Talk:Meantone: Difference between revisions

Line 65:

:::::: I agree with that. I am just saying that septimal meantone has not been completely absent historically. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 23:26, 25 August 2024 (UTC)

== Wrong Septimal Comma in tuning spectrum table? ==

I scratched my head for a while about the lines having septimal comma fractions in the tuning spectrum table. For a while I thought they were just there to have a spectrum of what some commnn foreign commas do when used to flatten the fifth, but then it occurred to me that whoever put them there was probably really trying to do the right thing for septimal meantone, but didn't understand which 7-limit comma to use. As observed in the current table, flattening the fifth by fractions of an Archytas septimal comma gives the Eigenmonzos listed (apart from one which was off by a typo that I fixed earlier), but they aren't very good Eigenmonzos, for the most part except as educational examples of why one should flatten fifths by fractions of a syntonic comma but sharpen fifths by fractions of an Arcyhtas comma and not the other way around (admittedly, they serve that purpose quite well). (For an additional use, see below.)

The comma the author was probably after is actually [[Harrison's comma]] = [-13 10 0 -1⟩ (as listed in [[Meantone family]], which does not have a "septimal meantone comma" alternate name (maybe it should?), although the page text does mention septimal meantone, and currently does not get any mention in the [[Septimal comma]] disambiguation page (maybe it should?). So if I put in some examples in the form of a hypothetical set of rows of the table (many lines from the actual table omitted to save space here):

{| class="wikitable center-all left-4"

! Edo<br>Generator

! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]

! Generator<br>(¢)

! Comments

|-

|

| [[27/20]]

| 680.449

| Full comma (syntonic comma; from here onwards "comma" without an adjective refers to syntonic comma)

|-

| '''[[7edo|4\7]]'''

|

| '''685.714'''

| '''Lower bound of 5-odd-limit diamond monotone'''

|-

|

| [[567/512]]

| 688.323

| 1/2 septimal comma

|-

|

| [[896/729]]

| 689.274

| 1/4 Harrison's comma

|-

|

| {{monzo| 16 -10 }}

| 690.225

| 1/2 Pythagorean comma, as M2.

|-

|

| [[51/38]]

| 690.603

| As P4.

|-

| [[33edo|19\33]]

|

| 690.909

| 33cddd val

|-

|

| {{monzo| -19 9 0 2 }}

| 691.049

| 2/5 septimal comma

|-

|

| [[9/5]]

| 691.202

| [[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9, a.k.a. lower bound of 9-odd-limit diamond tradeoff

|-

|

| [[243/224]]

| 691.810

| 1/5 Harrison's comma

|-

|

| [[112/81]]

| 693.501

| 1/6 Harrison's comma

|-

|

| [[28/27]]

| 694.709

| 1/7 Harrison's comma

|-

| '''[[19edo|11\19]]'''

|

| '''694.737'''

| '''Lower bound of 7- and 9-odd-limit diamond monotone'''

|-

|

| [[5/3]]

| 694.786

| [[1/3-comma meantone|1/3 comma]]

|-

|

| [[9/7]]

| 695.614

| 1/8 Harrison's comma

|-

|

| [[7/6]]

| 696.319

| 1/9 Harrison's comma

|-

|

| [[7/4]]

| 696.883

| 1/10 Harrison's comma

|-

|

| [[21/16]]

| 697.344

| 1/11 Harrison's comma

|-

|

| [[64/63]]

| 697.728

| 1/12 Harrison's comma

|}

As you can see, only the first Harrison's comma entry has a bad Eigenmonzo (that also does not currently have a Xenharmonic Wiki page); all the rest have Xenharmonic Wiki pages, and they keep getting simpler until 1/10 Harrison's comma, after which they get more complex again, with a pass through [[64/63]] as an Eigenmonzo rather than a tempering comma fraction (but not the same as the line with the 64/63 Eigenmonzo at a fifth of 702.272¢ that was removed on 2024-07-30T10:10:27 -- still haven't figured out where that one came from). Indeed, the actual table appears to contain a subset of the above entries, but with no explanation in the comments column.

While checking the above, I noticed that flattening the fifth by 1/7 of the Archytas septimal comma does have the additional use of generating as its Eigenmonzo the flattone comma [-17 9 0 1⟩ (does this comma have an official name? -- it doesn't seem to have its own page). Inspired by this, I also looked to see if other listed Archytas septimal comma fractions produced as their Eigenmonzos the Harrison's comma [-13 10 0 -1⟩ or the flattertone comma [-24 17 0 -1⟩, but I did not see either of these, so those must require Archytas comma fractions that are not listed (I have not yet tried to the math to figure out what they are).

So I would recommend integrating the above table into the septimal meantone tuning table, since Harrison's comma is the comma other than the syntonic comma that really fits in with the septimal meantone tuning spectrum, but maybe leaving in Archytas septimal comma fractional flattenings that produce as their Eigenmonzos commas that are used in 7-limit meantone extensions.

As an alternative, I could certainly go along with the idea of splitting septimal meantone into its own article with its own tuning spectrum table (as Bcmills has recommended) that has the 7-limit comma fractions (as has been done for [[flattone]], and reserving the tuning spectrum table in the main meantone article for 3-limit and 5-limit comma fractions and the EDO tunings in their vicinity. This table could reference the other tables, for instance noting at 19EDO that tunings flatter than this use the flattone 7-limit extension, while tunings sharper than this use septimal meantone (and like wise for 12EDO as the intersection between septimal meantone and dominant, and likewise for 26EDO as the intersection between flattone and flattertone -- and even likewise for 5EDO as the intersection between dominant and sharptone, if one accepts tunings with a negative size of diatonic semitone in the way-out northwest wilderness of the grand unified 5-limit meantone spectrum).

[[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:28, 29 August 2024 (UTC)

@@ Line 65: / Line 65: @@
 :::::: I agree with that.  I am just saying that septimal meantone has not been completely absent historically.  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 23:26, 25 August 2024 (UTC)
+== Wrong Septimal Comma in tuning spectrum table? ==
+I scratched my head for a while about the lines having septimal comma fractions in the tuning spectrum table.  For a while I thought they were just there to have a spectrum of what some commnn foreign commas do when used to flatten the fifth, but then it occurred to me that whoever put them there was probably really trying to do the right thing for septimal meantone, but didn't understand which 7-limit comma to use.  As observed in the current table, flattening the fifth by fractions of an Archytas septimal comma gives the Eigenmonzos listed (apart from one which was off by a typo that I fixed earlier), but they aren't very good Eigenmonzos, for the most part except as educational examples of why one should flatten fifths by fractions of a syntonic comma but sharpen fifths by fractions of an Arcyhtas comma and not the other way around (admittedly, they serve that purpose quite well).  (For an additional use, see below.)
+The comma the author was probably after is actually [[Harrison's comma]] = [-13 10 0 -1⟩ (as listed in [[Meantone family]], which does not have a "septimal meantone comma" alternate name (maybe it should?), although the page text does mention septimal meantone, and currently does not get any mention in the [[Septimal comma]] disambiguation page (maybe it should?).  So if I put in some examples in the form of a hypothetical set of rows of the table (many lines from the actual table omitted to save space here):
+{| class="wikitable center-all left-4"
+! Edo<br>Generator
+! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
+! Generator<br>(¢)
+! Comments
+|-
+|
+| [[27/20]]
+| 680.449
+| Full comma (syntonic comma; from here onwards "comma" without an adjective refers to syntonic comma)
+|-
+| '''[[7edo|4\7]]'''
+|
+| '''685.714'''
+| '''Lower bound of 5-odd-limit diamond monotone'''
+|-
+|
+| [[567/512]]
+| 688.323
+| 1/2 septimal comma
+|-
+|
+| [[896/729]]
+| 689.274
+| 1/4 Harrison's comma
+|-
+|
+| {{monzo| 16 -10 }}
+| 690.225
+| 1/2 Pythagorean comma, as M2.
+|-
+|
+| [[51/38]]
+| 690.603
+| As P4.
+|-
+| [[33edo|19\33]]
+|
+| 690.909
+| 33cddd val
+|-
+|
+| {{monzo| -19 9 0 2 }}
+| 691.049
+| 2/5 septimal comma
+|-
+|
+| [[9/5]]
+| 691.202
+| [[1/2-comma meantone|1/2 comma]], tunings flatter than this do not fit the original sense of meantone, since their whole tones are no longer between 9/8 and 10/9, a.k.a. lower bound of 9-odd-limit diamond tradeoff
+|-
+|
+| [[243/224]]
+| 691.810
+| 1/5 Harrison's comma
+|-
+|
+| [[112/81]]
+| 693.501
+| 1/6 Harrison's comma
+|-
+|
+| [[28/27]]
+| 694.709
+| 1/7 Harrison's comma
+|-
+| '''[[19edo|11\19]]'''
+|
+| '''694.737'''
+| '''Lower bound of 7- and 9-odd-limit diamond monotone'''
+|-
+|
+| [[5/3]]
+| 694.786
+| [[1/3-comma meantone|1/3 comma]]
+|-
+|
+| [[9/7]]
+| 695.614
+| 1/8 Harrison's comma
+|-
+|
+| [[7/6]]
+| 696.319
+| 1/9 Harrison's comma
+|-
+|
+| [[7/4]]
+| 696.883
+| 1/10 Harrison's comma
+|-
+|
+| [[21/16]]
+| 697.344
+| 1/11 Harrison's comma
+|-
+|
+| [[64/63]]
+| 697.728
+| 1/12 Harrison's comma
+|}
+As you can see, only the first Harrison's comma entry has a bad Eigenmonzo (that also does not currently have a Xenharmonic Wiki page); all the rest have Xenharmonic Wiki pages, and they keep getting simpler until 1/10 Harrison's comma, after which they get more complex again, with a pass through [[64/63]] as an Eigenmonzo rather than a tempering comma fraction (but not the same as the line with the 64/63 Eigenmonzo at a fifth of 702.272¢ that was removed on 2024-07-30T10:10:27 -- still haven't figured out where that one came from).  Indeed, the actual table appears to contain a subset of the above entries, but with no  explanation in the comments column.
+While checking the above, I noticed that flattening the fifth by 1/7 of the Archytas septimal comma does have the additional use of generating as its Eigenmonzo the flattone comma [-17 9 0 1⟩ (does this comma have an official name? -- it doesn't seem to have its own page).  Inspired by this, I also looked to see if other listed Archytas septimal comma fractions produced as their Eigenmonzos the Harrison's comma [-13 10 0 -1⟩ or the flattertone comma [-24 17 0 -1⟩, but I did not see either of these, so those must require Archytas comma fractions that are not listed (I have not yet tried to the math to figure out what they are).
+So I would recommend integrating the above table into the septimal meantone tuning table, since Harrison's comma is the comma other than the syntonic comma that really fits in with the septimal meantone tuning spectrum, but maybe leaving in Archytas septimal comma fractional flattenings that produce as their Eigenmonzos commas that are used in 7-limit meantone extensions.
+As an alternative, I could certainly go along with the idea of splitting septimal meantone into its own article with its own tuning spectrum table (as Bcmills has recommended) that has the 7-limit comma fractions (as has been done for [[flattone]], and reserving the tuning spectrum table in the main meantone article for 3-limit and 5-limit comma fractions and the EDO tunings in their vicinity.  This table could reference the other tables, for instance noting at 19EDO that tunings flatter than this use the flattone 7-limit extension, while tunings sharper than this use septimal meantone (and like wise for 12EDO as the intersection between septimal meantone and dominant, and likewise for 26EDO as the intersection between flattone and flattertone -- and even likewise for 5EDO as the intersection between dominant and sharptone, if one accepts tunings with a negative size of diatonic semitone in the way-out northwest wilderness of the grand unified 5-limit meantone spectrum).
+[[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:28, 29 August 2024 (UTC)