2601/2600: Difference between revisions

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{{Infobox Interval

~~| Ratio = 2601/2600~~

| Name = sextantonisma

~~| Monzo = -3 2 -2 0 0 -1 2~~

| Color name = 17oo3ugg2, sosothugugu 2nd,<br>Sosothugugu comma

~~| Cents = 0.66573~~

| Comma = yes

| Name = ~~septendecimal sixth-tones comma~~

| Color name =

| ~~Sound~~ =

}}

'''2601/2600''' is a [[17-limit]] (also 2.3.5.13.17 subgroup) [[superparticular]] [[comma]] measuring about 0.67 [[cent]]s. It may be properly described as the ''septendecimal sixth-tones comma'', since it is the difference between [[51/50]] and [[52/51]], the two 17-limit sixth-tones. ~~Another prominent identity is~~ the little gap between [[18/13]] and a stack of two [[20/17]]'s.

'''2601/2600''', the '''sextantonisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.3.5.13.17-[[subgroup]]) [[superparticular]] [[comma]] measuring about 0.666 [[cent]]s. It may be properly described as the ''septendecimal sixth-tones comma'', since it is the difference between [[51/50]] and [[52/51]], the two 17-limit sixth-tones. It also represents the little gap between [[18/13]] and a stack of two [[20/17]]'s.

In terms of commas, it is the difference between [[289/288]] and [[325/324]]~~, between~~ [[833/832]] and [[1225/1224]]~~, between~~ [[1156/1155]] and [[2080/2079]]~~, between~~ [[1275/1274]] and [[2500/2499]]~~, between~~ [[1701/1700]] and [[4914/4913]]~~, and between~~ [[2401/2400]] and [[31213/31212]].

== Commatic relations ==

In terms of commas, it is the difference between the following pairs:

* [[289/288]] and [[325/324]] *

* [[561/560]] and [[715/714]]

* [[833/832]] and [[1225/1224]]

* [[1156/1155]] and [[2080/2079]]

* [[1275/1274]] and [[2500/2499]]

* [[1701/1700]] and [[4914/4913]]

* [[2401/2400]] and [[31213/31212]]

* [[2431/2430]] and [[37180/37179]]

<nowiki/>* relation within the 2.3.5.13.17 subgroup

== Temperaments ==

Tempering out this comma results in [[26/25]] ~~being~~ split into two equal parts, each representing 51/50~52/51, and ~~enables~~ the ~~related essentially tempered chords~~.

[[Tempering out]] this comma in the 17-limit results in the rank-6 '''sextantonismic''' temperament, or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic''' temperament. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and [[sextantonismic chords]] are enabled.

If [[140625/140608]] is also added to the comma list, the sixth-tone above becomes literally a sixth of [[9/8]] and is tuned exactly middle of 51/50 and 52/51. This temperament, however, strongly suggests also tempering out [[9801/9800]] and/or [[12376/12375]] since 2601/2600 = (9801/9800)⋅(12376/12375)<sup>2</sup>(140625/140608).

=== Sextantonic ===

[[Subgroup]]: 2.3.5.13.17

{{Mapping|legend=2| 1 0 0 2 1 | 0 1 0 0 -1 | 0 0 0 0 1 | 0 0 0 2 1 }}

: mapping generators: ~2, ~3, ~5, ~51/20

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0165{{c}}, ~3/2 = 701.9109{{c}}, ~5/4 = 386.3400{{c}}, ~51/40 = 420.2767{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9128{{c}}, ~5/4 = 386.3554{{c}}, ~51/40 = 420.2871{{c}}

{{Optimal ET sequence|legend=1| 22f, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848, 3609g, 4069g, 4457g, 4917gg }}

[[Badness]] (Sintel): 0.0654

=== Sextantonismic ===

[[Subgroup]]: 2.3.5.7.11.13.17

[[Mapping]]: <br>

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || 1 || 2 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || -1 || ],

|-

| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 1 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 0 || 2 || 1 || ]]

|}

: mapping generators: ~2, ~3, ~5, ~7, ~11, ~51/20

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0165{{c}}, ~3/2 = 701.9109{{c}}, ~5/4 = 386.3400{{c}}, ~7/4 = 968.7969{{c}}, ~11/8 = 551.2685{{c}}, ~51/40 = 420.2767{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9128{{c}}, ~5/4 = 386.3554{{c}}, ~7/4 = 968.8019{{c}}, ~11/8 = 551.2883{{c}}, ~51/40 = 420.2871{{c}}

{{Optimal ET sequence|legend=1| 17cg, 22f, 26, 29g, 31, 38df, 43, 46, 60e, 65d, 68, 72, 103, 111, 140, 171, 183, 217, 243e, 282, 311, 354, 400, 422, 460, 494, 742, 814, 954, 1236, 1696, 2190g, 4069g }}<nowiki>*</nowiki>

<nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]]

[[Badness]] (Sintel): 0.689

== Etymology ==

The sextantonisma was named by [[Flora Canou]] in 2023. It is a contraction of ''sixth-tones comma'' into a single word consisting of Latin ''sextans'' ("sixth") and ''tonus'' ("tone"). This comma was chosen as the sixth-tones comma because the sixth-tones it separates lie in the middle of the harmonic series segment of sixth-tones, 48::54.

== See also ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

[[Category:~~17-limit~~]]

[[Category:Sextantonismic]]

[[Category:~~Unnoticeable comma]]~~

[[Category:Commas named for the intervals they stack]]

~~[[Category:Superparticular~~]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 2601/2600
+| Name = sextantonisma
-| Monzo = -3 2 -2 0 0 -1 2
+| Color name = 17oo3ugg2, sosothugugu 2nd,<br>Sosothugugu comma
-| Cents = 0.66573
+| Comma = yes
-| Name = septendecimal sixth-tones comma
-| Color name =
-| Sound =
 }}
-'''2601/2600''' is a [[17-limit]] (also 2.3.5.13.17 subgroup) [[superparticular]] [[comma]] measuring about 0.67 [[cent]]s. It may be properly described as the ''septendecimal sixth-tones comma'', since it is the difference between [[51/50]] and [[52/51]], the two 17-limit sixth-tones. Another prominent identity is the little gap between [[18/13]] and a stack of two [[20/17]]'s.
+'''2601/2600''', the '''sextantonisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.3.5.13.17-[[subgroup]]) [[superparticular]] [[comma]] measuring about 0.666 [[cent]]s. It may be properly described as the ''septendecimal sixth-tones comma'', since it is the difference between [[51/50]] and [[52/51]], the two 17-limit sixth-tones. It also represents the little gap between [[18/13]] and a stack of two [[20/17]]'s.
-In terms of commas, it is the difference between [[289/288]] and [[325/324]], between [[833/832]] and [[1225/1224]], between [[1156/1155]] and [[2080/2079]], between [[1275/1274]] and [[2500/2499]], between [[1701/1700]] and [[4914/4913]], and between [[2401/2400]] and [[31213/31212]].
+== Commatic relations ==
+In terms of commas, it is the difference between the following pairs:
+* [[289/288]] and [[325/324]] *
+* [[561/560]] and [[715/714]]
+* [[833/832]] and [[1225/1224]]
+* [[1156/1155]] and [[2080/2079]]
+* [[1275/1274]] and [[2500/2499]]
+* [[1701/1700]] and [[4914/4913]]
+* [[2401/2400]] and [[31213/31212]]
+* [[2431/2430]] and [[37180/37179]]
+<nowiki/>* relation within the 2.3.5.13.17 subgroup
 == Temperaments ==
-Tempering out this comma results in [[26/25]] being split into two equal parts, each representing 51/50~52/51, and enables the related essentially tempered chords.
+[[Tempering out]] this comma in the 17-limit results in the rank-6 '''sextantonismic''' temperament, or in the 2.3.5.13.17 subgroup, the rank-4 '''sextantonic''' temperament. In either case [[26/25]] is split into two equal parts, each representing 51/50~52/51, and [[sextantonismic chords]] are enabled.
+If [[140625/140608]] is also added to the comma list, the sixth-tone above becomes literally a sixth of [[9/8]] and is tuned exactly middle of 51/50 and 52/51. This temperament, however, strongly suggests also tempering out [[9801/9800]] and/or [[12376/12375]] since 2601/2600 = (9801/9800)⋅(12376/12375)<sup>2</sup>(140625/140608).
+=== Sextantonic ===
+[[Subgroup]]: 2.3.5.13.17
+{{Mapping|legend=2| 1 0 0 2 1 | 0 1 0 0 -1 | 0 0 0 0 1 | 0 0 0 2 1 }}
+: mapping generators: ~2, ~3, ~5, ~51/20
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0165{{c}}, ~3/2 = 701.9109{{c}}, ~5/4 = 386.3400{{c}}, ~51/40 = 420.2767{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9128{{c}}, ~5/4 = 386.3554{{c}}, ~51/40 = 420.2871{{c}}
+{{Optimal ET sequence|legend=1| 22f, 26, 31, 34, 72, 106, 137, 171, 183, 217, 277, 354, 388, 460, 494, 677, 814, 848, 3609g, 4069g, 4457g, 4917gg }}
+[[Badness]] (Sintel): 0.0654
+=== Sextantonismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 1 || 2 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 2 || 1 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11, ~51/20
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0165{{c}}, ~3/2 = 701.9109{{c}}, ~5/4 = 386.3400{{c}}, ~7/4 = 968.7969{{c}}, ~11/8 = 551.2685{{c}}, ~51/40 = 420.2767{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9128{{c}}, ~5/4 = 386.3554{{c}}, ~7/4 = 968.8019{{c}}, ~11/8 = 551.2883{{c}}, ~51/40 = 420.2871{{c}}
+{{Optimal ET sequence|legend=1| 17cg, 22f, 26, 29g, 31, 38df, 43, 46, 60e, 65d, 68, 72, 103, 111, 140, 171, 183, 217, 243e, 282, 311, 354, 400, 422, 460, 494, 742, 814, 954, 1236, 1696, 2190g, 4069g }}<nowiki>*</nowiki>
+<nowiki>*</nowiki> [[optimal patent val]]: [[2932edo|2932]]
+[[Badness]] (Sintel): 0.689
+== Etymology ==
+The sextantonisma was named by [[Flora Canou]] in 2023. It is a contraction of ''sixth-tones comma'' into a single word consisting of Latin ''sextans'' ("sixth") and ''tonus'' ("tone"). This comma was chosen as the sixth-tones comma because the sixth-tones it separates lie in the middle of the harmonic series segment of sixth-tones, 48::54.
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
-[[Category:17-limit]]
+[[Category:Sextantonismic]]
-[[Category:Unnoticeable comma]]
+[[Category:Commas named for the intervals they stack]]
-[[Category:Superparticular]]