41edo: Difference between revisions

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=== Prime harmonics ===

{{Harmonics in equal|41|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 41edo (continued)}}

{{Harmonics in equal|41|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 41edo (continued)}}

~~=== Mappings ===~~

~~A step of 41edo is close and consistently mapped to [[64/63]], the septimal comma.~~

=== As a tuning of other temperaments ===

41edo can be seen as a tuning of the [[garibaldi temperament|garibaldi]] temperament<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref>, the [[magic]] temperament, the [[superkleismic]] temperament and multiple temperaments in the [[tetracot family]].

Various 13-limit [[magic family|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]].

41edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of ~~tetracot~~[7] can also be more accurately interpreted as [[21/19]] – which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.

41edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of Tetracot[7] can also be more accurately interpreted as [[21/19]] – which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.

A step of 41edo is close and consistently mapped to [[64/63]], the septimal comma.

=== Subsets and supersets ===

41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]].

[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ~~±1~~ [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo.

[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo.

== Intervals ==

@@ Line 17: / Line 17: @@
 === Prime harmonics ===
-{{Harmonics in equal|41|columns=12}}
+{{Harmonics in equal|41|columns=9}}
-{{Harmonics in equal|41|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 41edo (continued)}}
+{{Harmonics in equal|41|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 41edo (continued)}}
-=== Mappings ===
-A step of 41edo is close and consistently mapped to [[64/63]], the septimal comma.
+=== As a tuning of other temperaments ===
 edo can be seen as a tuning of the [[garibaldi temperament|garibaldi]] temperament<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref>, the [[magic]] temperament, the [[superkleismic]] temperament and multiple temperaments in the [[tetracot family]].
 Various 13-limit [[magic family|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]].
-edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of tetracot[7] can also be more accurately interpreted as [[21/19]] – which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.
+edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of Tetracot[7] can also be more accurately interpreted as [[21/19]] – which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.
+A step of 41edo is close and consistently mapped to [[64/63]], the septimal comma.
 === Subsets and supersets ===
 edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]].
-[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning &#177;1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo.
+[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo.
 == Intervals ==