33/32: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = al-Farabi quarter tone, undecimal quarter tone, io comma, Alpharabian parachroma, Alpharabian ultraprime | |||
| Color name = 1o1, ilo unison | |||
| Name = al-Farabi | |||
| Color name = | |||
| Sound = jid_33_32_pluck_adu_dr220.mp3 | | Sound = jid_33_32_pluck_adu_dr220.mp3 | ||
| Comma = yes | |||
}} | }} | ||
'''33/32''', the '''al-Farabi | '''33/32''', the '''al-Farabi quarter tone'''<ref group="note">The name goes back to Abu Nasr Al-Farabi (in Western reception also Alpharabius), see [[Wikipedia: Al-Farabi]].</ref>, '''undecimal quarter tone''', or '''undecimal formal comma''', is a [[superparticular]] [[ratio]] which differs by a [[385/384|keenanisma (385/384)]], from the [[36/35|septimal quarter tone (36/35)]]. Raising a just [[4/3|perfect fourth (4/3)]] by the al-Farabi quarter-tone leads to the [[11/8|undecimal superfourth (11/8)]]. Raising it instead by 36/35 leads to the [[48/35|septimal superfourth (48/35)]] which approximates 11/8. Apart from this, it is also the interval between [[32/27]] and [[11/9]], and between [[9/8]] and [[12/11]]. | ||
Because of its close proximity to [[28/27]], from which it differs only by [[Pentacircle comma|896/891]], one could reasonably argue that 33/32 is the [[undecimal]] counterpart to 28/27 in a way, particularly if treated as an interval in its own right. However, despite this, 33/32 generally has properties more akin to a chromatic interval than to anything resembling a diatonic interval. In addition, 33/32 could arguably have been used as a melodic interval in the Greek Enharmonic Genus, and if so, there are several possibilities for the resulting [[tetrachord]]. The most obvious of these possibilities would be to include 32:33:34 within the interval of a perfect fourth, in which case this [[ancient Greek]] scale can be approximated in [[22edo]] and [[24edo]], with the comma [[1089/1088]] being tempered out so that 33/32 and 34/33 are equated. Another possibility, however, is that the semitone was [[16/15]], which, according to [[Wikipedia: Genus (music)|Wikipedia]], is indirectly attested to in the writings of [[Ptolemy]], and thus, if 33/32 was in fact used, it would have been paired with [[512/495]]. | |||
33/32 is significant in [[Functional Just System]] as the undecimal formal comma which translates a Pythagorean interval to a nearby undecimal interval. | It should be noted that in some significant respects, treating 33/32 as a comma rather than as an important musical interval in its own right sells it short, and results in the failure to correctly define the properties of certain intervals. Namely, a stack of two 33/32 intervals equals [[1089/1024]], a type of chromatic semitone that has [[128/121]] as its diatonic counterpart. Furthermore, 33/32 is one of two distinct 11-limit quartertone intervals required to add up to a whole tone, with [[4096/3993]] being the other—specifically, adding 4096/3993 to a stack of three 33/32 quartertones yields [[9/8]]. In addition to all this, 33/32 finds a special place in [[Alpharabian tuning]] and it is from this area of microtonal theory, among a select few others, that 33/32 acquires the names "'''Alpharabian parachroma'''" and "'''Alpharabian ultraprime'''", names that at this point are only used in said theoretical contexts. While many may be accustomed to thinking of 33/32 and [[729/704]] as "semiaugmented primes", this analysis is only completely accurate when [[243/242]] is tempered out. | ||
== Temperaments == | |||
If treated as a comma to be tempered out in the 2.3.11 [[subgroup]], it results in the [[no-fives subgroup temperaments #Io|io]] temperament, giving rise to the name '''io comma'''. The temperament is generated by a flat fifth ([[7edo]] and [[26edo]] being good tunings) which represents both [[3/2]] and [[16/11]]. | |||
== Approximation == | |||
[[22edo]] and [[23edo]]'s step sizes are good, albeit in[[consistent]] approximations of this interval. Since equal-step tuning of 33/32 is roughly equivalent to 22.5edo, 2 steps of [[45edo]] represent the interval with great accuracy. [[46edo]] inherits mapping from 23edo and does it consistently. | |||
== Notation == | |||
This interval is significant in [[Functional Just System]] and [[Helmholtz–Ellis notation]] as the undecimal formal comma which translates a Pythagorean interval to a nearby undecimal interval. | |||
=== Ben Johnston's notation === | |||
In [[Ben Johnston's notation]], this interval is denoted with ↑, and its reciprocal as ↓. If the base note is C, then 11/8 is represented by C–F↑. | |||
=== Sagittal notation === | |||
In the [[Sagittal]] system, this comma (possibly tempered) is represented by the sagittal {{sagittal | /|\ }} and is called the '''11 medium diesis''', or '''11M''' for short, because the simplest interval it notates is 11/1 (equiv. 11/8), as for example in C–F{{nbhsp}}{{sagittal | /|\ }}. The downward version is called '''1/11M''' or '''11M down''' and is represented by {{sagittal| \!/ }}. | |||
== See also == | == See also == | ||
* [[Gallery of just intervals]] | |||
* [[1ed33/32]] – equal multiplication of this interval | |||
* [[64/33]] – its [[octave complement]] | |||
* [[16/11]] – its [[fifth complement]] | |||
* [[32/31]] – the tricesimoprimal counterpart | |||
* [[:File:Ji-33-32-csound-foscil-220hz.mp3]] – alternative sound example | |||
== References == | |||
<references group="note" /> | |||
[[Category:Quartertone]] | [[Category:Quartertone]] | ||
[[Category: | [[Category:Alpharabian]] | ||
[[Category: | [[Category:Commas named after their color name]] | ||
[[Category:Commas named after polymaths]] | |||