Porcupine temperament modal harmony: Difference between revisions

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__FORCETOC__

=~~Intro To [[~~Porcupine]]=

Porcupine is the temperament splitting 4/3 into three equal parts representing 10/9 and 27/25, so that two steps represent 6/5. This results in a slightly flat 4/3 and a slightly sharp 6/5, meaning that 5/4 is accurately tuned.

== Porcupine scales ==

Porcupine ~~temperament~~ has ~~a number~~ of significant ~~properties that make~~ it ~~somewhat unique in~~ the ~~universe~~ of ~~temperaments~~.

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.

Porcupine is one of the few reasonably good 5-limit temperaments, alongside [[meantone]], which possesses a 7-note MOS scale 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 ~~ETs~~, ~~Porcupine~~ is an excellent candidate for providing a 7-nominal notational basis and 7-interval-class categorical framework ~~for those ETs~~. This is important, because if you want to use 15edo or 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. ~~Furthermore~~, *any* heptatonic scale in [[15edo]] or [[22edo]] can be viewed as a MODMOS of ~~the~~ porcupine-7 ~~MOS~~.

Lastly, porcupine temperament is extremely notable as being, in a certain sense, the simplest 5-limit temperament that is not supported by 12-EDO (and is thus inherently xenharmonic), but is at least as good intonationally. To be precise, porcupine temperament has the following properties:

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

2. In a [http://x31eq.com/cgi-bin/more.cgi?r=2&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.

Porcupine remains the "best" such temperament even if the maximum error is relaxed somewhat, until eventually being superceded by [[mavila]] temperament, which is simpler but much less accurate than porcupine temperament (using standard harmonic timbres).

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).

=Brief Intro to Modal Harmony and MODMOSes=

Porcupine is a temperament completely capable of diverse harmony. ~~Using it's~~ MOS ~~in conjunction with~~ its ~~modmos will~~ allow for some very diverse chordal movement that will still be tonally solid.

== Brief Intro to Modal Harmony and MODMOSes ==

Porcupine is a temperament completely capable of diverse harmony. Its MOS and its MODMOS scales (including zarlino) allow for some very diverse chordal movement that will still be tonally solid.

Because the generator of porcupine is simultaneously 10/9, 11/10, and 12/11, a chord made up of consecutive generators approximates 9:10:11:12; since -6 generators approximates 9/8, the chord 8:9:10:11:12 can be approximated by a series of consecutive 2nds in ~~the~~ 7~~-note MOS~~. Varying subsets of 8:9:10:11:12 provide fruitful ground for building harmonic movement in porcupine.

Because the generator of porcupine is simultaneously 10/9, 11/10, and 12/11, a chord made up of consecutive generators approximates 9:10:11:12; since -6 generators approximates 9/8, the chord 8:9:10:11:12 can be approximated by a series of consecutive 2nds in porcupine[7]. Varying subsets of 8:9:10:11:12 provide fruitful ground for building harmonic movement in porcupine.

Additionally, chords of stacked sevenths of varying sizes may be used to improve voice leading and add variety. Chords without a 'fifth' are recommended for use on the root for modes without a 3/2 from the root. Further, quartal triads, tetrads, and pentads are available in more variety than in Meantone, and make good use of Porcupine's 11-limit consonances.

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=Reasons Why Porcupine Will Make for The Best Modal Harmony Ever=

~~The~~ 7~~-note MOS of porcupine~~ is great, but it lumps all the consonant chords together at one end of the scale (see below). It also fails to contain a complete 11-limit otonal hexad (4:5:6:7:9:11), even though porcupine is not excessively complex in the 11-limit. Thankfully, unlike in meantone[7] where the MODMOSs all reduce the number of consonant triads available, there are several MODMOSs of porcupine that contain an equal or greater number of consonant triads to the MOS (these are highlighted in blue text below). These MODMOSs also conveniently alter the distribution of triads within the scale, providing much-needed variety of possible chord progressions. Even better, some single-alteration MODMOSs make at least one full 11-limit hexad available, meaning 6 of the 7 notes of the scale can be sounded simultaneously and produce a consonant otonal harmony. The fact that porcupine temperament provides not just one 7-note scale but several that are equally-rich in harmonic resources makes it extremely fertile for modal harmony.

Porcupine[7] is great, but it lumps all the consonant chords together at one end of the scale (see below). It also fails to contain a complete 11-limit otonal hexad (4:5:6:7:9:11), even though porcupine is not excessively complex in the 11-limit. Thankfully, unlike in meantone[7] where the MODMOSs all reduce the number of consonant triads available, there are several MODMOSs of porcupine that contain an equal or greater number of consonant triads to the MOS (these are highlighted in blue text below). These MODMOSs also conveniently alter the distribution of triads within the scale, providing much-needed variety of possible chord progressions. Even better, some single-alteration MODMOSs make at least one full 11-limit hexad available, meaning 6 of the 7 notes of the scale can be sounded simultaneously and produce a consonant otonal harmony. The fact that porcupine temperament provides not just one 7-note scale but several that are equally-rich in harmonic resources makes it extremely fertile for modal harmony.

=Porcupine[7] and its MODMOSs=

@@ Line 1: / Line 1: @@
 __FORCETOC__
-=Intro To [[Porcupine]]=
+Porcupine is the temperament splitting 4/3 into three equal parts representing 10/9 and 27/25, so that two steps represent 6/5. This results in a slightly flat 4/3 and a slightly sharp 6/5, meaning that 5/4 is accurately tuned.
+== Porcupine scales ==
 {{Todo|rework|inline=1|text=Explain in a non-MOS-centric way}}
-Porcupine temperament has a number of significant properties that make it somewhat unique in the universe of temperaments.
+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.
-Porcupine is one of the few reasonably good 5-limit temperaments, alongside [[meantone]], which possesses a 7-note MOS scale 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 ETs, Porcupine is an excellent candidate for providing a 7-nominal notational basis and 7-interval-class categorical framework for those ETs. This is important, because if you want to use 15edo or 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. Furthermore, *any* heptatonic scale in [[15edo]] or [[22edo]] can be viewed as a MODMOS of the porcupine-7 MOS.
 Lastly, porcupine temperament is extremely notable as being, in a certain sense, the simplest 5-limit temperament that is not supported by 12-EDO (and is thus inherently xenharmonic), but is at least as good intonationally. To be precise, porcupine temperament has the following properties:
 . It is the simplest 5-limit temperament that is at least as precise as 12-EDO, and which is not supported by 12-EDO.
 . In a [http://x31eq.com/cgi-bin/more.cgi?r=2&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.
-Porcupine remains the "best" such temperament even if the maximum error is relaxed somewhat, until eventually being superceded by [[mavila]] temperament, which is simpler but much less accurate than porcupine temperament (using standard harmonic timbres).
+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).
-=Brief Intro to Modal Harmony and MODMOSes=
-Porcupine is a temperament completely capable of diverse harmony. Using it's MOS in conjunction with its modmos will allow for some very diverse chordal movement that will still be tonally solid.
+== Brief Intro to Modal Harmony and MODMOSes ==
+Porcupine is a temperament completely capable of diverse harmony. Its MOS and its MODMOS scales (including zarlino) allow for some very diverse chordal movement that will still be tonally solid.
-Because the generator of porcupine is simultaneously 10/9, 11/10, and 12/11, a chord made up of consecutive generators approximates 9:10:11:12; since -6 generators approximates 9/8, the chord 8:9:10:11:12 can be approximated by a series of consecutive 2nds in the 7-note MOS. Varying subsets of 8:9:10:11:12 provide fruitful ground for building harmonic movement in porcupine.
+Because the generator of porcupine is simultaneously 10/9, 11/10, and 12/11, a chord made up of consecutive generators approximates 9:10:11:12; since -6 generators approximates 9/8, the chord 8:9:10:11:12 can be approximated by a series of consecutive 2nds in porcupine[7]. Varying subsets of 8:9:10:11:12 provide fruitful ground for building harmonic movement in porcupine.
 Additionally, chords of stacked sevenths of varying sizes may be used to improve voice leading and add variety. Chords without a 'fifth' are recommended for use on the root for modes without a 3/2 from the root. Further, quartal triads, tetrads, and pentads are available in more variety than in Meantone, and make good use of Porcupine's 11-limit consonances.
@@ Line 26: / Line 29: @@
 =Reasons Why Porcupine Will Make for The Best Modal Harmony Ever=
-The 7-note MOS of porcupine is great, but it lumps all the consonant chords together at one end of the scale (see below). It also fails to contain a complete 11-limit otonal hexad (4:5:6:7:9:11), even though porcupine is not excessively complex in the 11-limit. Thankfully, unlike in meantone[7] where the MODMOSs all reduce the number of consonant triads available, there are several MODMOSs of porcupine that contain an equal or greater number of consonant triads to the MOS (these are highlighted in blue text below). These MODMOSs also conveniently alter the distribution of triads within the scale, providing much-needed variety of possible chord progressions. Even better, some single-alteration MODMOSs make at least one full 11-limit hexad available, meaning 6 of the 7 notes of the scale can be sounded simultaneously and produce a consonant otonal harmony. The fact that porcupine temperament provides not just one 7-note scale but several that are equally-rich in harmonic resources makes it extremely fertile for modal harmony.
+Porcupine[7] is great, but it lumps all the consonant chords together at one end of the scale (see below). It also fails to contain a complete 11-limit otonal hexad (4:5:6:7:9:11), even though porcupine is not excessively complex in the 11-limit. Thankfully, unlike in meantone[7] where the MODMOSs all reduce the number of consonant triads available, there are several MODMOSs of porcupine that contain an equal or greater number of consonant triads to the MOS (these are highlighted in blue text below). These MODMOSs also conveniently alter the distribution of triads within the scale, providing much-needed variety of possible chord progressions. Even better, some single-alteration MODMOSs make at least one full 11-limit hexad available, meaning 6 of the 7 notes of the scale can be sounded simultaneously and produce a consonant otonal harmony. The fact that porcupine temperament provides not just one 7-note scale but several that are equally-rich in harmonic resources makes it extremely fertile for modal harmony.
 =Porcupine[7] and its MODMOSs=