159edo: Difference between revisions

Line 32:

=== Additional Properties of 159-tone equal temperament ===

The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and similarly this system can approximate the sounds of other systems such as [[10edo]], [[13edo]], [[22edo]] and [[31edo]]. Furthermore, the step size of 159edo is simultaneously above the average peak melodic JND and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another, even as it also allows one to also control the bandwidth of certain sounds.

The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and similarly this system can approximate the sounds of other systems such as [[10edo]], [[13edo]], [[22edo]] and [[31edo]]. Furthermore, the step size of 159edo is simultaneously above the average peak melodic JND and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another, even as it also allows one to also control the bandwidth of certain sounds. As a result of tempering out some of the commas, it allows [[essentially tempered chord]]s including [[marveltwin chords]], [[gentle chords]], [[keenanismic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit.

=== MOSes and other scales ===

Line 1,381:

=== Commas ===

Assuming the [[val]] {{val|159 252 369 446 550 588 650 }}, 159tet [[tempers out]] the following [[comma]]s.

Assuming the [[val]] {{val|159 252 369 446 550 588 650 }}, 159tet [[tempers out]] the following [[comma]]s in the 17-limit.

{| class="commatable wikitable center-all left-3 right-4 left-6"

Line 1,777:

| [[Sparkisma]]

|-

|}

* In the 17-limit, it tempers out [[273/272]], [[375/374]], [[595/594]], [[715/714]], [[833/832]], [[936/935]], [[1089/1088]], [[1701/1700]], [[2025/2023]], [[2431/2430]], 8624/8619, 11271/11264, 15379/15300, [[24576/24565]], 57375/57344 and [[248897/248832]]. In addition, it also tempers out the [[flashma]] and the [[sparkisma]], the latter of which is also tempered out by 53edo despite 53edo having a different mapping for 17.

* In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas.

* In the 23-limit, with the 19-limit skipped, this system is known to temper out 392/391, 460/459, 507/506, 529/528, 897/896, 1105/1104, 1288/1287, 2024/2023, 2025/2024, and 2646/2645 among others.

~~As a result of tempering out some of~~ the ~~commas~~, it ~~allows [[essentially tempered chord]]s including [[marveltwin chords]]~~, ~~[[gentle chords]], [[keenanismic chords]], [[werckismic chords]], and [[sinbadmic chords]]~~ in the 13-~~odd~~-limit, ~~in addition to [[island chords]] and [[nicolic chords]] in~~ the ~~15-odd~~-limit.

In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas. In the 23-limit, with the 19-limit skipped, this system is known to temper out 392/391, 460/459, 507/506, 529/528, 897/896, 1105/1104, 1288/1287, 2024/2023, 2025/2024, and 2646/2645 among others.

=== Rank-2 temperaments ===

@@ Line 32: / Line 32: @@
 === Additional Properties of 159-tone equal temperament ===
-The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and similarly this system can approximate the sounds of other systems such as [[10edo]], [[13edo]], [[22edo]] and [[31edo]].  Furthermore, the step size of 159edo is simultaneously above the average peak melodic JND and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another, even as it also allows one to also control the bandwidth of certain sounds.
+The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and similarly this system can approximate the sounds of other systems such as [[10edo]], [[13edo]], [[22edo]] and [[31edo]].  Furthermore, the step size of 159edo is simultaneously above the average peak melodic JND and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another, even as it also allows one to also control the bandwidth of certain sounds.  As a result of tempering out some of the commas, it allows [[essentially tempered chord]]s including [[marveltwin chords]], [[gentle chords]], [[keenanismic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit.
 === MOSes and other scales ===
@@ Line 1,381: / Line 1,381: @@
 === Commas ===
-Assuming the [[val]] {{val|159 252 369 446 550 588 650 }}, 159tet [[tempers out]] the following [[comma]]s.
+Assuming the [[val]] {{val|159 252 369 446 550 588 650 }}, 159tet [[tempers out]] the following [[comma]]s in the 17-limit.
 {| class="commatable wikitable center-all left-3 right-4 left-6"
@@ Line 1,777: / Line 1,777: @@
 | [[Sparkisma]]
 |-
 |}
-* In the 17-limit, it tempers out [[273/272]], [[375/374]], [[595/594]], [[715/714]], [[833/832]], [[936/935]], [[1089/1088]], [[1701/1700]], [[2025/2023]], [[2431/2430]], 8624/8619, 11271/11264, 15379/15300, [[24576/24565]], 57375/57344 and [[248897/248832]]. In addition, it also tempers out the [[flashma]] and the [[sparkisma]], the latter of which is also tempered out by 53edo despite 53edo having a different mapping for 17.
-* In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas.
-* In the 23-limit, with the 19-limit skipped, this system is known to temper out 392/391, 460/459, 507/506, 529/528, 897/896, 1105/1104, 1288/1287, 2024/2023, 2025/2024, and 2646/2645 among others.
-As a result of tempering out some of the commas, it allows [[essentially tempered chord]]s including [[marveltwin chords]], [[gentle chords]], [[keenanismic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit.
+In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas.  In the 23-limit, with the 19-limit skipped, this system is known to temper out 392/391, 460/459, 507/506, 529/528, 897/896, 1105/1104, 1288/1287, 2024/2023, 2025/2024, and 2646/2645 among others.
 === Rank-2 temperaments ===