User:Aura/Aura's Diatonic Scales: Difference between revisions
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== Introduction == | == 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 | 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 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 == | == Definitions of Scale Degree Names == | ||
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'''Dominant''' - As per the name, and as noted in the relevant Wikipedia article [https://en.wikipedia.org/wiki/Dominant_(music)], 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. | '''Dominant''' - As per the name, and as noted in the relevant Wikipedia article [https://en.wikipedia.org/wiki/Dominant_(music)], 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 | '''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- the latter interval being used in place of 5/3 for reasong which are detailed below. 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 [https://en.wikipedia.org/wiki/Subtonic]. | '''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 [https://en.wikipedia.org/wiki/Subtonic]. | ||
'''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 [https://en.wikipedia.org/wiki/Leading-tone], but do note that this article does refers to what I call a "Reverse Lead" by the term "Upper Leading-Tone". | '''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 [https://en.wikipedia.org/wiki/Leading-tone], but do note that this article does refers to what I call a "Reverse Lead" by the term "Upper Leading-Tone". | ||
== Why 27/16 instead of 5/3? == | |||
While the 5/3 Major 6th is 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 Perfect 5th between the Contramediant and the Mediant can lend itself to the tonicization of the Contramediant itself 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 virtual fundamental 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 == | == Diatonic Scales == | ||