User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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The idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees is one that has at least some roots in [[Wikipedia: Ivan Wyschnegradsky|Wyschnegradsky]]'s concept of [[Wikipedia: Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]]. However, I wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while I drew up sketches based loosely on [[24edo]] for early versions of this concept, I also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function. Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, I decided to takes stacks of 11/8 to form a second navigational axis which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this second axis. | The idea of extending Diatonic functional harmony to cover intervals between the standard scale degrees is one that has at least some roots in [[Wikipedia: Ivan Wyschnegradsky|Wyschnegradsky]]'s concept of [[Wikipedia: Major fourth and minor fifth|"Major Fourth" and "Minor Fifth"]]. However, I wanted to use LCJI as a basis for defining these intervals and thus decided to take [[11/8]] as being the just version of Wyschnegradsky's "Major Fourth", and while I drew up sketches based loosely on [[24edo]] for early versions of this concept, I also realized that that two instances of [[33/32]] added up to an interval smaller than [[2187/2048]] but which had a similar function. Furthermore, since two instances of 11/8 resulted in an interval in the vicinity of a Major seventh, I decided to takes stacks of 11/8 to form a second navigational axis which works together with the Diatonic Axis in order to define the microtonal functions positioned roughly halfway between the German and Viennese Diatonic functions, though there are a few other microtonal functions as well that are not immediately covered by this second axis. | ||
[[File:Diatonic_Function_Map.png|thumb|Initial diagram of paradiatonic function locations I made around the time of officially joining the Xenharmonic community. Note that a number of the functions listed on this page are missing, while the Contralead, the Semicontralead, the Intersubiant and | [[File:Diatonic_Function_Map.png|thumb|Initial diagram of paradiatonic function locations I made around the time of officially joining the Xenharmonic community. Note that a number of the functions listed on this page are missing, while the Contralead, the Semicontralead, the Intersubiant, the Interregnant, the Misoserviant and the Misodominant initially had different names.]] | ||
Most traditional music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. In [[Talk:159edo notation #My Second Idea for a Notation System|a conversation]] between myself and [[Kite Giedraitis]] about this topic, Kite mentioned that there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and the aforementioned 2187/2048- a chromatic semitone that is otherwise known as the Apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, Kite also mentioned how in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]]. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields [[27/25]]- two additional diatonic semitones. On the other hand, subtracting 81/80 from the Apotome yields [[135/128]], and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second. | Most traditional music theorists know that there are basically two types of semitones- the diatonic semitone or minor second, and the chromatic semitone or augmented prime. They also know that a diatonic semitone and a chromatic semitone add up to a whole tone. The same things are true in Just Intonation as well as in EDOs other than 12edo or even 24edo. In [[Talk:159edo notation #My Second Idea for a Notation System|a conversation]] between myself and [[Kite Giedraitis]] about this topic, Kite mentioned that there are two types of semitone in 3-limit tuning- a diatonic semitone of with a ratio of 256/243, and the aforementioned 2187/2048- a chromatic semitone that is otherwise known as the Apotome- which, when added together, add up to a 9/8 whole tone. Furthermore, Kite also mentioned how in 5-limit tuning, these same semitones exist alongside other semitones derived through alteration by [[81/80]]. On one hand, adding 81/80 to 256/243 yields 16/15, and adding another 81/80 yields [[27/25]]- two additional diatonic semitones. On the other hand, subtracting 81/80 from the Apotome yields [[135/128]], and subtracting another 81/80 yields 25/24- two additional chromatic semitones. When added up in the proper pairs- 16/15 with 135/128, and 27/25 with 25/24- the additional sets of semitones again yield a 9/8 whole tone. In light of all this, Kite argued that the familiar sharp signs and flat signs- which are used to denote the chromatic semitone- were never meant to denote exactly half of a whole tone, but rather, a whole tone minus a minor second. | ||