  • Ionic Aeolian
  • Ionic Locrian

Sample:

Dorian

Unlike the version that is present in 12edo- and perhaps even many other Just versions of this scale, my preferred version of this scale is not symmetrical, as having it be symmetrical would result in a less-than-ideal tuning for the interval between the Mediant and the Contramediant- instead, it is the first of two diatonic scales in this set that uses 256/243 as the interval between two of its notes. This scale consists of notes related to the Tonic by the following ratios:

As for the other modes of this scale, I give them the following names:

  • Doric Phrygian
  • Doric Lydian
  • Doric Mixolydian
  • Doric Aeolian
  • Doric Locrian
  • Doric Ionian

Sample:

Phrygian

Unlike the version that is present in 12edo- and likely even many other Just versions of this scale, my preferred version actually differs from the inverse of the Ionian scale. A circle progression in this scale requires not only a Suspended-4 chord on the Contramediant, but also both a Suspended-Sharp-4 chord on the Reverse Lead, and, a Half-Diminished chord with added Minor 9th on the Dominant. This scale consists of notes related to the Tonic by the following ratios:

As for the other modes of this scale, I give them the following names:

  • Phrygic Lydian
  • Phrygic Mixolydian
  • Phrygic Aeolian
  • Phrygic Locrian
  • Phrygic Ionian
  • Phrygic Dorian

Lydian

This scale is the one most closely associated with Ptolemy's Intense Diatonic Scale; in fact, Ptolemy's Intense Diatonic scale occurs as "Lydic Ionian"- the fifth mode of this scale. If you use the Sycophant Antitonic harmony, you are forced to follow it up with the Lead harmony if you want to maintain this tonality. Likewise, using the full Supertonic harmony is generally ill-advised unless you swiftly follow it up with the Tonic- otherwise, you need to omit the third of this chord if you want to maintain this tonality. This scale consists of notes related to the Tonic by the following ratios:

The other modes of this scale are as follows:

  • Lydic Mixolydian
  • Lydic Aeolian
  • Lydic Locrian (only ever distinct from Locrian proper if the Keenanisma is not tempered out)
  • Lydic Ionian (otherwise known as Ptolemy's Intense Diatonic Scale)
  • Lydic Dorian
  • Lydic Phrygian

Mixolydian

This scale is the second of two diatonic scales in this set that uses 256/243 as the interval between two of its notes. It consists of notes related to the Tonic by the following ratios:

As for the other modes of this scale, I give them the following names:

  • Myxic Aeolian
  • Myxic Locrian
  • Myxic Ionian
  • Myxic Dorian
  • Myxic Phrygian
  • Myxic Lydian

Aeolian

Aside from the substitution of the 77/64 Minor 3rd for the 6/5 Minor 3rd found in other Just versions of this scale, my preferred version of the Aeolian scale is pretty typical. It consists of notes related to the Tonic by the following ratios:

As for the other modes of this scale, I give them the following names:

  • Aeolic Locrian
  • Aeolic Ionian
  • Aeolic Dorian
  • Aeolic Phrygian
  • Aeolic Lydian
  • Aeolic Mixolydian

Locrian

One might think that Locrian is useless and not even worth the trouble, however, the actual problem is that people generally don't know how to handle a flattened fifth, and they assume that the harmony of Locrian must be strictly tertian when this is not the case at all. In truth, Locrian harmonies generally rely on the omission of either the fifth or the third- with the only consistent exceptions being the Tyrant Antitonic chord on the flattened fifth, and variants of the Tonic harmony that are used as a means of creating tension. If one is looking for resolution in this mode, they only have to create a chord consisting of the Tonic, the Mediant, and octave reduplications of the Tonic both above and below. Do note that because Locrian's strongest distal harmony is the Tyrant Antitonic harmony- which is unable to act as a proper anchor for melodies without chord support- the Tonic, Mediant, and Subtonic pick up the slack and are thus more common than they would be otherwise, with the end result being that Locrian chord progressions run a significant risk of stagnation, especially in the hands of unskilled composers. A circle progression in Locrian requires not only the omission of the third on the Reverse Lead chord, but also the displacement of the Tyrant Antitonic chord's third upwards by an octave. This scale is only distinct from the Lydian scale if the Keenanisma is not tempered out, as otherwise, the step patterns between the two scales are identical- with the Locrian scale consisting of notes related to the Tonic by the following ratios:

As for the other modes of this scale, I give them the following names:

  • Locric Ionian
  • Locric Dorian
  • Locric Phrygian
  • Locric Lydian (only ever distinct from Lydian proper if the Keenanisma is not tempered out)
  • Locric Mixolydian
  • Locric Aeolian