User:Aura/Aura's Ideas on Tonality: Difference between revisions

However, that just covers the 11-limit's quartertones.  Since two parachromatic quartertones add up to a chromatic semitone, and since 33/32 is the primary parachromatic quartertone, we can safely assume that two 33/32 intervals adds up to a chromatic semitone of some sort, and indeed, [[1089/1024]] ''is'' a chromatic semitone under this definition, but as it differs from the apotome by the rastma, and we don't have the benefit of the "primary" versus "secondary" distinction here due to the apotome being a 3-limit interval, we should instead look to another source for a name.  Since 33/32 is also called the "Al-Farabi Quartertone" and is the primary limma of the of the 11-limit, and, since Al-Farabi himself was also referred to as "Alpharabius" according to [https://en.wikipedia.org/wiki/Al-Farabi Wikipedia's article on him], we can use the term "Alpharabian" to refer to no-fives no-sevens just 11-limit tuning in the same way that we can use the "Pythagorean" to refer to just 3-limit tuning.  Therefore, we can use the term "Alpharabian" to refer to the 11-limit semitones in the same way that we use "Pythagorean" to refer to the 3-limit semitones.  Thus, just as the apotome can also be referred to as the "Pythagorean Chromatic Semitone", we can refer to 1089/1024 as the "Alpharabian Chromatic Semitone". Furthermore, just as there's an Alpharabian Chromatic Semitone, there is an "Alpharabian Diatonic Semitone" which adds up together with the Alpharabian Chromatic Semitone to create a 9/8 whole tone, and, when you take 9/8 and subtract 1089/1024, you arrive at [[128/121]] as the ratio for the Alpharabian Diatonic Semitone.  Not only that, but just as there's the [[Pythagorean comma]] which forms the difference between a stack of two Pythagorean Diatonic Semitones and a 9/8 whole tone, so there is an "[[Alpharabian comma|Alpharabian Comma]]" which forms the difference between a stack of two Alpharabian Diatonic Semitones and a 9/8 whole tone, and doing the math yields 131769/131072 as the ratio for the Alpharabian Comma.

With all of the aforementioned stuff about, one can easily go on to ask what all this means in terms of the classification of more familiar 11-limit ratios like [[11/8]], seeing as the 11/8 can be derived from [[4/3]]- the Just Perfect Fourth- through the addition of the primary parachromatic quartertone.  Since the addition of the primary parachromatic quartertone to the Perfect Unison results in the primary parachromatic quartertone, one would assume that this means that 11/8 would be classified as the "parachromatic superfourth".  In actuality, however, while one would be correct in asserting 11/8 is a parachromatic alteration of the perfect fourth, interpreting 11/8 as a derivative of 33/32 would in many respects be akin to interpreting [[3/2]]- the Just Perfect Fifth- as a derivation of the apotome, when in fact, it is the other way around.  Recall that the prime factorization of 33 is 3*11, so that means that, 33/32 is ''not'' a pure 11-limit interval.  Therefore, rather than assume the primary parachromatic quartertone to be the basic 11-limit interval, we instead must recognize that that title properly belongs to 11/8.  Furthermore we should take stock of the fact that two 11/8 intervals stacked on top of one another yields [[121/64]], the octave complement of the Alpharabian diatonic semitone.  Since 121/64 is arguably a form of major seventh as a diatonic semitone always has a major seventh as its octave complement, and since a stack of two fourths equals a seventh, what does that mean for 11/8?  Well, it means we need more terms, and we need to define those terms.