11/9: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 256749618 - Original comment: ** |
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{{Infobox Interval | |||
| Name = undecimal neutral third, Alpharabian artoneutral third | |||
| Color name = 1o3, ilo 3rd | |||
| Sound = jid_11_9_pluck_adu_dr220.mp3 | |||
}} | |||
In | In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]. As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. Since it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]]. It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings. | ||
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]]. | |||
== Approximation == | |||
{{Interval edo approximation|11/9}} | |||
== See also == | |||
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic | * [[7edo]] | ||
* [[24edo]] | |||
See | * [[18/11]] – its [[octave complement]] | ||
* [[27/22]] – its [[fifth complement]] | |||
* [[12/11]] – its [[fourth complement]] | |||
* [[Gallery of just intervals]] | |||
* [[Iceface tuning]] | |||
[[Category:Third]] | |||
[[Category:Neutral third]] | |||
[[Category:Alpharabian]] | |||
[[Category:Tritave-reduced harmonics]] | |||
Latest revision as of 13:05, 3 November 2025
| Interval information |
Alpharabian artoneutral third
[sound info]
In 11-limit just intonation, 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but it is only one of many, of which others include 16/13, 27/22, 39/32, 49/40 and 60/49. As this is the smaller of the two 11-limit neutral thirds obtained by modifying Pythagorean intervals by 33/32, it is dubbed the Alpharabian artoneutral third in Alpharabian tuning. Since it is nearly halfway between two intervals of 12edo, it implies that it is both very xenharmonic and well-represented in 24edo. It is approximated even more closely in 31edo and 38edo, where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including 7edo, 10edo, 17edo, 24edo, 31edo, 41edo, 58edo, miracle, harry, and sesquart, conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament jove.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 7 | 2\7 | 342.86 | -4.55 | -2.65 |
| 14 | 4\14 | 342.86 | -4.55 | -5.31 |
| 17 | 5\17 | 352.94 | +5.53 | +7.84 |
| 21 | 6\21 | 342.86 | -4.55 | -7.96 |
| 24 | 7\24 | 350.00 | +2.59 | +5.18 |
| 31 | 9\31 | 348.39 | +0.98 | +2.53 |
| 38 | 11\38 | 347.37 | -0.04 | -0.13 |
| 45 | 13\45 | 346.67 | -0.74 | -2.78 |
| 52 | 15\52 | 346.15 | -1.25 | -5.43 |
| 55 | 16\55 | 349.09 | +1.68 | +7.71 |
| 59 | 17\59 | 345.76 | -1.65 | -8.09 |
| 62 | 18\62 | 348.39 | +0.98 | +5.06 |
| 69 | 20\69 | 347.83 | +0.42 | +2.40 |
| 76 | 22\76 | 347.37 | -0.04 | -0.25 |