User:Aura/Aura's Diatonic Scales: Difference between revisions

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'''Supertonic''' - This is the note that occurs roughly at intervals between 567/512 and 256/255 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 [https://en.wikipedia.org/wiki/Supertonic], 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 7/6 and 165/128 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 [https://en.wikipedia.org/wiki/Mediant].

'''Mediant''' - This is the note that occurs roughly at intervals between 75/64 and 165/128 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 [https://en.wikipedia.org/wiki/Mediant].

'''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 45/32, 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 45/32 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 45/32, 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.

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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 64/45 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 64/45 and 17/11, 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 256/165 and 12/7 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.

'''Contramediant''' - This is my term for the note that occurs roughly at intervals between 256/165 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 [https://en.wikipedia.org/wiki/Subtonic].

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 '''Supertonic''' - This is the note that occurs roughly at intervals between 567/512 and 256/255 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 [https://en.wikipedia.org/wiki/Supertonic], 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 7/6 and 165/128 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 [https://en.wikipedia.org/wiki/Mediant].
+'''Mediant''' - This is the note that occurs roughly at intervals between 75/64 and 165/128 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 [https://en.wikipedia.org/wiki/Mediant].
 '''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 45/32, 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 45/32 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 45/32, 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.
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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 64/45 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 64/45 and 17/11, 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 256/165 and 12/7 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.
+'''Contramediant''' - This is my term for the note that occurs roughly at intervals between 256/165 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 [https://en.wikipedia.org/wiki/Subtonic].