Porcupine temperament modal harmony: Difference between revisions

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Porcupine is one of the few reasonably good 5-limit temperaments, alongside [[meantone]], which possesses a 7-note MOS scale ("porcupine[7]" or "onyx") that is well-supplied with 5-limit major and minor triads (specifically, it has two of each of them). What is perhaps more significant about it is how naturally it extends those triads to involve the 11th harmonic; for instance, the first five consecutive notes of the Lssssss mode approximate the consecutive harmonics 8:9:10:11:12. This makes it a prime candidate for using extended JI-style harmonies in a compact and intuitive scale structure.

It is also significant because it is supported by both [[15edo]] and [[22edo]], the smallest ETs which improve on the 11-limit damage of [[12edo]] (other than [[19edo]]). In those edos, porcupine is an excellent candidate for providing a 7-nominal notational basis and 7-interval-class categorical framework. This is important, because if you want to use 15edo (or to a lesser extent, 22edo) in a somewhat "heptatonic" fashion, where 5/4 and 6/5 are types of "~~third~~", 3/2 is a type of "fifth," and 5-limit major and minor triads share the same "1-3-5" triad shape, thinking of those ETs in terms of porcupine temperament is, in a certain sense, the natural approach. This most transparently leads to notation systems based upon porcupine[7], but 15edo and 22edo also contain a [[zarlino]] scale which can be naturally interpreted as a porcupine scale (what this means for zarlino is that the three sizes of step are equidistant). In fact, *any* heptatonic scale in [[15edo]] or [[22edo]] can be viewed as a MODMOS of porcupine[7].

It is also significant because it is supported by both [[15edo]] and [[22edo]], the smallest ETs which improve on the 11-limit damage of [[12edo]] (other than [[19edo]]). In those edos, porcupine is an excellent candidate for providing a 7-nominal notational basis and 7-interval-class categorical framework. This is important, because if you want to use 15edo (or to a lesser extent, 22edo) in a somewhat "heptatonic" fashion, where 5/4 and 6/5 are types of "thirds", 3/2 is a type of "fifth," and 5-limit major and minor triads share the same "{{nowrap|{{dash|1, 3, 5}}}}" triad shape, thinking of those ETs in terms of porcupine temperament is, in a certain sense, the natural approach. This most transparently leads to notation systems based upon porcupine[7], but 15edo and 22edo also contain a [[zarlino]] scale which can be naturally interpreted as a porcupine scale (what this means for zarlino is that the three sizes of step are equidistant). In fact, *any* heptatonic scale in [[15edo]] or [[22edo]] can be viewed as a [[MODMOS]] of porcupine[7].

~~Lastly, porcupine temperament is extremely notable because~~:

In summary:

# It is the simplest 5-limit temperament that ~~is at least as precise as~~ 12edo~~, and~~ which is not ~~supported by 12edo~~.

# Porcupine is the simplest 5-limit temperament that has less [[Tenney–Euclidean temperament measures#TE_error|tuning error]] than 12edo which 12edo does not support.

# In a [http://x31eq.com/pyscript/pregular.html?limit=5&error=5.0 temperament search] of 5-limit temperaments, using the standard search parameter of 5 ~~cents~~ of error, porcupine is the first temperament in the results that is not supported by 12.

# In a [http://x31eq.com/pyscript/pregular.html?limit=5&error=5.0 temperament search] of 5-limit temperaments, using the standard search parameter of 5{{cent}} of error, porcupine is the first temperament in the results that is not supported by 12.

Porcupine remains the "best" such temperament even if the maximum error is relaxed somewhat~~, until eventually being superceded by~~ [[mavila]] ~~temperament~~, ~~which~~ is ~~simpler in the 5-limit but~~ much ~~less accurate than porcupine temperament due to~~ its extremely flat ~~fifth of~~ 670-680 ~~cents. (using standard harmonic timbres~~).

Porcupine remains the "best" such temperament in the 5-limit even if the maximum error is relaxed somewhat. While [[mavila]] eventually supersedes it due to being simpler, mavila's accuracy is much lower as its fifths are extremely flat ({{nowrap|{{dash|670, 680{{c}}}}}}).

== Brief intro to modal harmony and MODMOSes ==

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 Porcupine is one of the few reasonably good 5-limit temperaments, alongside [[meantone]], which possesses a 7-note MOS scale ("porcupine[7]" or "onyx") that is well-supplied with 5-limit major and minor triads (specifically, it has two of each of them). What is perhaps more significant about it is how naturally it extends those triads to involve the 11th harmonic; for instance, the first five consecutive notes of the Lssssss mode approximate the consecutive harmonics 8:9:10:11:12. This makes it a prime candidate for using extended JI-style harmonies in a compact and intuitive scale structure.
-It is also significant because it is supported by both [[15edo]] and [[22edo]], the smallest ETs which improve on the 11-limit damage of [[12edo]] (other than [[19edo]]). In those edos, porcupine is an excellent candidate for providing a 7-nominal notational basis and 7-interval-class categorical framework. This is important, because if you want to use 15edo (or to a lesser extent, 22edo) in a somewhat "heptatonic" fashion, where 5/4 and 6/5 are types of "third", 3/2 is a type of "fifth," and 5-limit major and minor triads share the same "1-3-5" triad shape, thinking of those ETs in terms of porcupine temperament is, in a certain sense, the natural approach. This most transparently leads to notation systems based upon porcupine[7], but 15edo and 22edo also contain a [[zarlino]] scale which can be naturally interpreted as a porcupine scale (what this means for zarlino is that the three sizes of step are equidistant). In fact, *any* heptatonic scale in [[15edo]] or [[22edo]] can be viewed as a MODMOS of porcupine[7].
+It is also significant because it is supported by both [[15edo]] and [[22edo]], the smallest ETs which improve on the 11-limit damage of [[12edo]] (other than [[19edo]]). In those edos, porcupine is an excellent candidate for providing a 7-nominal notational basis and 7-interval-class categorical framework. This is important, because if you want to use 15edo (or to a lesser extent, 22edo) in a somewhat "heptatonic" fashion, where 5/4 and 6/5 are types of "thirds", 3/2 is a type of "fifth," and 5-limit major and minor triads share the same "{{nowrap|{{dash|1, 3, 5}}}}" triad shape, thinking of those ETs in terms of porcupine temperament is, in a certain sense, the natural approach. This most transparently leads to notation systems based upon porcupine[7], but 15edo and 22edo also contain a [[zarlino]] scale which can be naturally interpreted as a porcupine scale (what this means for zarlino is that the three sizes of step are equidistant). In fact, *any* heptatonic scale in [[15edo]] or [[22edo]] can be viewed as a [[MODMOS]] of porcupine[7].
-Lastly, porcupine temperament is extremely notable because:
+In summary:
-# It is the simplest 5-limit temperament that is at least as precise as 12edo, and which is not supported by 12edo.
+# Porcupine is the simplest 5-limit temperament that has less [[Tenney–Euclidean temperament measures#TE_error|tuning error]] than 12edo which 12edo does not support.
-# In a [http://x31eq.com/pyscript/pregular.html?limit=5&error=5.0 temperament search] of 5-limit temperaments, using the standard search parameter of 5 cents of error, porcupine is the first temperament in the results that is not supported by 12.
+# In a [http://x31eq.com/pyscript/pregular.html?limit=5&error=5.0 temperament search] of 5-limit temperaments, using the standard search parameter of 5{{cent}} of error, porcupine is the first temperament in the results that is not supported by 12.
-Porcupine remains the "best" such temperament even if the maximum error is relaxed somewhat, until eventually being superceded by [[mavila]] temperament, which is simpler in the 5-limit but much less accurate than porcupine temperament due to its extremely flat fifth of 670-680 cents. (using standard harmonic timbres).
+Porcupine remains the "best" such temperament in the 5-limit even if the maximum error is relaxed somewhat. While [[mavila]] eventually supersedes it due to being simpler, mavila's accuracy is much lower as its fifths are extremely flat ({{nowrap|{{dash|670, 680{{c}}}}}}).
 == Brief intro to modal harmony and MODMOSes ==