Talk:Quarter-comma meantone

message: I'm going to incorporate what is here into meantone temperament and make this a redirect there. I don't think it's a good idea to put quarter comma on its own page like this, especially when one considers that when people say "meantone temperament" they often mean quarter comma specifically. I suppose one day we might have lots and lots to say about each system of meantone (2/7 comma, 1/3 comma, etc) and then they can each have their own page, but I think for now, it's best that they all live in the same place. --Camembert
:I think that time has come. All the mathematical stuff about quarter-comma was cluttering up Meantone and making the stuff about meantone in general harder to find. I think it's fine to have separate articles as long as there are prominent links between them. —Keenan Pepper 20:40, 19 November 2005 (UTC)
thread_title: Redirect

message: Midis included. Hyacinth (talk) 15:59, 21 March 2010 (UTC)
thread_title: Reqmusic

message: I believe that Aaron defined the scale using D as the base note, then moving 5 fifths down to obtain E♭
and 6 fifths up to obtain G♯. This has two important consequences:
* The method is identical to Pythagorean tuning, except for the tampered size of the fifth.
* The most frequent value for each type of interval is observed when the interval starts for D.
* The wolf fifth starts from G♯
It does not matter to me that the method proposed in this article is based on a combination of S's and/or X's, rather than a much simpler stack of fifths. These methods are perfectly equivalent. I am not really concerned about that (the only problem, that can be easily solved, is that the article does not explain why that particular sequence of S's and X's was chosen; the reason is obvious: the sequence is automatically defined when you use the stack of fifths, otherwise it seems completely arbitrary)
frame|right|Approximate size of intervals in D-based quarter-comma meantone, as defined by Aron in 1523
Here is the size of all intervals in quarter-comma meantone (see table on the right). As you can see, the only row which is completely white is the first one, that shows intervals starting from D. The intervals starting from C are in the second last row.
In this article, however, the scale starts from C. Consequently, the sizes of the intervals starting from C are listed, but some of these values are different from those obtained when the intervals start from D. Thus, as you can see:
* not all of them are the most often observed (the least often observed are highlighted in yellow or red)
* hence, they cannot be meaningfully compared with the intervals published elsewhere for Pythagorean tuning, and 5-limit tuning.
Everywhere, in Wikipedia, you see tables and text showing the most often observed sizes: for Pythagorean tuning, these intervals start from D, for 5-limit tuning, they start from C, for quarter-comma meantone, they start from D. In other words, the note from which they start is always the base note used to construct the scale. I can prove that, because I have prepared similar tables for all these systems. For instance, this comparison table shows intervals from D for quarter-comma meantone.
 
I suggest to adjust the tables in this article to make them meaningfully comparable to those published elsewhere, and to those referring to other tuning systems, by showing intervals from D (the base note), rather than from C. Of course, this implies starting the scale from D, in the tables. If you don't mind, I can do it. The intervals will be expressed as a combination of S's and X's. Please let me know your opinion as soon as you can. Paolo.dL (talk) 19:53, 30 June 2010 (UTC)
replies: text: Not quite true. The interval between two named notes (say C–C#) depends on the root of the scale. It is X when you start from C, but becomes S when you start from some of the other notes. −Woodstone (talk) 10:41, 8 July 2010 (UTC)
thread_title:  Starting the scale from C has some drawbacks 

message: This sentence, which I just removed from the section "Construction of the chromatic scale", is misleading:
* "D based" displays the difference in cents from the quarter-comma meantone chromatic scale beginning on D rather than on C.
It should be clear, by reading my previous comment and studying my table, that the size of the interval depends on where the INTERVAL starts from, not on where the SCALE starts from.
That's a demonstration of the importance of the information conveyed by the table above. For instance I have read, in some articles on Wikipedia (included this one), that the quarter comma meantone tunes the major thirds to 5:4. That's not true! Only 8 out of twelve major thirds are tuned to that ratio! Fortunately, everybody knows that in Pythagorean tuning not all the fifths are tuned to their just ratio, although this systems focuses on that very purpose.
− Paolo.dL (talk) 20:57, 4 July 2010 (UTC)
replies: text: Not quite true. The interval between two named notes (say C–C#) depends on the root of the scale. It is X when you start from C, but becomes S when you start from some of the other notes. −Woodstone (talk) 10:41, 8 July 2010 (UTC)
thread_title:  Intervals may start from any note in the scale! 

message: Thank you, Woodstone. So, this is the scale you propose to "display consistently" in the above mentioned construction tables throughout the article, isn't it? Since when I wrote my first posting in this page, I have been simply warning you, and everybody (including, I hope, the author of this table) that this scale has some important drawbacks, clearly shown in my table.
Namely, a stack centered at D-A, which "shows the 1 in C" (typically called a C-based D-A centered stack) has these drawbacks:
# The ratio which defines G♯ (from the selected base note C) is a very badly dissonant wolf interval (which deviates of about twice a syntonic comma from the corresponding pure interval of 8/5 = 814 cents), and it is an augmented fifth (A5), rather than a minor sixth (m6).
# The ratio which defines C♯ (from the selected base note C) is an augmented first (A1), rather than a minor second (m2).
# The wolf A5's are observed only 4 times in my table (the same is true for your table), while the m6's, which appear in the same column (not coloured) are more frequent: they occur 8 times.
# The A1's are observed only 5 times in our tables, while the m2's, which appear in the same column (not coloured) are more frequent: they occur 7 times.
# The purpose of quarter-comma meantone is to provide pure major thirds (M3); thus, the m6, its inversion, is supposed to be pure as well (at least in a construction table). And in most cases it is, but not in this case; the ratios from C (which define the notes in this scale) do not happen to show the inversion of M3! On the contrary, they show C-G♯, which is not an m6, is definitely impure, and is a wolf. 
You can see this in my table, which shows pure intervals in bold font, wolfs in red, and augmented or diminished intervals in yellow or red (except for the augmented fourth, of course, because a perfect tritone cannot be produced with quarter-comma meantone).
Any "symmetric" scale would have similar drawbacks (where asymmetric means with "the 1", i.e. the base note, not coinciding with the left or rigth center of the stack of fifths). A "symmetric scale", on the other hand (where symmetric, or quasi-symmetric if you like, means with the base note at the left or rigth center of the stack), would not have these drawbacks. In this case, since the stack is centered at D-A, the "symmetric scales" are the D-based (D = 1/1) and A-based (A = 1/1) ones. For instance:
* D-based symmetric: E♭—B♭—F—C—G—D—A—E—B—F♯—C♯—G♯
You can clearly see in my table that this scale contains, with respect to its base-note D, no wolfs, and provides (from D) not only a pure M3 (ratio 5/4), but also its inversion, i.e. a pure m6 (ratio 8/5). Moreover, it is evident that its 12 intervals (from D) are statistically the most representative sample of the whole set of 156 intervals in our tables. This means their sizes are the most frequently observed sizes for intervals made up of 1 semitone, 2 semitones, 3 semitones, etc. (i.e. the most frequently observed in each column of either your table or mine). Obviously this is true neither for the C-based scale nor for any other asymmetric scale; see drawbacks 3 and 4, listed above.
So, I would suggest to use a symmetric scale. For instance
:1. A D-based symmetric scale (stack centered at D-A)
:2. A C-based symmetric scale (stack centered at C-G)
I prefer the D-based one, because D is the only note which, as you ponted out, divides a piano keyboard in two symmetrical parts, and a symmetric stack of 12 fifths centered at D produces as many flat as sharp notes. But if you like to use a scale starting from C, that's ok for me, provided it is symmetric.
 
In this article, we are supposed to show how the tuning of an instrument can be performed. The very first decision is selecting the note from which you start tuning. It does not matter that people like to play in C major, and are used to see scales starting from C. We are not describing the keyboard of a piano, we are describing a tuning system. In the construction tables, we are giving formulas to compute the frequency of twelve notes. It does not really matter that these frequencies are expressed relative to D, because after "repeating the octave" you have a fully tuned instrument which you can use to play whatever chords you like! This is not so difficult to understand for a reader. However, if you like to show a scale starting from C, you can select the C-based symmetric, rather than the asymmetric. 
Notice that the symmetric C-based scale is used everywhere. For instance, in this picture from the article circle of fifth.
In sum, I propose to show, in the construction tables, the "rule" (the 12 intervals most representative of the whole set of 156 intervals, i.e. a symmetric scale), rather than the "exception" (an asymmetric scale). Don't you agree that this is the most reasonable choice? At least, can you see now the reason why I do believe this is the most reasonable choice? 
I would love to know the opinion of Glenn L as well.
--Paolo.dL (talk) 12:44, 16 July 2010 (UTC)
replies: text: Sorry I haven't chimed in until now. I agree that, in terms of balancing flats and sharps, a D-based scale is preferable throughout (as the middle of the F-C-G-D-A-E-B set of fifths), with the single exception of Just intonation, where there would be a problem because of that unfortunately defective D-A fifth. -- Glenn L (talk) 06:14, 31 July 2010 (UTC)
thread_title:  Summary of my point, with explicit reference to the information provided in my table 

message: :
 	A1            	A5 	A2 	A6             	A3 	P1	P5 	M2 	M6 	M3 	M7  	A4
E	  76	773	269	  966	462	0 	697	193	890	386	1083	579
B	  76	773	269	  966	503	0 	697	193	890	386	1083	579
F	  76	773	269	1007           	503	0 	697	193	890	386	1083	579
C	  76	773	310	1007           	503	0 	697	193	890	386	1083	579
G	  76	814	310	1007           	503	0 	697	193	890	386	1083	579
D	117           	814	310	1007           	503	0 	697	193	890	386	1083	579
A	117           	814	310	1007           	503	0 	697	193	890	386	1083	621
E	117           	814	310	1007           	503	0 	697	193	890	386	1124	621
B	117           	814	310	1007           	503	0 	697	193	890	427	1124	621
F	117           	814	310	1007           	503	0 	697	193	931	427	1124	621
C	117           	814	310	1007           	503	0 	697	234	931	427	1124	621
G	117           	814	310	1007           	503	0 	738	234	931	427	1124	621
 	m2            	m6 	m3 	m7             	P4 	P1	d6 	d3 	d7 	d4 	d8  	d5
This table rearrangement, suggested by Woodstone and adjusted by Glenn L, is great. I will soon change all my tables, in Pythagorean tuning, this article, and 5-limit tuning according to this suggestion. Of course, as repeated above, I believe it is important to also show the pure intervals in bold and the wolfs in red. My tables are automatically generated in Excel, so I am not going to turn them into Wikitables. 
− Paolo.dL (talk) 10:47, 27 July 2010 (UTC)
thread_title:  Size of the 144 intervals 

message: When you edit this article, please always check whether these edits make it inconsistent with respect to other articles about tuning systems, mainly Pythagorean tuning and Five-limit tuning. In this case, I strongly suggest you to edit as soon as possible the other articles as well, to make them consistent with this one (or undo your edits here if you prefer). This is also valid for some recent edits by Woodstone and Glenn L. 
As far as I know, consistency between articles is not always required in Wikipedia, but valuable, and in this case a great effort has been made by everybody (including the above mentioned two editors) to obtain it at least between this article, Pythagorean tuning and Five-limit tuning. Thank you. 
− Paolo.dL (talk) 17:04, 8 August 2010 (UTC)
thread_title:  Consistent edits 

message: Is there a Wikipedia article about sixth-comma meantone temperament? 173.88.246.138 (talk) 20:21, 31 December 2021 (UTC)
thread_title: Sixth-comma meantone