User:Aura/Aura's Ideas on Functional Harmony (Part 1): Difference between revisions
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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, the notes at 99/64 and 16/11 above the Tonic relate to each other in much the same way as major and minor intervals do, except that this relationship occurs in a context where the note halfway between them is actually part of the base scale rather than the two notes in question, and there's a different interval between said two notes than the base scale's chroma. | 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, the notes at 99/64 and 16/11 above the Tonic relate to each other in much the same way as major and minor intervals do, except that this relationship occurs in a context where the note halfway between them is actually part of the base scale rather than the two notes in question, and there's a different interval between said two notes than the base scale's chroma. | ||
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 | 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 MOS generator or its period-inverse) and "paraminor" (the result of subtracting a parachroma from a MOS generator or its period-inverse), which I had previously come up with on an informal basis. | ||
=== Basic Paradiatonic Functions === | === Basic Paradiatonic Functions === | ||