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

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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 ==~~

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:Tonic (music)]].

'''Contralead''' - 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 Contralead 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:Supertonic]], but do note that this article does not distinguish between a Supertonic and a Contralead.

'''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: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, and has a function opposite that of the Dominant relative to the Tonic. 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 [[Wikipedia: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 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- 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 [[Wikipedia: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 [[Wikipedia:Leading-tone]], but do note that this article does refers to what I call a "Contralead" by the term "Upper Leading-Tone".

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

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 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 ==
-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:Tonic (music)]].
-'''Contralead''' - 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 Contralead 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:Supertonic]], but do note that this article does not distinguish between a Supertonic and a Contralead.
-'''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: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, and has a function opposite that of the Dominant relative to the Tonic.  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 [[Wikipedia: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 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- 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 [[Wikipedia: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 [[Wikipedia:Leading-tone]], but do note that this article does refers to what I call a "Contralead" by the term "Upper Leading-Tone".
 == Why 27/16 instead of 5/3? ==