Alpharabian tuning

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The Alpharabian tuning is an 11-limit version of just intonation – specifically the version that is limited to the 2.3.11 subgroup – that is currently being pioneered in large part by Aura, with significant parts of this research having been made almost two years earlier by Spt3125.

Basis

Many, if not most, musicians who are not microtonalists are acquainted with standard music notation, with its clefs and staves, key signatures and time signatures. However, when you take all of this into the microtonal realm, it becomes readily apparent that – in all of the most intuitive systems – it is the 3-limit that defines both the standard location and structure of the various standard notes and key signatures that one finds in 12edo. Not only are the traditional key signatures all related to each other along a navigational axis formed by the 3-limit, but the standard sharp and flat accidentals modify the base note by an apotome, 2187/2048, and the double sharp and double flat accidentals modify the base note by two apotomes. Furthermore, it is the Pythagorean Diatonic Scales that arise as the standard variants for the various key signatures, as these are the simplest diatonic scales that can be formed with the 3-limit. Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems, and its pivotal role in laying the groundwork for key signatures means that its significance is widely accepted.

In addition to all this, most 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. In 3-limit tuning, the diatonic semitone has a ratio of 256/243, and the corresponding chromatic semitone is the apotome – two intervals adding up to a 9/8 whole tone. Furthermore, 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. Similarly, 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 this logic, we can then 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, it should be noted that for quartertones, there are sometimes multiple correct options, and thus, things are more complicated. We shall begin to define the musical functions of quartertones by drawing a distinction between the terms "Parachromatic" and "Paradiatonic" for purposes of classifying quartertone intervals. For starters, paradiatonic quartertones are analogous to diatonic semitones in that they are denoted as seconds, albeit subminor seconds, while parachromatic quartertones are denoted as superprimes of some sort. 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, we can deduce 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, we have to choose 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.

When one checks the 11-limit's representation of quartertones against those of the other rational intervals called "quarter tones" on Wikipedia's list of pitch intervals, one will find the 11-limit's 33/32 and 4096/3993 to be better pairing than any of the other options in terms of ratio simplicity. Furthermore, just as a stack of 3/2 perfect fifths forms a sequence in which every other octave-reduced pitch is a whole tone apart, a stack of 11/8 paramajor fourths forms a sequence in which every other octave-reduced pitch is the octave complement of a stack of 128/121 diatonic semitones. As the 11-limit handles stacks of 128/121 diatonic semitones in much the same way that the 3-limit handles stacks of 256/243, conserving interval arithmetic, it can thus be argued that the 11-limit meets the standards set by the 3-limit in this respect. In fact, since the 11-limit semitones are actually closer to half of a whole tone that either one of the 3-limit semitones- and especially since the 11-limit's version of a double sharp fifth only differs from the 3-limit's major sixth by the unnoticeable nexus comma – it can be argued that the 11-limit makes for good semitone representation as well. With this information in hand, we can now safely assume that the 11-limit does is fact, carry the function of a navigational axis. It is this foundation on which the idea of Alpharabian tuning rests.

Interval Naming Scheme

In the current interval naming scheme, there are several basic premises of Alpharabian tuning:

  • Intervals that are in the 2.11 subgroup are all considered Axirabian intervals as 2.11 forms a core navigational axis of Alpharabian tuning.
  • The intervals 3/2, 4/3, 9/8, 16/9, and so forth, have the same functions as in Pythagorean tuning.
  • The interval 33/32 is the standard Alpharabian quartertone due to not only being the simplest quartertone in the 2.3.11 subgroup, but also due to the fact that stacking three of these and subtracting the resulting interval from 9/8 yields the simplest possible interval that can result from such as process; furthermore, modification of a Pythagorean interval by this quartertone generally results in an Alpharabian interval- the only two known exceptions to this being 11/8 and 16/11, which differ from 4/3 and 3/2 respectively by this interval.
  • The rastma, 243/242, is functionally the simplest type of Alpharabian subchroma, and, since three instances of 243/242 are almost equal to 81/80 in JI, one can closely approach just 5-limit intervals simply by moving three rastmas away from Pythagorean intervals; furthermore, modification of a Pythagorean interval by this subchroma generally results in an Alpharabian interval- the only two known exceptions to this being 121/64 and 128/121, which differ from 243/128 and 256/243 respectively by this interval.

The following rules are directly derived from the above premises:

  • Generally, intervals that result from the modification of a Pythagorean interval by 33/32 take either the 'ultra' or 'infra' prefixes- for example, 891/512, which is the Alpharabian Ultramajor Sixth, and 512/297, which is the Alpharabian Inframinor Seventh- however, there are a number of special cases...
  • Augmentation of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paramajor interval
  • Dimunition of a Perfect Fourth or Perfect Fifth by 33/32 results in a Paraminor interval
  • Augmentation of a Pythagorean Minor interval by 33/32 results in an Artoneutral interval
  • Dimunition of a Pythagorean Major interval by 33/32 results in a Tendoneutral interval

The following rules are more questionable as they rely on the aforementioned questionable premise:

  • Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take either the 'super' or 'sub' prefixes, with these prefixes generally being stacked where multiple such modifications occur, however, there are some significant caveats...
  • Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a Supraminor interval, but a second such augmentation results in a Betarabian Major interval due to said interval differing from the nearby Alpharabian Major (covered under modifications by 1089/1024) by a rastma.
  • Dimunition of a Pythagorean Major interval by a single 1331/1296 results in a Submajor interval, but a second such dimunition results in a Betarabian Minor interval due to said interval differing from the nearby Alpharabian Minor (covered under modifications by 1089/1024) by a rastma.