Porcupine: Difference between revisions
No edit summary |
Unhighlight "porkypine". Correct mapping (adopt full 11-limit cuz otherwise it can be confusing as "2.3.5.11" isn't specified) |
||
| Line 4: | Line 4: | ||
| es = | | es = | ||
| ja = | | ja = | ||
}}{{Infobox regtemp|Title=Porcupine | }} | ||
{{Infobox regtemp | |||
| Title = Porcupine | |||
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.7.11 | |||
| Comma basis = [[250/243]] (2.3.5); <br> [[55/54]], [[100/99]] (2.3.5.11) | |||
| Edo join 1 = 7 | Edo join 2 = 15 | |||
| Generator = 11/10 | |||
| Generator tuning = 164 | |||
| Optimization method = CWE | |||
| MOS scales= [[1L 6s]], [[7L 1s]], [[7L 8s]] | |||
| Mapping = 1; -3 -5 6 -4 | |||
| Pergen = (P8, P4/3) | |||
| Color name = Triyoti | |||
| Odd limit 1 = 5 | Mistuning 1 = ? | Complexity 1 = 15 | |||
| Odd limit 2 = (2.3.5.11) 15 |Mistuning 2 = ? |Complexity 2 = 22 | |||
}} | |||
[[File:porcupine.png|thumb|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]] | |||
[[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]] | [[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]] | ||
| Line 11: | Line 27: | ||
One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]). | One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]). | ||
Porcupine can be thought of as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament (sometimes called | Porcupine can be thought of as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the [[6/5]] minor third, due to the extreme flatness of 10/9. This also means that the [[27/20]] acute fourth of the JI diatonic scale is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at -4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy. | ||
It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990 cents), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720 cents, with the best tunings around 711–712 cents, which roughly splits the damage on 7/4 and 9/7. | It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990 cents), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720 cents, with the best tunings around 711–712 cents, which roughly splits the damage on 7/4 and 9/7. | ||