32/27: Difference between revisions
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Hotcrystal0 (talk | contribs) Mention alteraugment |
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{{Infobox Interval | {{Infobox Interval | ||
| Name = Pythagorean minor third | | Name = Pythagorean minor third | ||
| Color name = w3, wa 3rd | | Color name = w3, wa 3rd | ||
| Line 9: | Line 5: | ||
}} | }} | ||
The '''Pythagorean minor third''' of '''32/27''' is the interval between [[9/8]] and [[4/3]] which arises naturally in [[3-limit]] just intonation. | The '''Pythagorean minor third''' of '''32/27''' is the interval between [[9/8]] and [[4/3]] which arises naturally in [[3-limit]] [[just intonation]]. Compared to the more typical [[6/5]]- with which it is conflated in [[meantone]]- this interval is more dissonant, with a [[harmonic entropy]] level roughly on par with that of 9/8. | ||
It is 352/351 sharp of [[13/11]], and tempering 352/351 out equates it with 13/11 and leads to [[minthmic chords]]. | |||
== Temperaments == | |||
32/27 is treated as a comma in edos 3 & 6, where the best approximation of a perfect 5th is the 800 cent interval that wraps around to the octave again after only three iterations, producing [[alteraugment]]. Temperaments it can be interpreted as if used as a generator include [[Kleismic_family#Kleiboh|Kleiboh]] or [[Gariberttet]]. | |||
== Approximation == | |||
{{Interval edo approximation|32/27}} | |||
== See also == | == See also == | ||
* [[ | * [[27/16]] – its [[octave complement]] | ||
* [[ | * [[81/64]] – its [[fifth complement]] | ||
* [[9/8]] – its [[fourth complement]] | |||
* [[Gallery of just intervals]] | |||
* [[Pythagorean tuning]] | |||
[[Category:Third]] | [[Category:Third]] | ||
[[Category: | [[Category:Minor third]] | ||
Latest revision as of 20:52, 1 June 2026
| Interval information |
reduced subharmonic
[sound info]
The Pythagorean minor third of 32/27 is the interval between 9/8 and 4/3 which arises naturally in 3-limit just intonation. Compared to the more typical 6/5- with which it is conflated in meantone- this interval is more dissonant, with a harmonic entropy level roughly on par with that of 9/8.
It is 352/351 sharp of 13/11, and tempering 352/351 out equates it with 13/11 and leads to minthmic chords.
Temperaments
32/27 is treated as a comma in edos 3 & 6, where the best approximation of a perfect 5th is the 800 cent interval that wraps around to the octave again after only three iterations, producing alteraugment. Temperaments it can be interpreted as if used as a generator include Kleiboh or Gariberttet.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 4 | 1\4 | 300.00 | +5.87 | +1.96 |
| 8 | 2\8 | 300.00 | +5.87 | +3.91 |
| 12 | 3\12 | 300.00 | +5.87 | +5.87 |
| 16 | 4\16 | 300.00 | +5.87 | +7.82 |
| 20 | 5\20 | 300.00 | +5.87 | +9.78 |
| 33 | 8\33 | 290.91 | -3.23 | -8.87 |
| 37 | 9\37 | 291.89 | -2.24 | -6.92 |
| 41 | 10\41 | 292.68 | -1.45 | -4.96 |
| 45 | 11\45 | 293.33 | -0.80 | -3.01 |
| 49 | 12\49 | 293.88 | -0.26 | -1.05 |
| 53 | 13\53 | 294.34 | +0.20 | +0.90 |
| 57 | 14\57 | 294.74 | +0.60 | +2.86 |
| 61 | 15\61 | 295.08 | +0.95 | +4.81 |
| 65 | 16\65 | 295.38 | +1.25 | +6.77 |
| 69 | 17\69 | 295.65 | +1.52 | +8.72 |