512/507: Difference between revisions

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{{Infobox Interval

~~| Ratio = 512/507~~

~~| Monzo = 9 -1 0 0 0 -2~~

~~| Cents = 16.98968~~

| Name = tridecimal neutral thirds comma

| Color name =

| Color name = 3uu1, thuthu 1sn,<br>Thuthu comma

~~| FJS name = A1<sub>169~~<~~/sub~~>

| Comma = yes

| ~~Sound~~ =

}}

'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also 2.3.13 subgroup) comma with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].

'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also [[2.3.13 subgroup|2.3.13-subgroup]]) [[comma]] with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].

== Temperaments ==

[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. It can thus be seen as the 2.3.13 analogue of 2.3.11 [[Neutral (temperament)|neutral]], which tempers out [[243/242]] and equates [[11/9]] with its fifth complement, but instead equating 16/13 with 39/32, and preferring sharper fifths over flatter ones.

However, it has significantly more damage than neutral, similarly to how [[archy]] has more damage than [[meantone]]. For example, the neutral third in 2.3.13 CWE tuning is 7.7{{C}} flat of 16/13, but also 4.4{{C}} sharp of 11/9. This makes it ambiguous, but since 11/9 is simpler and closer, the neutral third is likely to be heard as 11/9 rather than 16/13, which shows the higher damage of harmoneutral. Using a tuning sharper than "optimal", such as [[17edo]] or [[61edo|78edo]], makes the tridecimal nature of the intervals more apparent, but only to an extent. Compare that to in neutral, where the neutral third in 2.3.11 CWE tuning is 3.1{{C}} sharp of 11/9 and 8.9{{C}} flat of 16/13, so it is much more clearly 11/9. Harmoneutral also has much greater errors, with a (2.3.13) 13-odd-limit [[minimax]] error of 6.80{{C}} compared to neutral's (2.3.11) 11-odd-limit minimax error of 2.86{{C}}.

Another possible option is to temper out both 512/507 and 243/242, and therefore [[144/143]], leading to [[namo]]. This temperament is generated by a neutral third like harmoneutral and neutral, except all of the neutral intervals are now ambiguously 11-limit or 13-limit.

Tempering it out along with 4096/3993 and 361/352 in 2.3.11.13.19 leads to a temperament proposed to be called ''deltahexacot'' after its ploidacot.

== See also ==

* [[Small comma]]

[[Category:~~13-limit]]~~

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

~~[[Category:Small comma~~]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 512/507
-| Monzo = 9 -1 0 0 0 -2
-| Cents = 16.98968
 | Name = tridecimal neutral thirds comma
-| Color name =
+| Color name = 3uu1, thuthu 1sn,<br>Thuthu comma
-| FJS name = A1<sub>169</sub>
+| Comma = yes
-| Sound =
 }}
-'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also 2.3.13 subgroup) comma with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].
+'''512/507''', the '''tridecimal neutral thirds comma''', is a [[13-limit]] (also [[2.3.13 subgroup|2.3.13-subgroup]]) [[comma]] with a value of roughly 17.0 [[cent]]s. It is significant in [[just intonation]] as the amount by which the tridecimal neutral third, [[16/13]], differs from its [[fifth complement]], [[39/32]], as well as the amount by which a stack of two [[1053/1024]] quartertones fall short of the [[apotome]].
+== Temperaments ==
+[[Tempering out]] this comma equates both tridecimal neutral thirds to the true neutral third [[sqrt(3/2)]], and is proposed to be called '''harmoneutral''' temperament by [[User:VectorGraphics|Vector]] in 2025, as it tempers the subharmonic neutral third together with its fifth complement. It can thus be seen as the 2.3.13 analogue of 2.3.11 [[Neutral (temperament)|neutral]], which tempers out [[243/242]] and equates [[11/9]] with its fifth complement, but instead equating 16/13 with 39/32, and preferring sharper fifths over flatter ones.
+However, it has significantly more damage than neutral, similarly to how [[archy]] has more damage than [[meantone]]. For example, the neutral third in 2.3.13 CWE tuning is 7.7{{C}} flat of 16/13, but also 4.4{{C}} sharp of 11/9. This makes it ambiguous, but since 11/9 is simpler and closer, the neutral third is likely to be heard as 11/9 rather than 16/13, which shows the higher damage of harmoneutral. Using a tuning sharper than "optimal", such as [[17edo]] or [[61edo|78edo]], makes the tridecimal nature of the intervals more apparent, but only to an extent. Compare that to in neutral, where the neutral third in 2.3.11 CWE tuning is 3.1{{C}} sharp of 11/9 and 8.9{{C}} flat of 16/13, so it is much more clearly 11/9. Harmoneutral also has much greater errors, with a (2.3.13) 13-odd-limit [[minimax]] error of 6.80{{C}} compared to neutral's (2.3.11) 11-odd-limit minimax error of 2.86{{C}}.
+Another possible option is to temper out both 512/507 and 243/242, and therefore [[144/143]], leading to [[namo]]. This temperament is generated by a neutral third like harmoneutral and neutral, except all of the neutral intervals are now ambiguously 11-limit or 13-limit.
+Tempering it out along with 4096/3993 and 361/352 in 2.3.11.13.19 leads to a temperament proposed to be called ''deltahexacot'' after its ploidacot.
 == See also ==
 * [[Small comma]]
-[[Category:13-limit]]
+[[Category:Commas named for the intervals they stack]]
-[[Category:Small comma]]