View source for User:Aura/Aura's Diatonic Scales

== Introduction ==

Just about every non-microtonal musician these days is acquainted with the 12edo [[5L 2s|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 dissonant, 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 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.

== Why 27/16 instead of 5/3? ==

While most microtonalists are keenly aware of the power of small rational intervals between notes, they don't seem to be as aware of some of the implications of such power- namely the implication that when such intervals exist between the wrong two notes in the scale, they can undermine tonal stability.  Think of it like this- frequency resonance between structures and the wind can wreak havoc, as was the case with [[Wikipedia:Tacoma Narrows Bridge (1940)|the original Tacoma Narrows Bridge]].   Even though tonality is largely based on musical context, this context is in part shaped by low-complexity frequency resonances.  Thus, it stands to reason that tonal systems are subject to that same phenomenon when certain resonances exist between the wrong two notes- from what I'm hearing from others, this "resonance in the wrong place" can lead to a sense of disorientation, and while this can indeed be exploited musically, there are times when such a sense of disorientation is not good, because it doesn't fit what one is trying to convey through the music.  

Take for example the 5/3 Major 6th- the traditional Contramediant.  The note at 5/3 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 as the resulting 3/2 Perfect 5th between the Contramediant and the Mediant can lend itself to the tonicization of the Contramediant where one might otherwise want a clear sense that "we're not done yet"- a weakness of Ptolemy's Intense Diatonic Scale, and a persistent weakness of the traditional 12edo major scale.  Furthermore, the note at 5/3 away from the Tonic 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.  Worse, 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 due to a strong difference-tone-based effect that is present even when the Serviant itself is not a direct component of the chord as demonstrated in the audio file below- an even more devastating weaknesses of Ptolemy's Intense Diatonic Scale.  Meanwhile, the first chord played in the same audio file uses 27/16 as the Contramediant, and thus has C3 as a virtual fundamental- this gives it a much more stable configuration relative to the Key of C major.

[[File:Interval_Test.mp3|thumb|none|First chord: CM6 as 1/1-5/4-3/2-27/16 <br> Second chord: CM6 as 1/1-5/4-3/2-5/3]]

== Diatonic Scales ==

All of the chief Diatonic scales listed here are named for their most useful mode.

=== 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 Contramediant, conspicouously, this arrangement also features two identical tetrachords on either side of the disjunct between the fourth and the fifth.  Therefore, it consists of notes related to the Tonic by the following ratios:

* [[1/1]]
* [[9/8]]
* [[5/4]]
* [[4/3]]
* [[3/2]]
* [[27/16]]
* [[15/8]]
* [[2/1]]

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:

[[File:Study in Ionian.mp3|thumb|none|Study in Ionian]]

=== 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:

* [[1/1]]
* [[9/8]]
* [[77/64]]
* [[4/3]]
* [[3/2]]
* [[27/16]]
* [[16/9]]
* [[2/1]]

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:

[[File:Study in Dorian.mp3|thumb|none|Study in Dorian]]

=== 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.  This scale consists of notes related to the Tonic by the following ratios:

* [[1/1]]
* [[16/15]]
* [[77/64]]
* [[4/3]]
* [[3/2]]
* [[8/5]]
* [[16/9]]
* [[2/1]]

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.  This scale consists of notes related to the Tonic by the following ratios:

* [[1/1]]
* [[9/8]]
* [[5/4]]
* [[45/32]]
* [[3/2]]
* [[27/16]]
* [[15/8]]
* [[2/1]]

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:

* [[1/1]]
* [[9/8]]
* [[5/4]]
* [[4/3]]
* [[3/2]]
* [[27/16]]
* [[16/9]]
* [[2/1]]

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

* Mixolydic Aeolian
* Mixolydic Locrian
* Mixolydic Ionian
* Mixolydic Dorian
* Mixolydic Phrygian
* Mixolydic 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:

* [[1/1]]
* [[9/8]]
* [[77/64]]
* [[4/3]]
* [[3/2]]
* [[8/5]]
* [[16/9]]
* [[2/1]]

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.  This scale is only distinct from the Lydian scale by means other than simple mode 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:

* [[1/1]]
* [[16/15]]
* [[77/64]]
* [[4/3]]
* [[64/45]]
* [[8/5]]
* [[16/9]]
* [[2/1]]

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