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== Introduction ==
== Introduction ==
It seems that some people in the community want to know how my system relates to the more well-known approach of Aaron Hunt- a simple question with a complicated answer.  Although this page is late in coming, I think it is time to really begin to let people see the underpinnings of my approach to music in general and microtonal music in particular, and hopefully begin to answer this question.
It seems that some people in the community want to know how my system relates to the more well-known approach of Aaron Hunt- a simple question with a complicated answer.  Although this page is late in coming, I think it is time to really begin to let people see the underpinnings of my approach to music in general and microtonal music in particular, and hopefully begin to answer this question.


== A Strange New World ==
== A Strange New World ==
 
From looking things up, it seems that Hunt and I have both been influenced by music theory of [https://en.wikipedia.org/wiki/Harry_Partch Harry Partch]. However, there are significant differences, as while Hunt has been influenced by the work of [https://de.wikipedia.org/wiki/Martin_Vogel_(Musikwissenschaftler) Martin Vogel] and Dr. Patrick Ozzard-Low, I, for my part, have been influenced rather heavily by what little I know of the works of [https://en.wikipedia.org/wiki/Hugo_Riemann Hugo Riemann], and I've even picked up a few tricks concerning Locrian mode from [https://alexanderlafollett.com/site/the-locrian-mode-no-its-not-unusable/ Alexander LaFollett], as well as learning from my own experimentations with Locrian.  While I can't remember which came first, I must say that the influence of Riemann's concept of Harmonic Duality on my work is strongly connected to my discovery that [https://music.stackexchange.com/questions/11274/what-are-the-greek-modes-and-how-do-they-differ-from-modern-modes Ancient Greek modes were built from the Treble downwards], and, because when the Ancient Romans borrowed the Greek terminology, they evidently made the mistake of assuming that the Greek note names were built from the Bass upwards, [[wikipedia:el:%CE%A4%CF%81%CF%8C%CF%80%CE%BF%CF%82_(%CE%BC%CE%BF%CF%85%CF%83%CE%B9%CE%BA%CE%AE)|resulting in a disconnect]] between the Ancient Greek musical system and Modern Western Music Theory.   
From looking things up, it seems that Hunt and I have both been influenced by music theory of [https://en.wikipedia.org/wiki/Harry_Partch Harry Partch]. However, there are significant differences, as while Hunt has been influenced by the work of [https://de.wikipedia.org/wiki/Martin_Vogel_(Musikwissenschaftler) Martin Vogel] and Dr. Patrick Ozzard-Low, I, for my part, have been influenced rather heavily by what little I know of the works of [https://en.wikipedia.org/wiki/Hugo_Riemann Hugo Riemann], and I've even picked up a few tricks concerning Locrian mode from [https://alexanderlafollett.com/site/the-locrian-mode-no-its-not-unusable/ Alexander LaFollett], as well as learning from my own experimentations with Locrian.  While I can't remember which came first, I must say that the influence of Riemann's concept of Harmonic Duality on my work is strongly connected to my discovery that [https://music.stackexchange.com/questions/11274/what-are-the-greek-modes-and-how-do-they-differ-from-modern-modes Ancient Greek modes were built from the Treble downwards], and, because when the Ancient Romans borrowed the Greek terminology, they evidently made the mistake of assuming that the Greek note names were built from the Bass upwards, [https://el.wikipedia.org/wiki/%CE%A4%CF%81%CF%8C%CF%80%CE%BF%CF%82_(%CE%BC%CE%BF%CF%85%CF%83%CE%B9%CE%BA%CE%AE) resulting in a disconnect] between the Ancient Greek musical system and Modern Western Music Theory.   


In light of this information, and in light of the development of Western Music Theory since the time of the Romans, I think it would be a good idea to also build on the more historically accurate version of the Ancient Greek modes and Treble-Down tonality in general to the same extent as has been done for Bass-Up tonality.  However, doing this involves discarding the commonly-held dogmatic assumption in Modern Western Music Theory that all music is built from the Bass Upwards.  Furthermore, it involves renaming some of the diatonic functions encountered in Modern Western Music Theory to be better accommodating to Treble-Down Tonality, something which Hunt's system fails to do.  So, in order to do this, what sort of foundation shall we use?  Well, for one thing, I propose we take Riemann's concept of harmonic duality- as well as Partch's argument that the Overtone Series and the Undertone Series are equally fundamental- much more seriously.  Nevertheless, Hunt has done a fantastic job in integrating the ancient idea of a comma and the modern idea of the Just-Noticeable Difference in pitch perception, and I have even taken from [http://musictheory.zentral.zone/huntsystem2.html#2 the research on his site in this area] to establish core aspects of my standards in terms of pitch representation quality.  However, I differ significantly with him in terms of what intervals can be regarded as "commas" as opposed to "chromas", as due to my prior experience with [[24edo]], I can only assume that chromas, in addition to the standard definitions, can also be intervals that are less than 50 cents, yet greater than 25 cents.
In light of this information, and in light of the development of Western Music Theory since the time of the Romans, I think it would be a good idea to also build on the more historically accurate version of the Ancient Greek modes and Treble-Down tonality in general to the same extent as has been done for Bass-Up tonality.  However, doing this involves discarding the commonly-held dogmatic assumption in Modern Western Music Theory that all music is built from the Bass Upwards.  Furthermore, it involves renaming some of the diatonic functions encountered in Modern Western Music Theory to be better accommodating to Treble-Down Tonality, something which Hunt's system fails to do.  So, in order to do this, what sort of foundation shall we use?  Well, for one thing, I propose we take Riemann's concept of harmonic duality- as well as Partch's argument that the Overtone Series and the Undertone Series are equally fundamental- much more seriously.  Nevertheless, Hunt has done a fantastic job in integrating the ancient idea of a comma and the modern idea of the Just-Noticeable Difference in pitch perception, and I have even taken from [http://musictheory.zentral.zone/huntsystem2.html#2 the research on his site in this area] to establish core aspects of my standards in terms of pitch representation quality.  However, I differ significantly with him in terms of what intervals can be regarded as "commas" as opposed to "chromas", as due to my prior experience with [[24edo]], I can only assume that chromas, in addition to the standard definitions, can also be intervals that are less than 50 cents, yet greater than 25 cents.
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== Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications ==
== Delving into the 11-Limit: Alpharabian Tuning and Additional Interval Classifications ==


With the 11-limit established as perhaps the best p-limit for representing quartertones in terms of ratio simplicity, we need to look more closely at its properties- something which most microtonal systems at the time of this writing have yet to do.  First off, one will notice that a stack of two 33/32 quartertones falls short of an apotome by a comma called the [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]].  One will also notice that because two parachromatic quartertones equals a chromatic semitone, and because two 33/32 quartertones fall short of an apotome by a rastma, we can deduce that other parachromatic quartertones- or, more generally "parachromas"- can be derived from 33/32 by adding or subtracting the rastma.  When one adds the rastma to 33/32, one arrives at [[729/704]]- the parachromatic quartertone that adds up together with 33/32 to form an apotome.  Since 33/32 is smaller than 729/704, and both can be described as "undecimal quartertones", we need to disambiguate them somehow, not to mention acknowledge the 11-limit's newfound status as a navigational prime.  Thus, for purposes of continuing this discussion at the moment, we'll start referring to 33/32 as the "primary parachromatic quartertone", and, we'll refer to 729/704 as the "secondary parachromatic quartertone".  However, the rastma doesn't only derive other parachromatic intervals from 33/32, but also derives other paradiatonic intervals from [[4096/3993]]- which we shall refer to for the moment as the "primary paradiatonic quartertone".  For instance, if one subtracts the rastma from the primary paradiatonic quartertone, we get [[8192/8019]], which, when added to 33/32, yields [[256/243]]- the Pythagorean Diatonic Semitone.  Because of this, and because 8192/8019 is derived from the primary paradiatonic quartertone, we shall refer to 8192/8019 as the "secondary paradiatonic quartertone".   
With the 11-limit established as perhaps the best p-limit for representing quartertones in terms of ratio simplicity, we need to look more closely at its properties- something which most microtonal systems at the time of this writing have yet to do.  First off, one will notice that a stack of two 33/32 quartertones falls short of an apotome by a comma called the [[243/242|rastma]], which means that the rastma, when not tempered out, is a very important "parasubchroma"- a term derived from [[Diatonic, chromatic, enharmonic, subchromatic|subchroma]].  One will also notice that because two parachromatic quartertones equals a chromatic semitone, and because two 33/32 quartertones fall short of an apotome by a rastma, we can deduce that other parachromatic quartertones- or, more generally "parachromas"- can be derived from 33/32 by adding or subtracting the rastma.  When one adds the rastma to 33/32, one arrives at [[729/704]]- the parachromatic quartertone that adds up together with 33/32 to form an apotome.  Since 33/32 is smaller than 729/704, and both can be described as "undecimal quartertones", we need to disambiguate them somehow, not to mention acknowledge the 11-limit's newfound status as a navigational prime.  Thus, for purposes of continuing this discussion at the moment, we'll start referring to 33/32 as the "primary parachromatic quartertone", and, we'll refer to 729/704 as the "secondary parachromatic quartertone".  However, the rastma doesn't only derive other parachromatic intervals from 33/32, but also derives other paradiatonic intervals from [[4096/3993]]- which we shall refer to for the moment as the "primary paradiatonic quartertone".  For instance, if one subtracts the rastma from the primary paradiatonic quartertone, we get [[8192/8019]], which, when added to 33/32, yields [[256/243]]- the Pythagorean Diatonic Semitone.  Because of this, and because 8192/8019 is derived from the primary paradiatonic quartertone, we shall refer to 8192/8019 as the "secondary paradiatonic quartertone".   


However, the "primary" versus "secondary" distinction is temporary at best, as in truth there is more nuance to consider.  Furthermore, we have to contend with the idea of a "Subchroma" from earlier, as well as the idea of a "Diesis", and define how these concept relate to the idea of "Parachromatic" and "Paradiatonic" intervals, and for this we should begin by looking at the distinction between a "Paradiatonic" interval and a "Diesis".  In order to define a "Paradiatonic" interval as it contrasts with a "Diesis", we need to consider that "Paradiatonic" consists of the prefix "Para-" and the word "Diatonic", with "Para-" meaning "alongside" in this case, as paradiatonic intervals are those that are relatively easy to use as accidentals in otherwise diatonic keys.  More importantly, we need to consider that diatonic intervals- as the term "diatonic" pertains to intervals other than semitones- are the intervals found in those heptatonic scales consisting of five whole tones and two semitones in each octave in which the semitones are [https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedia's article on the Diatonic Scale].  In light of all this, it should follow that we can define "Paradiatonic" intervals as being those microtonal intervals which are as distant from the boundaries of the nearest semitone-based intervals as possible.  Furthermore, since quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches, this means that Paradiatonic intervals are inevitably quartertone-based, as are their "Parachromatic" counterparts.  In contrast, terms such as "Subchroma" and "Diesis" are a bit broader, as they are not restricted to quartertone-based intervals- in fact, they are most often used to refer to intervals ''smaller'' than a quartertone.  That said, there are such things as "Subchromatic Quartertones", which contrast with their Paracrhomatic counterparts in that they have more complicated ratios whereas the Parachromatic quartertones have fairly simple ratios.
However, the "primary" versus "secondary" distinction is temporary at best, as in truth there is more nuance to consider.  Furthermore, we have to contend with the idea of a "Subchroma" from earlier, as well as the idea of a "Diesis", and define how these concept relate to the idea of "Parachromatic" and "Paradiatonic" intervals, and for this we should begin by looking at the distinction between a "Paradiatonic" interval and a "Diesis".  In order to define a "Paradiatonic" interval as it contrasts with a "Diesis", we need to consider that "Paradiatonic" consists of the prefix "Para-" and the word "Diatonic", with "Para-" meaning "alongside" in this case, as paradiatonic intervals are those that are relatively easy to use as accidentals in otherwise diatonic keys.  More importantly, we need to consider that diatonic intervals- as the term "diatonic" pertains to intervals other than semitones- are the intervals found in those heptatonic scales consisting of five whole tones and two semitones in each octave in which the semitones are [https://en.wikipedia.org/wiki/Maximal_evenness spread out as much as possible], as per the more strict definition of "Diatonic" listed on [https://en.wikipedia.org/wiki/Diatonic_scale Wikipedia's article on the Diatonic Scale].  In light of all this, it should follow that we can define "Paradiatonic" intervals as being those microtonal intervals which are as distant from the boundaries of the nearest semitone-based intervals as possible.  Furthermore, since quartertones are the smallest musical intervals that can be used in musical capacities without being considered a variation of one of the surrounding pitches, this means that Paradiatonic intervals are inevitably quartertone-based, as are their "Parachromatic" counterparts.  In contrast, terms such as "Subchroma" and "Diesis" are a bit broader, as they are not restricted to quartertone-based intervals- in fact, they are most often used to refer to intervals ''smaller'' than a quartertone.  That said, there are such things as "Subchromatic Quartertones", which contrast with their Paracrhomatic counterparts in that they have more complicated ratios whereas the Parachromatic quartertones have fairly simple ratios.
Line 61: Line 59:
!| Interval
!| Interval
!| Cents
!| Cents
!| Shorthand Designations
!| Names
!| Names
|-
|-
| [[1331/1296]]
| [[1331/1296]]
| 46.133824
| 46.133824
| Alpharabian Superprime, Alpharabian Subchromatic Quartertone
| rU1
|
|-
|-
| [[33/32]]
| [[33/32]]
| 53.272943
| 53.272943
| U1
| Alpharabian Ultraprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone
| Alpharabian Ultraprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone
|-
|-
| [[729/704]]
| [[729/704]]
| 60.412063
| 60.412063
| Alpharabian Parasubaugmented Unison, Alpharabian Parasubapotomic Quartertone
| uA1
| Alpharabian Infraaugmented Prime, Alpharabian Infraapotomic Quartertone
|-
|-
| [[1089/1024]]
| [[1089/1024]]
| 106.545886
| 106.545886
| Alpharabian Augmented Unison, Alpharabian Chromatic Semitone
| rA1
| Alpharabian Chromatic Semitone
|-
|-
| [[161051/147456]]
| [[161051/147456]]
| 152.679710
| 152.679710
| Alpharabian Superaugmented Unison
| rrUA1
| Alpharabian Superaugmented Prime
|-
|-
| [[35937/32768]]
| [[35937/32768]]
| 159.818830
| 159.818830
| Alpharabian Sesquiaugmented Unison
| rUA1
|
|-
| [[72171/65536]]
| 166.957949
| UA1
| Alpharabian Ultraaugmented Prime
|-
|-
| [[8192/8019]]
| [[8192/8019]]
| 36.952052
| 36.952052
| Alpharabian Inframinor Second, Alpharabian Diesis
| um2
| Alpharabian Inframinor Second
|-
|-
| [[4096/3993]]
| [[4096/3993]]
| 44.091172
| 44.091172
| Rum2
| Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone
| Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone
|-
|-
| [[128/121]]
| [[128/121]]
| 97.364115
| 97.364115
|
| Alpharabian Minor Second, Alpharabian Diatonic Semitone
| Alpharabian Minor Second, Alpharabian Diatonic Semitone
|-
|-
| [[21296/19683]]
| [[21296/19683]]
| 136.358819
| 136.358819
|
| Alpharabian Supraminor Second
| Alpharabian Supraminor Second
|-
|-
| [[88/81]]
| [[88/81]]
| 143.497938
| 143.497938
|
| Lesser Alpharabian Neutral Second
| Lesser Alpharabian Neutral Second
|-
|-
| [[12/11]]
| [[12/11]]
| 150.637059
| 150.637059
|
| Greater Alpharabian Neutral Second
| Greater Alpharabian Neutral Second
|-
|-
| [[1458/1331]]
| [[1458/1331]]
| 157.776178
| 157.776178
|
| Alpharabian Submajor Second
| Alpharabian Submajor Second
|-
|-
| [[1331/1152]]
| [[1331/1152]]
| 250.043825
| 250.043825
|
| Alpharabian Supermajor Second
| Alpharabian Supermajor Second
|-
|-
| [[297/256]]
| [[297/256]]
| 257.182945
| 257.182945
|
| Alpharabian Parasupermajor Second
| Alpharabian Parasupermajor Second
|-
|-
| [[1024/891]]
| [[1024/891]]
| 240.862054
| 240.862054
|
| Alpharabian Parasubminor Third
| Alpharabian Parasubminor Third
|-
|-
| [[1536/1331]]
| [[1536/1331]]
| 248.001174
| 248.001174
|
| Alpharabian Subminor Third
| Alpharabian Subminor Third
|-
|-
| [[144/121]]
| [[144/121]]
| 301.274117
| 301.274117
|
| Alpharabian Minor Third
| Alpharabian Minor Third
|-
|-
| [[2662/2187]]
| [[2662/2187]]
| 340.268821
| 340.268821
|
| Alpharabian Supraminor Third
| Alpharabian Supraminor Third
|-
|-
| [[11/9]]
| [[11/9]]
| 347.407941
| 347.407941
|
| Lesser Alpharabian Neutral Third
| Lesser Alpharabian Neutral Third
|-
|-
| [[27/22]]
| [[27/22]]
| 354.547060
| 354.547060
|
| Greater Alpharabian Neutral Third
| Greater Alpharabian Neutral Third
|-
|-
| [[6561/5324]]
| [[6561/5324]]
| 361.686180
| 361.686180
|
| Alpharabian Submajor Third
| Alpharabian Submajor Third
|-
|-
| [[121/96]]
| [[121/96]]
| 400.680884
| 400.680884
|
| Alpharabian Major Third
| Alpharabian Major Third
|-
|-
| [[1331/1024]]
| [[1331/1024]]
| 453.953827
| 453.953827
|
| Alpharabian Supermajor Third
| Alpharabian Supermajor Third
|-
|-
| [[2673/2048]]
| [[2673/2048]]
| 461.092947
| 461.092947
|
| Alpharabian Parasupermajor Third
| Alpharabian Parasupermajor Third
|-
|-
| [[128/99]]
| [[128/99]]
| 444.772056
| 444.772056
|
| Alpharabian Paraminor Fourth
| Alpharabian Paraminor Fourth
|-
|-
| [[1728/1331]]
| [[1728/1331]]
| 451.911176
| 451.911176
|
| Alpharabian Subfourth
| Alpharabian Subfourth
|-
|-
| [[11/8]]
| [[11/8]]
| 551.317942
| 551.317942
|
| Alpharabian Paramajor Fourth, Just Paramajor Fourth
| Alpharabian Paramajor Fourth, Just Paramajor Fourth
|-
|-
| [[1331/972]]
| [[1331/972]]
| 544.178823
| 544.178823
|
| Alpharabian Superfourth
| Alpharabian Superfourth
|-
|-
| [[363/256]]
| [[363/256]]
| 604.590886
| 604.590886
|
| Alpharabian Augmented Fourth
| Alpharabian Augmented Fourth
|-
|-
| [[512/363]]
| [[512/363]]
| 595.409114
| 595.409114
|
| Alpharabian Diminished Fifth
| Alpharabian Diminished Fifth
|-
|-
| [[16/11]]
| [[16/11]]
| 648.682058
| 648.682058
|
| Alpharabian Paraminor Fifth, Just Paraminor Fifth
| Alpharabian Paraminor Fifth, Just Paraminor Fifth
|-
|-
| [[99/64]]
| [[99/64]]
| 755.227944
| 755.227944
|
| Alpharabian Paramajor Fifth
| Alpharabian Paramajor Fifth
|-
|-
| [[4096/2673]]
| [[4096/2673]]
| 738.907053
| 738.907053
|
| Alpharabian Parasubminor Sixth
| Alpharabian Parasubminor Sixth
|-
|-
| [[2048/1331]]
| [[2048/1331]]
| 746.046173
| 746.046173
|
| Alpharabian Subminor Sixth
| Alpharabian Subminor Sixth
|-
|-
| [[192/121]]
| [[192/121]]
| 799.319116
| 799.319116
|
| Alpharabian Minor Sixth
| Alpharabian Minor Sixth
|-
|-
| [[44/27]]
| [[44/27]]
| 845.452940
| 845.452940
|
| Lesser Alpharabian Neutral Sixth
| Lesser Alpharabian Neutral Sixth
|-
|-
| [[18/11]]
| [[18/11]]
| 852.592059
| 852.592059
|
| Greater Alpharabian Neutral Sixth
| Greater Alpharabian Neutral Sixth
|-
|-
| [[121/72]]
| [[121/72]]
| 898.725883
| 898.725883
|
| Alpharabian Major Sixth
| Alpharabian Major Sixth
|-
|-
| [[1331/768]]
| [[1331/768]]
| 951.998826
| 951.998826
|
| Alpharabian Supermajor Sixth
| Alpharabian Supermajor Sixth
|-
|-
| [[891/512]]
| [[891/512]]
| 959.137946
| 959.137946
|
| Alpharabian Parasupermajor Sixth
| Alpharabian Parasupermajor Sixth
|-
|-
| [[512/297]]
| [[512/297]]
| 942.817055
| 942.817055
|
| Alpharabian Parasubminor Seventh
| Alpharabian Parasubminor Seventh
|-
|-
| [[2304/1331]]
| [[2304/1331]]
| 949.956175
| 949.956175
|
| Alpharabian Subminor Seventh
| Alpharabian Subminor Seventh
|-
|-
| [[11/6]]
| [[11/6]]
| 1,049.362941
| 1,049.362941
|
| Lesser Alpharabian Neutral Seventh
| Lesser Alpharabian Neutral Seventh
|-
|-
| [[81/44]]
| [[81/44]]
| 1,056.502061
| 1,056.502061
|
| Greater Alpharabian Neutral Seventh
| Greater Alpharabian Neutral Seventh
|-
|-
| [[121/64]]
| [[121/64]]
| 1,102.635885
| 1,102.635885
|
| Alpharabian Major Seventh
| Alpharabian Major Seventh
|-
|-
| [[3993/2048]]
| [[3993/2048]]
| 1,155.908828
| 1,155.908828
|
| Alpharabian Supermajor Seventh
| Alpharabian Supermajor Seventh
|-
|-
| [[8019/4096]]
| [[8019/4096]]
| 1,163.047948
| 1,163.047948
|
| Alpharabian Ultramajor Seventh
| Alpharabian Ultramajor Seventh
|-
|-
| [[2048/1089]]
| [[2048/1089]]
| 1,093.454114
| 1,093.454114
|
| Alpharabian Diminished Octave
| Alpharabian Diminished Octave
|-
|-
| [[64/33]]
| [[64/33]]
| 1,146.727057
| 1,146.727057
|
| Alpharabian Infraoctave
| Alpharabian Infraoctave
|-
|-
| [[2592/1331]]
| [[2592/1331]]
| 1,153.866176
| 1,153.866176
|
| Alpharabian Suboctave
| Alpharabian Suboctave
|-
|-
Line 275: Line 330:
* Intervals that result from the modification of a Pythagorean interval by [[1089/1024]] are labeled similarly to those modified in the equivalent fashion by [[2187/2048]], the only difference being that modification by 1089/1024 results in an Alpharabian interval rather than a Pythagorean interval.
* Intervals that result from the modification of a Pythagorean interval by [[1089/1024]] are labeled similarly to those modified in the equivalent fashion by [[2187/2048]], the only difference being that modification by 1089/1024 results in an Alpharabian interval rather than a Pythagorean interval.
* Since 1089/1024 is (33/32)^2, modifying a Pythagorean interval by [[33/32]] always results in an interval that is considered "Alpharabian".
* Since 1089/1024 is (33/32)^2, modifying a Pythagorean interval by [[33/32]] always results in an interval that is considered "Alpharabian".
* As both the [[243/242|rastma]] and [[1331/1296]] are [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchromas]] that form differences between members of the 2.11 subgroup and Pythagorean intervals, both of these subchromas belong to a set of intervals defining different interval sets within Alpharabian tuning, and subchromas within this particular interval set help define the differences between Pythagorean, Alpharabian and Betarabian intervals.
* As both the [[243/242|rastma]] and [[1331/1296]] are [[diatonic, chromatic, enharmonic, subchromatic|subchromas]] that form differences between members of the 2.11 subgroup and Pythagorean intervals, both of these subchromas belong to a set of intervals defining different interval sets within Alpharabian tuning, and subchromas within this particular interval set help define the differences between Pythagorean, Alpharabian and Betarabian intervals.


The following rules are directly derived from the above premises:
The following rules are directly derived from the above premises:
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== Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma ==
== Delving into the 11-Limit: Betarabian Intervals and the Nexus Comma ==
(Todo: rewrite this section based on further modification of the rules for defining subsets of Alpharabian tuning)
(Todo: rewrite this section based on further modification of the rules for defining subsets of Alpharabian tuning)


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