# Talk:Quarter-comma meantone

## Redirect

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)

## Reqmusic

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

## Starting the scale from C has some drawbacks

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)

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)

Here is what happens if you eventually (after the construction from D) rearrange the D-based scale so that it starts from C (as in this article):

Approximate size of intervals in D-based quarter-comma meantone (rearranged scale starting from C).

I published a similar occurrence table for the Pythagorean tuning system:

As you can see, the white row is always the first one.

--Paolo.dL (talk) 10:55, 3 July 2010 (UTC)

### Intervals may start from any note in the scale!

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)

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)

You did not read my previous posting ("Starting the scale from C has some drawbacks"). There's a difference between the note from which the scale STARTS, and the base-note from which the scale is COMPUTED. In this article, all scales are D-based, i.e. COMPUTED from D, but some of them START from C. I proved my statement by comparing two graphs (see above). So, the sentence I removed was severely misleading! And the values in the table column labeled "D-based" were obviously wrong.

Paolo.dL (talk) 22:31, 11 July 2010 (UTC)

## Using semitones in definition?

Hi Paolo, I see you have reworked some of this article, but I'm not sure all is for the better. In my view it is ill advised to express the definitions in terms of semitones. The construction just uses (perfect) fifths and (major) thirds. That is enough to define the diatonic scale. Basing the construction on semitones is methodologically incorrect. −Woodstone (talk) 10:26, 8 July 2010 (UTC)

I did not change the definition originally given, I just made it clearer. I addressed this point in my previous posting ("Starting the scale from C has some drawbacks"). Please read it. Since nobody answered, I just cleaned the previously existing text. I accepted the definition "in terms of semitones", because the semitones are clearly defined, in the main section of the article, based on the tempered size of the fifth. The only thing that is not explained using this approach is the reason why the X and S semitones are arranged in the particular and appearently arbitrary sequence shown in the article.
However, several days ago (June 29 and July 3), I added the following two sentences (the first in the introduction), which might be enough to satisfy both you and the author of this article:
1. "This method is a variant of Pythagorean tuning. The difference is that in this system the perfect fifth (an interval composed of 7 semitones) is flattened by one quarter of a syntonic comma, with respect to its just intonation used in Pythagorean tuning (frequency ratio 3:2)."
2. "Exactly the same scale was originally defined and produced by using a sequence of tempered fifths, ranging from E♭(five fifths below D) to G♯ (six fifths above D), rather than a sequence of semitones. This method is similar to the D-based Pythagorean tuning system, and explains the reason why the X and S semitones are arranged in the particular and appearently arbitrary sequence shown above."
By the way, I like the section "Construction", because it shows how, starting from the ratio 5/4, you can first compute the size of a tempered fifth, then the size of a tempered tone, and the size of a semitone. The logic structure of this section was not substantially modified by my edits. It was only improved.
Paolo.dL (talk) 22:25, 11 July 2010 (UTC)
I added a table explaining the original method based on a stack of fifths. I moved the other construction method to a section labeled "Alternative construction".
Paolo.dL (talk) 10:48, 12 July 2010 (UTC)

I am going to entirely rewrite this section unless someone can propose a better plan. The article starts out by saying that in meantone, the fifth is composed of seven semitones. It isn't, unless you distinguish the two different sizes of semitone, chromatic and diatonic. The fifth is composed of four diatonic and three chromatic semitones. Similarly, the tone is composed of a diatonic and a chromatic semitone, the major third of two of each, the fourth three diatonic and two chromatic semitones, and the octave seven diatonic and five chromatic semitones. A "chromatic semitone" means the same as an augmented unison, and the diatonic semitone is the semitone between E and F, and B and C', in the diatonic scale. Gene Ward Smith (talk) 11:30, 30 May 2011 (UTC)

## D-based scale starting from C (or C-based asymmetric)

The alternative construction method (which, I repeat, was not written, but only made more easily readable by me) still shows a D-based scale starting from C! This decision by the original authors of this article is questionable, in my opinion.

I insist, as in my very first comment in this discussion ("Starting the scale from C has some drawbacks"), that it would be wiser to consistently show, throughout the article, either a D-based scale starting from D, or a C-based scale starting from C. And of course, the scale should be symmetric, which means built by moving 6 fifths up and 6 fifths down from the base note. − Paolo.dL (talk) 10:48, 12 July 2010 (UTC)

Hi Paolo, I'm on a very shaky rural mobile connection. So must be short. There are not independent choices "start from", "compute from" and "symmetric". Actually there is only one choice: where do you cut the "just" chain. The resulting scale is then suitable for a few keys (minor/major).
Easiest for Pythagorean: where is the wolf. The same intervals not crossing the wolf are all equal, the ones crossing also. Where you put the 1 has no influence on the intervals created.
Also easy for quartertone: the row of tempered fifths must be cut, but will no create new conflicts with the thirds.
More difficult for 5-limit; there you must cut in 2 dimensions, and much closer to the start. The place of the cut determines the symmetry. You can take thirds up and down or only up. Again, where you put the 1 has no real meaning (all is relative only, multiply by some frequency to get a real scale).
Woodstone (talk) 08:12, 14 July 2010 (UTC)

I agree with what you wrote, except for the conclusion. When I say that this scale is D-based, I actually mean D-based symmetric (as I wrote above). And this, togheter with the arbitrary decision of discarding one of the two enharmonic notes at the ends of the symmetric sequence, determines univocally where are the "cuts". But you are right: it can be also interpreted as an asymmetric C-based scale, if you like, provided the cuts (and hence the wolf) are in the same position. That would not change my point at all. Whether you call it

• a D-based symmetric scale, which you arbitrarily rearrange to start from C, or
• a C-based asymmetric, starting from C

or whatever, it does not matter, as they are the same scale. My point is another.

Approximate size in cents of the 156 intervals in quarter-comma meantone tuning. Interval names are given in their standard shortened form. Pure intervals are shown in bold font.

My point. In short, for such a scale, whatever you call it, whatever "base note" you select for building it, it is most reasonable to show intervals from D (the most often observed ones), rather than intervals from C (which include the least often observed sizes for m2 and m6). This was my point in my very first posting, and I proved it by showing the sizes in cents of all the 156 intervals determined by this scale (see on the right).

Now we need only two very simple logical steps to reach the correct conclusion:

First step. Think about it, please: showing intervals from D means starting the scale from D in the construction tables (here and here), and assigning to D the frequency ratio 1/1, right?
Second step. Would it make sense to call "C-based asymmetric" a scale that starts with D, in which D is 1/1? Of course, theoretically it is possible (in my table, the 12 rows actually show 12 scales, only one of which starts from the base note...), but is it a reasonable choice?

If your answer is "no" (i.e., it does not make sense, it is not reasonable to call C-based a 12 tone scale starting from D), then you will agree that, as I wrote before, you have only two reasonable possibilities:

1. D-based symmetric starting from D
2. C-based symmetric starting from C

3. C-based asymmetric starting from C (see here and here)

Is that reasonable? − Paolo.dL (talk) 10:50, 14 July 2010 (UTC)

Can you see now that, contrary to what you wrote, it is relevant "where you put the 1"? For a given stack of fifths (with given left and right ends, i.e. given "cuts"), the scale in which "the 1" (or base note) is at the center of the stack (i.e the symmetric one) is statistically more representative of the whole population of intervals (the 156 intervals in my table) than the other 11 scales in which "the 1" is off-center. Please just take a look at my table and compare the 12 scales (rows): what would you select? I am not in a hurry, please take your time, go back home and read everything with a stable connection before answering.

Paolo.dL (talk) 20:25, 14 July 2010 (UTC)

Hi, Paolo, ignoring your advice to return home first, and not able to see your new pictures (I'm on a low quality GPRS line), I have one counteradvice:
Reorganise your tables by sorting the rows and columns by the circle of fifths. Assuming the one in the article, with C# thrown out, take the rows to be in sequence for:
Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#
and reorder the columns as:
P1, P5, M2, M6, M3, M7, TT, m2, m6, m3, m7, P4, P8
You will see a perfect counterdiagonal matrix, showing what I meant by an interval "crossing the wolf" or not.
It works best for Pyhagorean (and quarter comma mean), but also shows much more structure for the 5-limit.
The first few rows will have good major intervals, the last few good minor intervals. In the middle is the sweet spot where a lot makes sense.
The reason I think publications are often D centered is because then both flats and sharps will be equally far. And think of it, it is the only white note around which the keyboard is symmetric.
Woodstone (talk) 06:46, 15 July 2010 (UTC)

You did not answer my questions! :-) It is worth to repeat here one of them: even if I reorganized my table as you suggested, what row would you select as the construction sequence, to be shown here and here? Although your comment is quite interesting (thank you for your suggestion), it does not address the problem, which is here and here, not in my table. The reason is repeatedly explained above.

By the way, as I repeatedly showed above, and contrary to what you wrote in your latest posting, the sequence of fifths selected in the article is not C-based symmetric, but C-based asymmetric. As you perfectly know (I write it here so that others can understand as well), these are the two sequences:

• C-based asymmetric: A♭—E♭—B♭—F—C—G—D—A—E—B—F♯—C♯—G♯
• C-based symmetric: G♭—D♭—A♭—E♭—B♭—F—C—G—D—A—E—B—F♯

And they are not equivalent, as the wolf fifth is not in the same place. Have a good time.

Paolo.dL (talk) 11:39, 15 July 2010 (UTC)

Hi, Paolo, It's not really clear what exactly your question is. The construction of the intervals in the scale is now perfectly clear. Everything is defined in the circle of fifths. For QCM there is only one set of 12 ratios possible (based on a closed stack 11 tempered fifths and a wolf fifth). Since there are 11 good fifths, there is no real center, just a left and a right leaning center. The only remaining question is: if you are going to play a piece in a certain key, how do you map the scale to your keyboard.
If you have a piece in C major, the notes you will use mostly are the white keys (F C G D A E B). So you would like the intervals between those to be as just as possible. That implies a non-symmetric choice for the mapping. So actually the symmetric choice around D (surprise!) might be best (not the symmetric one around C!). If you want to play in A minor, you still want the white keys, but preferably also the F# and G#, needed for the melodic scale (F C G D A E B F# C# G#), so for that key you would go for a symmetric choice around or A.or E Since C major and A minor are related keys, there is good chance that you would need both in a piece. For the combination a choice with (D A) in the (left/right) center seems justified. These remarks are actually more valid for 5-limit than for QCM, but nevertheless give some insight here as well.
Now to the presentation in the article. In view of the above, I would propose to display consistently the diagrams with D center left and the 1 in C. That way the readers can think in C (which is always easiest for non-professional musicians, such as me and most of them) and will see the intervals from C, and the constructed scale is suitable for C and Am.
The diagrams showing the inner intervals between all notes should be ordered by the circle of fifths both by start of the interval and length of the interval, since that gives the clearest reflection of the relations in the scale. Note that the wolf fifth (e.g. G#–Eb) is actually a diminished sixth. On the last note before the cut there is no fifth available in the 12 note scale. The 12-note scale forces the spiral of fifths into a circle, and we can only fill many of the intervals in the table by using an enharmonic note. I would propose to render their values in italics. It will reveal even more structure.
P.S. This is all quite intriguing, being in Thailand's Isan region, where much of the traditional music is justly intonated. It really takes some getting used to. Furthermore the tonality is often Dorian, so it is in your terms rightly D centered and starting from D.
Woodstone (talk) 19:20, 15 July 2010 (UTC)

If you really want to interact, you should understand my questions first. I suspect that reading with more attention the short paragraph marked "My point" would suffice. Everything else is just an easy deduction. If this is not enough, please help me to figure out what prevents you from understanding my point, and let me help you to understand it. Would you mind to help me by answering this simple question, formulated using your own terminology? Do you agree that the scale shown in this section of the article is defined by the following sequence of 11 fifths centered at D-A, but with C = 1/1?

• C-based asymmetric stack (11 fifths): E♭—B♭—F—C—G—D—A—E—B—F♯—C♯—G♯

I assume that you understand that, in my terminology, since C = 1/1, this is "C-based", and since C is neither center-left nor center-right, this must come from a stack of 12 fifths where the "base note" C is off-center (i.e an "asymmetric" stack). For instance, from a C-based stack centered at D:

• C-based asymmetric stack (12 fifths): A♭—E♭—B♭—F—C—G—D—A—E—B—F♯—C♯—G♯

Let's forget that this section (called "Alternative construction") defines the scale using a stack of semitones: This is not relevant, because the article repeatedly explains and shows that the scale is traditionally defined using a stack of fifths. − Paolo.dL (talk) 20:43, 15 July 2010 (UTC)

Hi, Paolo. Don't think I have not tried to understand "your point". I read it many times, but it still escapes me. And just as you some time earlier, I agree with almost everything you write, except the conclusion. If you replace the value 1/1 as start of the construction by anything else (e.g. 3/2 or 4/3 or 9/8 or π), the resulting table of intervals between each of the notes will be exactly the same. Of course there is a difference in the presentation to the reader in the bare construction diagram.
Conclusion: there is only one possible scale (in abstracto, as a sequence of intervals), constructed by starting at an arbitrary point in the circle of fifths (CoF) and multiplying by 51/4 while reducing octaves as needed, till the 12th note. There is no independent choice for "start from", "based on" and "symmetric.
The single remaining point is where you map it to the keyboard: what is the start in the CoF (or equivalently, where is it centered). That wholly depends on what musical piece you want to play. Every mapping is suitable for pieces in only a few keys. For example if you play in C, you would like F-C-G-D-E-A-B to be good, so you might want to cut symmetric from D (Eb till G# avaible). If you want to play in Eb, you will need Eb-Bb-F-C-G-D-E and you might cut symmetric from C.
If you reorder everything by the CoF (and to fix thoughts map from Eb to G#) , you will get:
+1 +5 +2 +6 +3 P1 P5 M2 M6 M3 M7 +4
Eb 76 773 269 966 462 0 697 193 890 386 1083 579
Bb 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 -6 -3 -7 -4 -8 -5
Here you see that some keys are good, more distant keys (in sequence of CoF) add more and more deviant intervals. The top header shows the interval towards the top of the table, the bottom header towards the bottom.
Hopefully this clarifies the situation. By the way, I'm still on the rural site, but a borrowed SIM card from another service provider works somewhat better, so I could at least lookup your "here" refs.
Woodstone (talk) 12:45, 17 July 2010 (UTC)
After you tuned your keyboard in D-A centric, you can play in D having all minor, major and perfect intervals plus the augmented fourth. If you play a piece in A you get a diminished fifth instead of the +4 (nice symmetry for this D-A centric map). If you play in E you lose the M7 and get a -8 instead, in B you also lose the M3 and have to use a -4 instead, etc. On the other side: play in G, lose the m2 for the +1; in C you also lose the m6 for a +5. You can still play, but the enharmonics are not tuned as well as the originals. −Woodstone (talk) 17:06, 17 July 2010 (UTC)
Regarding your table above: I suggest it might look better by substituting A (Aug) for + and d (dim) for - which produces (with spacing):
A1 A5 A2 A6 A3 P1 P5 M2 M6 M3 M7 A4
ETemplate:Music   76 773 269   966 462 0 697 193 890 386 1083 579
BTemplate:Music   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
FTemplate:Music 117 814 310 1007 503 0 697 193 931 427 1124 621
CTemplate:Music 117 814 310 1007 503 0 697 234 931 427 1124 621
GTemplate:Music 117 814 310 1007 503 0 738 234 931 427 1124 621
m2 m6 m3 m7 P4 P1 d6 d3 d7 d4 d8 d5
Glenn L (talk) 18:25, 17 July 2010 (UTC)

Main topic. I started this discussion. Let me explain that this discussion should be focused on selecting the scale which will be consistently used in these three tables: here, here and here. We already reached some agreement. We agreed that this scale should be constructed from a D-A centered sequence of 11 fifths, but that was obvious since my very first posting in this whole talk page. We need to reach an agreement about whether we will use C or D as the base-note of this stack of fifths. Translated into your terminology, we need to decide "where to put the 1" in the three above mentioned tables (more exactly, in their colums containing frequency ratios). I am sure you agree that deciding the center of the sequence (i.e. D-A) is not enough to build these three tables. Was I clear enough? It is essential that you let me know if you understand this paragraph. Please, do not ignore my request of feedback.

(By the way, it is true that "in abstracto" deciding the center of the sequence of 11 fifths is the only independent choice, but we need to take a practical decision here, and "in concreto" the selection of the base-note is independent of the selection of the center, otherwise this discussion would be pointless.)

A new strategy. You stated that you did not understand my main point, and you did not really understand my questions. On the contrary, I did understand what you wrote. Wouldn't it be wise to let me help you to understand my point? My main point is based on my table, which you cannot see, because you are on a vacation. Your table is similar, but the relevant information is given in my table by a color code and font emphasis which is more complex that the one you used. Your strategy to overcome the problem seems to be somewhat randomic. You try to explain whatever you think fit to this discussion, hoping it may be useful to reach an agreement. Of course, I am honoured by your patience, and enjoy reading your postings, but you don't need to repeat what you wrote, because I have read it several times and perfectly understood. Besides, I am not on a vacation.

So, let me suggest another strategy.

Being the only one who knows both your point and mine, I ask you to trust my judgement: we won't reach an agreement, unless you let me help you to understand my point, because I am convinced you cannot reach a wise conclusion if you ignore my point, which I consider crucial. If you are interested to understand my point, please let me help you. Please accept both these suggestions, or at least the second one:

1. Wait until you come back home, so that you can see my tables
2. At least answer my simple question, so that I can base my explanation on a common ground.

Here is my question again: do you agree that the scale shown in this section of the article is defined by the following sequence of 11 fifths centered at D-A, but with C = 1/1?

• C-based asymmetric stack (11 fifths): E♭—B♭—F—C—G—D—A—E—B—F♯—C♯—G♯

By the way, this is the same scale you proposed to "display consistently" throughout the article.

My explanation will be much easier, if you agree that the answer to this question is yes, as I hope. Right now, I cannot be sure what your answer will be, as I am really confused. I can't really understand why you keep not grasping a point which seems absolutely obvious to me, and which I explained at least three times. However, your messages helped to build a common terminology. So, if you answer to my question, I am sure I'll be able to make my point clear to you, by briefly summarizing it with your own terminology. Your messages contain interesting information and suggestions which are not relevant to this specific discussion. Don't worry, I have read everything with attention and promise they will not be forgotten. But first, please let's reach an agreement on the main topic of this discussion, ok?

Paolo.dL (talk) 17:31, 17 July 2010 (UTC)

I have verified that the table below is indeed stacked from Eb to G#, is thus D-A centered, and shows the 1 in C.
Note Formula Ratio Cents 12TET Delta 1/4-commas
C 1 1.0000 0.0 0 0.0 0
CTemplate:Music X 1.0449 76.0 100 −24.0 7
D T 1.1180 193.2 200 −6.8 2
ETemplate:Music T S 1.1963 310.3 300 +10.3 -3
E T 2 1.2500 386.3 400 −13.7 4
F T 2 S 1.3375 503.4 500 +3.4 -1
FTemplate:Music T 3 1.3975 579.5 600 −20.5 6
G P 1.4953 696.6 700 −3.4 1
GTemplate:Music P X 1.5625 772.6 800 −27.4 8
A P T 1.6719 889.7 900 −10.3 3
BTemplate:Music P T S 1.7889 1006.8 1000 +6.8 -2
B P T 2 1.8692 1082.9 1100 −17.1 5
C' P T 2 S 2.0000 1200.0 1200 0.0 0
As requested I await your further (patient) guidance. −Woodstone (talk) 03:55, 18 July 2010 (UTC)

### Summary of my point, with explicit reference to the information provided in my table

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:

1. 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).
2. The ratio which defines C♯ (from the selected base note C) is an augmented first (A1), rather than a minor second (m2).
3. 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.
4. 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.
5. 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.

Circle of fifths showing major and minor keys

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)

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)

### Conclusion

Hi Paolo. Excellent exposé. Totally understandable. And agree with the conclusion. But three remarks.
1. quote: "... after "repeating the octave" you have a fully tuned instrument which you can use to play whatever chords you like!"; not really: that's the whole problem, many of the keys (and chords) will sound awful.
2. quote: "its 12 intervals (from D) are statistically the most representative sample of the whole set of 156 intervals in our tables."; yes true, but because of the above, the tuning was never meant to use all the 156 intervals; this is hardly a valid argument.
3. At all places where you say "my table", actually the (my) CoF resorted tables show this even clearer.
Now to the consequences. Clearly we now agree that we should show a quasisymmetric table with the 1 in the center. I hesitate between C and D.
C: most readers will be most at ease with C; many other displays (as shown CoF) center on C
D: divides more evenly between flats and sharps
I have no strong opinion and will go with general opinion. Perhaps we can do a quick check what center dominates in the other music articles (and in literature).
Woodstone (talk) 06:19, 19 July 2010 (UTC)

Thank you for your support, and thank you for accepting my strategy to reach a conclusion. About your three remarks:

1. I agree, of course. That was implicit. I can't reapeat every concept every time, expecially when this it is evident in my table, which highlights 16 wolfs in red! As I wrote to you in another discussion (perhaps in Talk:Five-limit tuning), the main reason why I included my tables in Pythagorean tuning, quarter-comma meantone, and 5-limit tuning was showing that many of the 156 intervals differ from the 12 ones obtained in the respective construction tables! In this article, the text I added to comment my table, for instance, includes this crucial paragraph, which summarizes my main point: "Surprisingly, although this tuning system was designed to produce pure major thirds, only 8 of them are pure (5:4 ≈ 386.314 cents)."
2. In my opinion, if you cancel from the 156 intervals the ones which you don't like, my argument becomes even more valid! If you cancel the wolf intervals, the symmetric scales become even more representative of the remaining population of intervals. And if you cancel every augmented or diminished interval (included the wolfs), except for A4 or d5, the symmetric scale becomes perfectly representative. But if you want, you can suggest another way to say that the interval sizes defining the symmetric scale are the "rule", or the "desired outcome", while the different sizes produced by asymmetric scales are "the exception", or the "undesired outcome". It would be quite bizarre a tuning system in which the unwelcome outcome is more frequent than the welcome one, and indeed fortunately no tuning system is so bizarre.
3. This remark truly surprises me. Your table does not show wolfs, and does not show pure intervals. And my point is based on this information, as I repeatedly pointed out in my summary! Your table (as edited by Glenn L) has only one advantage with respect to my table, as far as the information needed in this specific discussion is concerned: it calls augmented or diminished the intervals that are coloured in yellow or red in my table; a terminology that, although widely used in Western music, is nonetheless questionable and difficult to understand. I would rather call "rules" the intervals produced by the symmetric scale, and "exceptions" the intervals of different sizes produced by the asymmetric scales (highlighted in colour both in your and my table). Anyway, the name (and colour) of the "exceptions" is not important in itself, but only because it is different from the name (and colour) of the "rules". The main advantage of your table is showing a pattern, but this is not relevant at all to this specific discussion, and will be discussed in a separate section. In short, I definitely don't need a pattern to show the 5 drawbacks listed in my previous comment, and your table would not provide enough information to compile that list.

I was hoping you could give an expert advice about whether we should select a D-based or a C-based scale. My advice is what I gave above, based on the fact that the D-based symmetric scale shows an additional symmetry (a symmetric stack of 12 fifths centered at D produces as many flat as sharp notes). But this, as you correctly point out, is only one side of the coin.

By the way, the D-based scale is used in Pythagorean tuning and, as far as I know, was used by Aron as well (but I can't find the reference anymore).

Glenn L, can you help us? (apparently, we are the only three editors involved; the main author of this article, User:AugPi, does not seem to be on line anymore)

Paolo.dL (talk) 10:26, 19 July 2010 (UTC)

I did some sampling and found:
Just informational, no conclusion yet. Let's look outside WP as well for sounder basis. −Woodstone (talk) 12:12, 19 July 2010 (UTC)

If you don't mind, I would not list the articles which just show one or more examples of a scale or interval. We already know that scales or intervals from C are most often used for examples. C- or D-based means, in this context, tuning with respect to C or D, not just starting a scale from C or D. We need examples on tuning systems or stacks of fifths. For instance, your example about list of meantone intervals is quite useful.

Paolo.dL (talk) 13:54, 19 July 2010 (UTC)

Here is an updated list (feel free to modify it):

Paolo.dL (talk) 13:54, 19 July 2010 (UTC)
Paolo.dL (talk) 18:21, 26 July 2010 (UTC)

Feel free to modify the list above. I added three references into it (Midicode.com, Techincal Library, Wolf fifth). It is obvious that everybody uses a D-A centered stack when describing quarter-comma meantone. Even when they decide to start from C (= 1/1), they do not use the symmetric (C-G centered) stack, but they keep using the D-A centered one (i.e. an asymmetric C-based stack). This is probably because, traditionally, the quarter comma meantone was D-A centered (as defined by Aron), with the wolf fifth in the same position as in Pythagorean tuning (from GTemplate:Music to ETemplate:Music).

Hypothesis. Most likely, musicians were so used not to play the GTemplate:Music-ETemplate:Music interval, that Aron decided to keep the wolf fifth in the same position, by using exactly the same (D-A centered) stack of fifths used by Pythagoras. Thus, I believe using a C-G centered stack (i.e. a C-based symmetric stack, with wolf fifth in a different position) in this article is not advisable.

The article Wolf interval uses a D-A centered stack as an example specific for meantone temperament: "If the meantone fifths are tuned from ETemplate:Music to GTemplate:Music, the anomalous interval will be between GTemplate:Music and ETemplate:Music." This is consistent with my hypothesis.

Paolo.dL (talk) 18:21, 26 July 2010 (UTC)

Let's not forget a previous comment by Woodstone supporting a D-A centered stack: "If you have a piece in C major, the notes you will use mostly are the white keys (F C G D A E B). So you would like the intervals between those to be as just as possible. That implies a non-symmetric choice for the mapping. So actually the symmetric choice around D (surprise!) might be best (not the symmetric one around C!). If you want to play in A minor, you still want the white keys, but preferably also the F# and G#, needed for the melodic scale (F C G D A E B F# C# G#), so for that key you would go for a symmetric choice around or A.or E Since C major and A minor are related keys, there is good chance that you would need both in a piece. For the combination a choice with (D A) in the (left/right) center seems justified. These remarks are actually more valid for 5-limit than for QCM, but nevertheless give some insight here as well."

I believe the conclusion should be:

1. C-based symmetric (C-G centered): Not used in the literature (according to my hypothesis, the reason is that it has the wolf fifth in an unusual position, i.e. FTemplate:Music-DTemplate:Music rather than GTemplate:Music-ETemplate:Music)
2. C-based asymmetric (D-A centered): Several drawbacks, listed in the previous subsection (e.g. it contains an unexpected wolf interval in the construction table; a very badly dissonant A5, instead of the expected justly intonated m6, the pure inversion of M3)
3. D-based symmetric (D-A centered): The less questionable, most symmetric, and most reasonable choice.

Do you agree?

Paolo.dL (talk) 22:35, 26 July 2010 (UTC)

I do not understand your second remark just above here. A C-based asymmetric is totally equal to D based symmetric. All intervals between pairs of notes are identical. Only the presentation of the construction (with C=1) shows different numbers.
Aside from that, I agree.
Woodstone (talk) 03:43, 27 July 2010 (UTC)

Thank you. Closing a discussion with such a clear agreement is a great and precious result. It would not have been possible without your excellent contributions. By the way, my second remark is just a short summary of the six drawbacks of the C-based asymmetric scale, that I listed in my summary (see previous subssection). Right at the beginning of this subsection, you wrote that my summary was "Excellent" and "Totally understandable." And agreed with my conclusion. Notice that C-based means C=1 in the construction table. Thus, my point coincides with yours:

• Your point. "Only the presentation of the construction (with C=1) shows different numbers."
• My point. "...it contains an unexpected wolf interval in the construction table..." (in which, by definition, C=1)

Paolo.dL (talk) 10:47, 27 July 2010 (UTC)

I still cannot comprehend why you say that C asymmetric is in any way different from D symmetric. For x=5(1/4) and ignoring octaves, you get:
• with D=1, C=x-2 and G#=x6, and the interval is x6/x-2=x8.
• with C=1, G#=x8
In both cases the interval is the same. −Woodstone (talk) 05:35, 28 July 2010 (UTC)
I never ever maintained the contrary! My two tables showed this equivalence since my very first posting on this talk page. Thus, your point and mine perfectly coincide: "Only the presentation of the construction (with C=1) shows different numbers." To be even clearer: the interval C-G# is shown only in the C-based asymmetric construction table. Since it starts from C, it is not shown in the D-based symmetric construction table.
So, my point (and yours) is simple: There's only one difference between C-based asymmetric and D-based symmetric: the base note! − Paolo.dL (talk) 07:34, 28 July 2010 (UTC)

I edited the article according to this conclusion, and trying to avoid this kind of misunderstanding. See the new section "C-based construction tables". − Paolo.dL (talk) 09:25, 28 July 2010 (UTC)

## Size of the 144 intervals

A1 A5 A2 A6 A3 P1 P5 M2 M6 M3 M7 A4
ETemplate:Music   76 773 269   966 462 0 697 193 890 386 1083 579
BTemplate:Music   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
FTemplate:Music 117 814 310 1007 503 0 697 193 931 427 1124 621
CTemplate:Music 117 814 310 1007 503 0 697 234 931 427 1124 621
GTemplate:Music 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)

## Precision

Quoting from the article pitch:

"The just-noticeable difference ... depends on the tone's frequency and is about 4.3 cents (1 cent = 1 hundredth of a semitone) or about 0.36 Hz in frequency within the octave of 1,000–2,000 Hz. ... The jnd is typically tested by playing two tones in quick succession with the listener asked if there was a difference in their pitches.[3] The jnd becomes smaller if the two tones are played simultaneously as the listener is then able to discern beat frequencies. The total number of perceptible pitch steps in the range of human hearing is about 1,400. "

The current article expresses intervals at various points in different precisions. In view of the above, a representation in whole cents seems accurate enough. One cent is a factor of 1.0006. Even in harmony, the beating frequency of one cent on 440 Hz is about 1/4 Hz, barely noticeable. Accordingly, factors with at most 5 significant places are sufficient.

Rounding the numbers would reduce clutter and improve consistency. The same goes for other articles concerning intervals and tuning. −Woodstone (talk) 05:01, 31 July 2010 (UTC)

You can choose the rounding precision you like. But in my opinion "about" is not implied, especially when you round to a smaller precision. Either the term "about" or "approximately" or the symbol "≈" is needed. Notice that, I rounded to integers the values in cents in the section "Size of intervals", and Glenn L corrected to three decimal places (see this revision by Glenn L). Since I wrote that the average of the values was exactly 700 for fifths, Glenn's edit made sense. Also, the value for ε, in the same section, must be more precise, because the wolf fifth is computed by adding 11ε to the mean value of 700 cents, and if you write that "ε is 3.4 cents", this would give a rounded value of 737 cents (737.4) rather than 738 (737.637). I originally used three decimal places, Glenn L proposed 5 (see this revision). Moreover, I would prefer rounding the syntonic comma (21.506) to 21.5 cents, rather than 22.
Also, in the "construction" section there are equations in which the syntonic comma and its quarter are computed. There, since the article is called quarter comma meantone, I believe we need a greater precision. I suggest three decimal places for cents and five for ratios. This precision is consistently used in many other articles. This was also suggested in some edit summary by Glenn L, as far as I can remember.
So, the needed precision is not always the same. In some cases, rounding to integer the values in cents is perfectly ok, in other cases (definitions, computations), you need a greater precision.
Paolo.dL (talk) 12:35, 1 August 2010 (UTC)
Differences of less than one cent (factor 1.0006) are not perceived by the human ear. Also, measuring the frequency of a real physical musical instrument cannot be performed to much more, since the frequency has some variation depending on power and other circumstances. So any precision greater than that in the ratios has no practical meaning. In the calculations, the exact values are clear and the numeric approximations indicated. In running text and tables, it is generally not needed to explicitly indicate approximation, since that can be assumed. Small rounding errors are always possible in tabulated sums and are acceptable. On the other hand, the tables and text have become more fluid and attractive. In a few places I have maintained slightly more precision, where otherwise points being made would be invisible, actually showing that these points are only of theoretical importance. Rounding a specific number to a different precision, just because it is close to halfway, is quite unusual. −Woodstone (talk) 15:44, 1 August 2010 (UTC)
I agree with you about using, in most cases, integer values for cents. I already mentioned that my inital text used integers, and I used them exactly for making the text more fluid and quickly readable. But I used "about", and my ε had three decimal places (have you seen this revision page, showing my original text on the left?). And I insist that we need greater precision in definitions of values used as a criterion for tuning or as a reference for computing other intervals (such as that of syntonic comma, quarter comma, and ε).
Sorry, I do not agree that "about" is always assumed. You assume it, I assume it, most readers do not. Especially when you say, for instance, that the average of 12 values is exactly 700 or 400, or 300,...
Some edits by Glenn L added the word "about" or the symbol "≈" where it was missing. Some others increased the precision (only in some specific cases, in the text not in the tables), rather than decreasing it. It is not necessary for you to agree with me and Glenn L, but at least you should consider that your opinion is not (yet) shared by others. When I disagreed with you and Glenn, even when I strongly disagreed, and even when you did not answer to my critic comments (as for the two diagrams published in 5-limit tuning), I did not try to impose my opinion.
You should also consider the opinion of the original authors of this text. Most of the occurrences of the term "about" and the symbol "≈" were there before my edits, and the precision of most numbers were decided by them.
Paolo.dL (talk) 17:33, 1 August 2010 (UTC)
On a sideline, about the graphic diagrams, although your comments are generally very polite, in that case there was a drift towards personal attack, which made me decide not to react. Perhaps we can pick that up later.
On the precision subject here, I did offer my proposal on talk first, and only after no reaction came, I decided to be bold (as advocated by WP). We have jointly been improving this article and what old editors put is now only of secondary importance.
On the facts, I think we agree (again) to a great extent. In tables and routine text, precision is best limited to what is essential. In some individual cases, like the ε and perhaps the defining 5(1/4) we can give a little more to avoid rounding in further calculations.
Woodstone (talk) 17:58, 1 August 2010 (UTC)

It is too easy not to consider the opinions of the original authors of this article. Even if they are not participating to this discussion and not editing this article anymore, I believe their opinion has the same value as yours. Also, the edits by Glenn L seemed to go in the opposite direction, with respect to yours.

Anyway, it is true that we agree on something, but you did not consider my opinion and Glenn's edits about the term "about".

Paolo.dL (talk) 23:05, 1 August 2010 (UTC)

I've agreed to leave out the "" sign, but I feel at least one decimal place is necessary, with three for the small intervals < 10 cents such as "ε". As stated in my edit remark, I do try to be reasonable. − Glenn L (talk) 04:09, 2 August 2010 (UTC)
Glenn, as I wrote above, I really don't mind about the rounding precision in most cases. We agreed with Woodstone to use, in most cases, integers, but I do not mind if you decided to use 1 decimal place. In some specific cases, however, I do care about the precision. A greater precision is useful, in my opinion, for values used as a criterion for tuning or as a reference for computing other intervals (such as syntonic comma, quarter comma, and ε). In these cases, three decimal places are ok, as you suggest.
I am not saying that we need the symbol "" embedded in text, I am saying that, not always, but at least in some critical parts of the text where calculations are explained, we need either that or the terms "about", or "approximately". Think about it: if ε has three decimal places, and another value is defined as 700-11ε, the reader expects to see the result with three decimal places (unless you say the result is approximate!). We do need to explain what we mean, we cannot assume the reader knows.
Paolo.dL (talk) 14:04, 2 August 2010 (UTC)
I edited a little, I hope you agree. I could not stand the inconsistencies in these equations, which resulted from Glenn's edit:
${\displaystyle \approx 702.0-696.6\approx 5.377\ {\hbox{cents}},}$
${\displaystyle \approx 4\left(5.377\right)\approx 21.5\ {\hbox{cents}}.}$
I am waiting to know how you want to solve the above mentioned problem regarding computations with ε. I would prefer to reinsert the "" symbols, but you can also increase the precision. Although I am the author of that part, I do not want to impose my decision.
Paolo.dL (talk) 14:26, 2 August 2010 (UTC)
I'll let you and Woodstone decide on what to do with the "" symbols. I didn't want to start an edit war so I didn't reinsert them. As for "ε", I still found three decimals were necessary to guarantee one decimal in all the computations that followed, the case of 100 − 3ε ≈ 76.0 being the reason for ε ≈ 3.422 cents rather than just 3.42. I made one fix where the difference was stated as 5 cents to 5.4, otherwise I am satisfied with your most recent edits. − Glenn L (talk) 15:48, 2 August 2010 (UTC)
I'll wait for an advice (or edit) to solve the above mentioned problem regarding computations with ε.
Paolo.dL (talk) 20:18, 2 August 2010 (UTC)

I doubt if you guys realise what 0.001 cent is. The highest A on a piano, almost the last key, is 3520 Hz. I suppose you know how difficult it is even to hear any mistuning in that high register (real sonority ends at about 2000 Hz). At that frequency, a difference of 0.001 cent is 0.002 Hz, or about one cycle per ten minutes. Hopefully, this makes you realise that precision at that level is meaningless.

The way to look at the mentioned ε is to define its value exactly (using fractions and roots) and only for information give an approximate decimal value. All calculations are done exactly and only the results are shown in decimal form. It is quite usual that totals or other compound values in tables or text are not quite consistent in the accuracy shown. Everybody that ever did VAT invoicing is acutely aware of that issue. So I still maintain that 1 cent (well below pitch distinction of the human ear) is accurate enough. The only exceptions can be made in the initial calculated values of exact fomulas.

Woodstone (talk) 13:33, 3 August 2010 (UTC)

And again, I am not rigidly maintaining that "≈" or "about" is needed everywhere (although I'd rather keep it where it was before your edit). I am only proposing to use it where it is currently used, plus the paragraph about computations involving ε.
By the way, if you know the frequency ratio of ε, expressed as a fraction, in Pythagorean tuning and quarter comma meantone, it would be nice to insert this info in the respective articles, as you suggested. This would be a great improvement to my text. However, you certainly understand that the paragraph needs to be based on values in cents, because the average in cents is the most meaningful value, in this case.
Paolo.dL (talk) 17:35, 4 August 2010 (UTC)
I am not opposed to using "≈" in an equation, but in running text it is unusual and awkward. Sparing use of "about" is agreed. However that all intervals of the same name average to a multiple of 100 cents is true for any 12-tone scale, because they form cycles within the repeating octaves. It does not really deserve explicit mention here (do not construe this as my opposing it here). −Woodstone (talk) 23:33, 4 August 2010 (UTC)

## Deviations from reference interval sizes as a multiple of a constant ε

Thank you for accepting my edit. I am very surprised that you don't regard as useful to mention here that all intervals of the same type average to a multiple of 100 cents. It is true that this information is not specific to this tuning system, but the paragraph showing that the 144 interval sizes are all deviations by from an average size is extremely insightful. It is not unusual to start a logic statement with a generic axiom, from which a more interesting specific conclusion is reached. That's greatly educational.

I consider this specific conclusion crucial. It reveals a simple pattern, which lets you understand, for instance, why the wolf fifth is the interval which deviates more than any other from the corresponding "reference value" (neither augmented nor diminished) shown in the construction table. And this specific information about deviation from reference is based on the specific concept of deviation from fixed average, which in turn is based on the generic concept of fixed average. Can you see the logic sequence?

Explaining the deviation from reference as a multiple of a fixed value ε is the most useful piece of information after the construction table, which in turn explains the "deviation" of each reference pitch from D.... This explanation provides the reader with a powerful tool to interpret my table: the larger the number of coloured (augmented or diminished) intervals, the smaller the deviation from reference value. Hence, the largest deviation is observed for the wolf fifth. Isn't that amazingly insightful? In my opinion, it is as insightful as the pattern shown in my tables, after the rearrangement you suggested.

By the way, the above mentioned logical conclusion is easy to understand, and its logical foundation, i.e. the concept of fixed average, is also easy to understand (at least for semitones), but nevertheless both these concepts were not obvious to me, until I discovered them, with great surprise, by reading Wolf interval. I think you sometimes tend to overestimate the reader's ability to understand (except when you removed from Pythagorean tuning the internally linked terms "octave" and "fifth"; in that case, you seemed to underestimate it, not because the terminology is simple, but because it is the appropriate conventional terminology, it is used everywhere as frequently as the word "note", and internal links were purposely provided to explain it to the beginner). Consider that the concept of fixed average is not exactly self evident. I accept it as an intuitively appealing proposition, but I am not sure I really and deeply understand it. Why a multiple of 100? Why shouldn't it be just any number commensurable with 1200 (e.g. 120)? I can easily understand it only for semitones and fifths. By definition 12 semitones make up exactly 1 octave. The average of 700 cents for the fifth is less obvious, but I understand it as I studied the circle of fifths: ascending by 12 fifths from any note, one is supposed to return to a note exactly in the same pitch class, and exactly 7 octaves above it. What about the other intervals? It is not as obvious at it seems (even the circle of fifth is not obvious).

Eventually, let me remind you that the value of ε is constant for the entire tuning system, and the search for constants has always been one of the main objectives in the history of physics. Understanding is sometimes synonymous of finding a constant, and the formula based on that constant (e.g., the universal gravitational constant discovered by Isaac Newton, and his formula for computing gravitational attraction). − Paolo.dL (talk) 09:33, 6 August 2010 (UTC)

The biggest problem is that there is a different ε for each regular tuning system. Basically, in each system, ε = 100 log2(diesis), where diesis is the ratio between two enharmonic notes, such as between GTemplate:Music and ATemplate:Music. To illustrate, in each of the following five tuning systems (chosen intentionally), the difference between the dieses of two consecutive systems is the syntonic comma of ratio 81:80 or ≈ 21.506 cents, so ε changes by one-twelfth of this comma, or ≈ 1.7922 cents:
• For third-comma meantone, the great diesis is 648:625 ≈ 62.565 cents, so ε ≈ 5.2138 cents.
• For quarter-comma meantone, the (lesser) diesis is 128:125 ≈ 41.059 cents, so ε ≈ 3.4216 cents.
• For sixth-comma meantone, the diaschisma is 2048:2025 ≈ 19.553 cents, so ε ≈ 1.6294 cents.
• For twelfth-comma "meantone", the schisma is 32768:32805 ≈ -1.954 cents, so ε ≈ -0.1628 cent.
• For Pythagorean tuning, the Pythagorean comma is 524288:531441 ≈ -23.460 cents, so ε ≈ -1.9550 cents.
Note that the twelfth-comma system is not a true meantone, just as eleventh-comma or 12-tone equal temperament isn't. I added it to complete the illustration, as the successive systems differ from each other by 1/12 of a comma. − Glenn L (talk) 16:58, 6 August 2010 (UTC)
Great job! Thank you very much. This makes perfectly sense, because it is obvious from what I wrote in the article that the deviation from reference (construction brick) fifth to wolf fifth is 12ε, so the value of ε in cents is 12 times smaller than that deviation. And the deviation from reference to wolf fifth is another way to define the diesis (or diaschisma, or schisma, or Pythagorean comma). − Paolo.dL (talk) 09:22, 8 August 2010 (UTC)
Glad you appreciate this. I made a minor correction and have made the dieses stand out more. − Glenn L (talk) 09:32, 8 August 2010 (UTC)
Indeed, I greatly appreciated your posting. One of the reasons why I did is that it is not obvious in the article Diesis that the diesis can be viewed as the deviation from reference to wolf fifth in quarter comma meantone. But I am not suggesting to mention this in Diesis. That article is not yet perfect, but already quite exhaustive. By the way, since it was too messy in some parts, I just decided to edit it, mainly by rearranging the existing text. − Paolo.dL (talk) 14:46, 8 August 2010 (UTC)

## Consistent edits

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)