Alpharabian tuning: Difference between revisions

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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 [[~~User:Aura|~~Aura]], with significant parts of this research having been made almost two years earlier by [[User:Spt3125|Spt3125]].

The '''Alpharabian tuning''' is the version of [[just intonation]] limited to the '''2.3.11 subgroup''' (a.k.a. ''ila'' in [[color notation]]). It consists of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[11/1|11]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3 and 13; this makes it a rank-3 system.

It is currently being pioneered in large part by [[Aura]], with significant parts of this research having been made almost two years earlier by [[User:Spt3125|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 tuning|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.

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 tuning|Pythagorean]] [[5L 2s|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 3 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.

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== 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 [[subchromatic|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 rastma, [[243/242]], is functionally the simplest type of Alpharabian [[subchromatic|subchroma]], and, since three instances of 243/242 are almost equal to [[81/80]] in JI, differing by only the [[parimo]] (which is less than 0.1 cents in size), 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

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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…

* 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.

{| class="mw-collapsible mw-collapsed wikitable center-1"

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! [[Cent]]s

! Interval ~~Name~~(s)

! Interval name(s)

! Notes

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! Interval ~~Name~~(s)

! Interval name(s)

! Notes

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! Interval name(s)

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[[Category:Alpharabian| ]]

[[Category:~~Subgroup~~]]

[[Category:Just intonation subgroups|#]]

[[Category:11-~~limit~~]]

[[Category:Rank-3 temperaments|#]]

[[Category:~~Rank 3~~]]

[[Category:11-limit|#]]

@@ Line 1: / Line 1: @@
-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 [[User:Aura|Aura]], with significant parts of this research having been made almost two years earlier by [[User:Spt3125|Spt3125]].
+The '''Alpharabian tuning''' is the version of [[just intonation]] limited to the '''2.3.11 subgroup''' (a.k.a. ''ila'' in [[color notation]]). It consists of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[11/1|11]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3 and 13; this makes it a rank-3 system.
+It is currently being pioneered in large part by [[Aura]], with significant parts of this research having been made almost two years earlier by [[User:Spt3125|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 tuning|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.
+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 tuning|Pythagorean]] [[5L 2s|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 3 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.
@@ Line 12: / Line 14: @@
 == 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 [[subchromatic|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 rastma, [[243/242]], is functionally the simplest type of Alpharabian [[subchromatic|subchroma]], and, since three instances of 243/242 are almost equal to [[81/80]] in JI, differing by only the [[parimo]] (which is less than 0.1 cents in size), 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
@@ Line 35: / Line 35: @@
 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…
-* 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.
+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.
 {| class="mw-collapsible mw-collapsed wikitable center-1"
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 ! Ratio
 ! [[Cent]]s
-! Interval Name(s)
+! Interval name(s)
 ! Notes
 |-
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 ! Ratio
 ! [[Cent]]s
-! Interval Name(s)
+! Interval name(s)
 ! Notes
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+! Interval name(s)
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 [[Category:Alpharabian| ]] <!-- main article -->
-[[Category:Subgroup]]
+[[Category:Just intonation subgroups|#]]
-[[Category:11-limit]]
+[[Category:Rank-3 temperaments|#]]
-[[Category:Rank 3]]
+[[Category:11-limit|#]]