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Alpharabian tuning: Difference between revisions

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When one checks the 11-limit's representation of quartertones against those of the other rational intervals called "quarter tones" [https://en.wikipedia.org/wiki/List_of_pitch_intervals 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 comma|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.
When one checks the 11-limit's representation of quartertones against those of the other rational intervals called "quarter tones" [https://en.wikipedia.org/wiki/List_of_pitch_intervals 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 comma|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 ==
== Interval naming scheme ==


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


== 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"
|+ style=white-space:nowrap | Table of Class I Axirabian Intervals
|-
! Ratio
! [[Cent]]s
! Interval Name(s)
! Notes
|-
| [[11/8]]
| 551.31794
| Axirabian paramajor fourth, just paramajor fourth
| This interval is the octave-reduced 11rd 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 11rd subharmonic, and it's one of two basic generator intervals that are not Pythagorean intervals.
|-
|}
{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style=white-space:nowrap | Incomplete Table of Class I Alpharabian Intervals
|-
! Ratio
! [[Cent]]s
! 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.
|-
|}


[[Category:Tuning]]
[[Category:Tuning]]
[[Category:Alpharabian| ]] <!-- main article -->
[[Category:Alpharabian| ]] <!-- main article -->
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