Alpharabian tuning
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 a 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 4096/3993 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, as a general rule, modification of a Unison or Octave by this subchroma results in a rastmic interval, otherwise, modification by this subchroma 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 Parachromatic Semilimma, 1331/1296, is slightly over half of 256/243, the Pythagorean Limma, with the remainder being 4096/3993, and since 1331/1296 differs from 33/32 by the rastma, modification of a Pythagorean interval by this quartertone often results in an Alpharabian interval—the principle exceptions to this being 1331/1024 and 2048/1331, which differ from 81/64 and 128/81 respectively by this interval, though there are others.
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
- Diminution 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
- Diminution of a Pythagorean Major interval by 33/32 results in a Tendoneutral interval
- Modification by 33/32 generally results in a Class I Alpharabian interval
- Generally, intervals that result from the modification of a Pythagorean interval by a single instance of 243/242 retain the same functionality as their Pythagorean counterparts, much like with the syntonic comma, however, there are a few special cases...
- Augmentation of a Perfect Fourth or Perfect Fifth by a single instance of 243/242 results in an Alpharabian wide interval
- Diminution of a Perfect Fourth or Perfect Fifth by a single instance of 243/242 results in an Alpharabian narrow interval
- Augmentation of a Perfect Unison or Perfect Octave by a single instance of 243/242 results in a Rastmic wide interval
- Diminution of a Perfect Octave by a single instance of 243/242 results in a Rastmic narrow interval
- Rastmic intervals are considered a type of Class II Alpharabian interval, as modifying other intervals by single instances of the rastma results in some sort of Class II Alphrabian interval.
The following rules have not yet been finalized in their entirety due to lack of details:
- Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take both the 'ultra' or 'infra' prefixes along with the "semilimmic" modifier before the word with the prefix, since 1331/1296 is the chromatic half of a Pythagorean Limma, however, there are some significant caveats...
- Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a semilimmic Supraminor interval, but a second such augmentation results in a Class IV Alpharabian Major interval due to said interval differing from the nearby Class II Alpharabian Major (covered under modifications by 1089/1024) by a rastma.
- Diminution of a Pythagorean Major interval by a single 1331/1296 results in a semilimmic Submajor interval, but a second such diminution results in a Class IV Alpharabian Minor interval due to said interval differing from the nearby Class II Alpharabian Minor (covered under modifications by 1089/1024) by a rastma.
Important intervals
This section contains a few charts of the most important intervals in Alpharabian tuning. Note that modifications of augmented and diminished intervals are not included in these charts for sake of relative simplicity.
| Ratio | Cents | Interval Name(s) | Notes |
|---|---|---|---|
| 11/8 | 551.31794 | Axirabian paramajor fourth, just paramajor fourth |
This interval is the octave-reduced 11th harmonic, and, it's one of two basic generator intervals that are not Pythagorean intervals. |
| 16/11 | 648.68206 | Axirabian paraminor fifth, just paraminor fifth |
This interval is the octave-reduced 11th subharmonic, and, it's one of two basic generator intervals that are not Pythagorean intervals. |
| Ratio | Cents | Interval Name(s) | Notes |
|---|---|---|---|
| 33/32 | 53.272943 | Alpharabian ultraprime, Alpharabian parachroma, al-Farabi quartertone |
This interval is the octave-reduced 33rd harmonic, and, it's the basic modifier interval for arriving at intervals of this class. |
| 8192/8019 | 36.952052 | Alpharabian inframinor second | This interval is reached by starting at 256/243 and moving downwards by 33/32. |
| 88/81 | 143.49794 | Alpharabian artoneutral second | This interval is reached by starting at 256/243 and moving upwards by 33/32. |
| 12/11 | 150.63706 | Alpharabian tendoneutral second | This interval is reached by starting at 9/8 and moving downwards by 33/32. |
| 297/256 | 257.18294 | Alpharabian ultramajor second | This interval is reached by starting at 9/8 and moving upwards by 33/32. |
| 1024/891 | 240.86205 | Alpharabian inframinor third | This interval is reached by starting at 32/27 and moving downwards by 33/32. |
| 11/9 | 347.40794 | Alpharabian artoneutral third | This interval is reached by starting at 32/27 and moving upwards by 33/32. |
| 27/22 | 354.54706 | Alpharabian tendoneutral third | This interval is reached by starting at 81/64 and moving downwards by 33/32. |
| 2673/2048 | 461.09295 | Alpharabian ultramajor third | This interval is reached by starting at 81/64 and moving upwards by 33/32. |
| 128/99 | 444.77206 | Alpharabian paraminor fourth, just paraminor fourth |
This interval is the paraminor counterpart of 11/8 and is reached by starting at 4/3 and moving downwards by 33/32. |
| 99/64 | 755.22794 | Alpharabian paramajor fifth, just paramajor fifth |
This interval is the paramajor counterpart of 16/11 and is reached by starting at 3/2 and moving upwards by 33/32. |
| 4096/2673 | 738.90705 | Alpharabian inframinor sixth | This interval is reached by starting at 128/81 and moving downwards by 33/32. |
| 44/27 | 845.45294 | Alpharabian artoneutral sixth | This interval is reached by starting at 128/81 and moving upwards by 33/32. |
| 18/11 | 852.59206 | Alpharabian tendoneutral sixth | This interval is reached by starting at 27/16 and moving downwards by 33/32. |
| 891/512 | 959.13795 | Alpharabian ultramajor sixth | This interval is reached by starting at 27/16 and moving upwards by 33/32. |
| 512/297 | 942.81706 | Alpharabian inframinor seventh | This interval is reached by starting at 16/9 and moving downwards by 33/32. |
| 11/6 | 1049.3629 | Alpharabian artoneutral seventh | This interval is reached by starting at 16/9 and moving upwards by 33/32. |
| 81/44 | 1056.5021 | Alpharabian tendoneutral seventh | This interval is reached by starting at 243/128 and moving downwards by 33/32. |
| 8019/4096 | 1163.0479 | Alpharabian ultramajor seventh | This interval is reached by starting at 243/128 and moving upwards by 33/32. |
| 64/33 | 1146.7271 | Alpharabian infraoctave | This interval is the octave-reduced 33rd subharmonic. |
| Ratio | Cents | Interval Name(s) | Notes |
|---|---|---|---|
| 121/64 | 1102.6359 | Axirabian tendomean major seventh | This interval is the octave-reduced 121st harmonic. |
| 128/121 | 97.364115 | Axirabian artomean minor second, Axirabian limma, Axirabian diatonic semitone, |
This interval is the octave-reduced 121st subharmonic. |
| Ratio | Cents | Interval Name(s) | Notes |
|---|---|---|---|
| 243/242 | 7.1391196 | Rastma | This interval is the comma slash subchroma that separates 128/121 from 256/243, and, it's the basic modifier interval for arriving at intervals of this class. |
| 121/108 | 196.77088 | Alpharabian tendomean major second | This interval is the major counterpart of 128/121 and is reached by starting at 9/8 and moving downwards by 243/242. |
| 144/121 | 301.27412 | Alpharabian artomean minor third | This interval is reached by starting at 32/27 and moving upwards by 243/242. |
| 121/96 | 400.68088 | Alpharabian tendomean major third | This interval is reached by starting at 81/64 and moving downwards by 243/242. |
| 968/729 | 490.90588 | Alpharabian narrow fourth | This interval is reached by starting at 4/3 and moving downwards by 243/242. |
| 162/121 | 505.18412 | Alpharabian wide fourth | This interval is reached by starting at 4/3 and moving upwards by 243/242. |
| 121/81 | 694.81588 | Alpharabian narrow fifth | This interval is reached by starting at 3/2 and moving downwards by 243/242. |
| 729/484 | 709.09412 | Alpharabian wide fifth | This interval is reached by starting at 3/2 and moving upwards by 243/242. |
| 192/121 | 799.31912 | Alpharabian artomean minor sixth | This interval is reached by starting at 128/81 and moving upwards by 243/242. |
| 121/72 | 898.72588 | Alpharabian tendomean major sixth | This interval is reached by starting at 27/16 and moving downwards by 243/242. |
| 216/121 | 1003.2291 | Alpharabian artomean minor seventh | This interval is the minor counterpart of 121/64 and is reached by starting at 16/9 and moving upwards by 243/242. |
| 484/243 | 1192.8609 | Rastmic narrow octave | This interval is the result of taking a rastma from an octave. |
| Ratio | Cents | Interval Name(s) | Notes |
|---|---|---|---|
| 1331/1024 | 453.95383 | Axirabian semilimmic ultramajor third | This interval is the octave-reduced 1331st harmonic |
| 2048/1331 | 746.04617 | Axirabian semilimmic inframinor sixth | This interval is the octave-reduced 1331st subharmonic |
| Ratio | Cents | Interval Name(s) | Notes |
|---|---|---|---|
| 1331/1296 | 46.133824 | Alpharabian semilimmic ultraprime, Alpharabian parachromatic semilimma |
This is the larger, parachormatic half of a Pythagorean Limma, and, it's the basic modifier interval for arriving at intervals of this class. |
| 4096/3993 | 44.091172 | Alpharabian semilimmic inframinor second, Alpharabian paralimma, Alpharabian paradiatonic semilimma |
This interval is smaller, paradiatonic half of a Pythagorean Limma |
| 21296/19683 | 136.35882 | Alpharabian semilimmic supraminor second | |
| 1458/1331 | 157.77618 | Alpharabian semilimmic submajor second | |
| 1331/1152 | 250.04383 | Alpharabian semilimmic ultramajor second | This is one of two semifourths in the 2.3.11 subgroup. |
| 1536/1331 | 248.00117 | Alpharabian semilimmic inframinor third | This is one of two semifourths in the 2.3.11 subgroup. |