User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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[[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 Interregnant initially had different names.]] | [[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 Interregnant 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: | 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. | ||
Building on Kite's logic, I then decided to apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, the catch was that that for quartertones, there are sometimes multiple correct options, making things more complicated. I decided to define the musical functions of quartertones initially on an informal basis by drawing a distinction between the terms "'''Parachromatic'''" (from the prefix ''para-'' in both the senses of ''alongside'' and ''resembling''<ref>[[Wiktionary: para- #Etymology 1]]</ref>, and the word ''chromatic'') and "'''Paradiatonic'''" (from the same two senses of the prefix ''para-'' and the word ''diatonic'') for purposes of classifying quartertone intervals. | Building on Kite's logic, I then decided to apply similar distinctions among quartertones, and thus make the argument that quartertones don't have to denote exactly one fourth of a whole tone in as of themselves, but rather, they only have to add up to a whole tone when paired up correctly. However, the catch was that that for quartertones, there are sometimes multiple correct options, making things more complicated. I decided to define the musical functions of quartertones initially on an informal basis by drawing a distinction between the terms "'''Parachromatic'''" (from the prefix ''para-'' in both the senses of ''alongside'' and ''resembling''<ref>[[Wiktionary: para- #Etymology 1]]</ref>, and the word ''chromatic'') and "'''Paradiatonic'''" (from the same two senses of the prefix ''para-'' and the word ''diatonic'') for purposes of classifying quartertone intervals. This was easy, since after I found that two instances of 33/32 added up to [[1089/1024]] rather than 2187/2048, I had already informally added the "para-" prefix in the same senses to both "major" and "minor" to create the terms '''paramajor''' and '''paraminor''' to better describe how 11/8 and 16/11 related to 128/99 and 99/64 respectively- namely to describe how, for instance, 99/64 and 16/11 relate to each other in much the same way as major and minor intervals do, except in a context where the interval between them is actually part of the base scale rather than the two intervals in question. | ||
The way I see it, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, Aura ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. As a result of multiple factors, I ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to both Diatonic theory and [[Alpharabian tuning]], and, through interactions with others in the Xenharmonic community, I was later influenced by others on Discord to take [[MOS]]-based structural considerations into account. This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and later, the "'''parastep'''" (the interval that remains after subtracting as many parachromas from a Major MOS-step as possible without resulting in a negative interval). | The way I see it, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit inframinor seconds by default, while parachromatic quartertones are analogous to chromatic semitones in that they are denoted as primes, albiet as ultraprimes by default. However, the distinction goes further than that- a parachromatic quartertone and a paradiatonic quartertone add up to a diatonic semitone, while two parachromatic quartertones add up to a chromatic semitone. Given both these definitions for "paradiatonic" and "parachromatic", and given that a diatonic semitone and a chromatic semitone add up to a whole tone when paired correctly, it can be deduced that a whole tone can be assembled from three parachromatic quartertones and one paradiatonic quartertone. Because there are sometimes multiple correct options for assembling parachromatic and paradiatonic intervals to make a 9/8 whole tone, Aura ended up choosing the simplest configuration of paradiatonic and parachromatic intervals to assemble in order to create a 9/8 whole tone- a configuration that only requires one type of parachromatic quartertone and one type of paradiatonic quartertone. As a result of multiple factors, I ended up choosing the combination of three 33/32 parachromatic quartertones and one [[4096/3993]] paradiatonic quartertone as the JI basis for this in regards to both Diatonic theory and [[Alpharabian tuning]], and, through interactions with others in the Xenharmonic community, I was later influenced by others on Discord to take [[MOS]]-based structural considerations into account. This eventually resulted in the first formal definition of a "'''parachroma'''" (an interval that can be easily tempered to equal half of a MOS-chroma), and later, the "'''parastep'''" (the interval that remains after subtracting as many parachromas from a Major MOS-step as possible without resulting in a negative interval). Finally, drawing from the concept of "parachromas" as applied to MOS-based contexts, I was able to finally give a formal definition of terms like "paramajor" (the result of adding a parachroma to either a perfect MOS generator or it's period-inverse) and "paraminor" (the result of subtracting a parachroma from a perfect MOS generator or it's period-inverse), which I had previously come up with on an informal basis after noticing that two instances of 33/32 didn't add up to 2187/2048, but rather added up to [[1089/1024]]. | ||
=== Basic Paradiatonic Functions === | === Basic Paradiatonic Functions === | ||