Talk:Wolf interval

Sign of ε

The sign of ε in this article is positive only for meantone temperaments where the fifth is narrowed by 1/11-comma (where ε ≈ +1/8593 cent) or more. It is exactly zero for equal temperament (where the fifth is exactly 700 cents) and negative where the fifth exceeds 700 cents, such as in 1/12-comma and Pythagorean tuning. — Glenn L (talk) 23:00, 8 August 2010 (UTC)

But why did you delete the reference to Pythagorean tuning? Notice that we need to be general in this section lead. Specific examples about Meantone and Pythagorean tuning are given in the subsection.

Also, in the article about Pythagorean tuning, we say that the Pythagorean fifth is 700+ε, and in the article about 1/4-comma meantone, we say that the meantone fifth is 700−ε, which is the opposite of what you wrote above (was that a typo, or am I missing something?). So, let's be consistent.

Paolo.dL (talk) 00:56, 9 August 2010 (UTC)

Again, the problem is that in most meantone temperaments (up to eleventh-comma), the ε is subtracted from the 700-cent fifth, whereas in twelfth-comma and Pythagorean tuning it is added to the fifth. That is precisely why I defined the greater/lesser diesis, diaschisma (all above unison), schisma & Pythagorean comma (both below unison) the way I did. The ε, in effect, changes sign when crossing the 700-cent line a micrometer (!) from the eleventh-comma mark. Let's see if this poorly-edited number line (the letter "e" for equal temperament should be almost precisely in the middle of the digits "11" for 11th-comma, otherwise nearly to scale) can help illustrate the problem:

Pythag||||||||||||||||||||||||||||||12|e11|||||||||||||||||||||||||6th|||||||||||||||||||||||||||||||||4th|||||||||||||||||||||||||||||||||3rd

On the number line, Pythagorean ≈ 701.95500 cents; 12[th-comma] ≈ 700.16281; "e[t]" = 700.00000; 11[th-comma] ≈ 699.99988; 6th[-comma] ≈ 698.37062; 4th[-comma] ≈ 696.57843; and 3rd[-comma] ≈ 694.78624. I'm using 5 decimals only because ET and 11th-comma are so close together; otherwise, 1 decimal is sufficient (702.0, 700.2, 700.0 twice, 698.4, 696.6 and 694.8 cents respectively).

The distances between Pythagorean & 12th-c, 12th-c & 6th-c, 6th-c & 4th-c, and 4th-c & 3rd-c are precisely 1/12-comma ≈ 1.79219 ≈ 1.8 cents by definition. For all systems to the right of the "e" mark, the corresponding fifth is narrower than 700 cents, so ε must be subtracted from 700. for those systems to the left of "e", the corresponding fifth is wider than 700 cents, so ε must be added to 700. Thus we need to present ε in a different way on either side of the line. — Glenn L (talk) 03:14, 9 August 2010 (UTC)

We agree completely, I guess. And that was already clear to me (because you had explained it to me before), except that I did not realize that the fifths in the tuning systems you listed differed by exactly 1/12 comma (interesting!). By the way, notice that I would not say that ε "changes its sign". This would mean that ε may be negative, i.e. ε may be equal to a negative number, which is not the same as stating that ε may be subtracted, as stated in Quarter-comma meantone (where ε is subtracted, but obviously its value is positive). I mean, we are not exactly discussing the "Sign of ε" as in the section title, are we? That would be an option (which I tried to explore, with little success, in one of my previous edits that you did not like), but I guess that's not what you meant exactly, because your first sentence in your posting above says:

• "The problem is that in most meantone temperaments (up to eleventh-comma), the ε is subtracted from the 700-cent fifth, whereas in twelfth-comma and Pythagorean tuning it is added to the fifth".

This sentence of yours implies that ε is always positive. That's ok, not really a problem, because:

1. it is perfectly consistent with what was written here before my edits,
2. it is perfectly consistent with what is currently written here, and
3. it is perfectly consistent with what we wrote in Pythagorean tuning, Quarter-comma meantone, and Interval (music).

So I am puzzled, because I cannot see the problem in the article, or the inconsistency with other articles where you accepted that ε is always seen as a positive number (subtracted when needed). If you want to explain in the article the difference between PT, 12th-c, 12tet, 11th-c, 6th-c, 4th-c, 3rd, please do it. This is good stuff, and I believe it is relevant, although most people do not need to know details about rarely used tuning systems. But please let me know whether, in your opinion, there's something wrong in my latest edit, because I cannot see the inconsistency with respect to what you wrote.

− Paolo.dL (talk) 08:50, 9 August 2010 (UTC)

Perhaps, I misinterpreted your words. It seemed too unbelievable to me that you stated something different from everything we wrote before (see points 1, 2, and 3 above), and from everything was written here before my edits. But perhaps you have been maintaining that,

1. being ε a fraction (exactly 1/12) of the diesis, it should have the same sign as the diesis, and
2. the size in cents of the Pythagorean comma is negative (as you showed a ratio < 1:1 in Diesis).

Is that what you meant? But it is not correct that the Pythagorean comma is defined as a negative number in cents (or a ratio < 1:1). On the contrary, it is defined by a ratio larger than 1:1. The ratio smaller than 1:1 (inverse of comma) that you used in Diesis means just you were referring to a descending comma, rather than an ascending one, but the comma stays positive. For instance, a descending pure fifth is 2/3, but the pure fifth is always defined as a positive number (3/2). Descending by an interval is conceptually equivalent to subtracting a positive size (or dividing by a ratio > 1:1), not to adding a negative size (or multiplying by a ratio < 1:1). Also, it would be bad to write that the Pythagorean fifth is 700-ε! Since it is larger than 700, the readers would not understand.

My point in short. In other words, as far as I know, interval sizes in cents are always positive (ratio larger than 1:1). And the diminished second is an interval. And ε is one twelfth of a diminished second. Thus ε is also an interval and always positive as well. In some tuning systems you subtract it, in others you add it, but its sign is always +.

As you can see, I am puzzled. Please let me know what you meant exactly. Paolo.dL (talk) 09:42, 9 August 2010 (UTC)

Your point in short hits the nail on the head. "The diminished second is an interval." It is above unison (1:1) when the fifth is tuned below 700 cents, as in meantone temperaments of eleventh-comma and above. It is unison when the fifth is tuned at exactly 700 cents in twelve-tone equal temperament. However, d2 is tuned below unison when the fifth is tuned above 700 cents, as in twelfth-comma and Pythagorean tuning; in that case, D is tuned below C, just as A♭ is tuned below G♯. To me it appears that d2 can be negative. That is the problem I've been trying to explain in this section. And that is also why I define the diesis and diaschisma as positive but the schisma and Pythagorean comma as negative intervals. − Glenn L (talk) 19:24, 9 August 2010 (UTC)

What about everything else I wrote? :-) I have already given detailed answers to everything you wrote. It was hard to write all that stuff! And I did it for you. Notice that I am not even English mother tongue (but I did not write just to train myself!). In short, if you can prove that the Pythagorean comma is typically defined as a ratio smaller than 1:1 (which is not consistent with the definition in Pythagorean comma), then we are authorized to use a negative ε (which would be, mathematically, a more versatile approach). Are you proposing to change the definition in Pythagorean comma? But please, read everything I wrote above. There's a lot more you should know before answering. Paolo.dL (talk) 23:53, 9 August 2010 (UTC)

I have read what you have written to me, and I appreciate it, having only good things to say about the information as well as your latest edits.

However, now that you ask me, I suppose that I am proposing that both the Pythagorean comma and the schisma be redefined as negative intervals. I suppose that Pythagorean tuning is considered strange by our standards today, because it results in, for example, G♯ > A♭ and B♯ > C > D instead of the other way around. The suggestion to redefine the Pythagorean comma and the schisma certainly appears to be revolutionary, but I think it makes sense in a way. I think I'll go to the Talk pages of both to make the suggestion. Thanks for the idea! :-) − Glenn L (talk) 02:59, 10 August 2010 (UTC)

Yes, it makes a lot of sense indeed, but I think we are not free to use a revolutionary definition here. We are bound to follow the literature. Original contributions are forbidden in Wikipedia.

Not only it makes sense, but I am on your side because I like people using their mind creatively, (provided they produce something useful). My very first contribution in Wikipedia was about the title of an article which I wanted to be changed (Exterior algebra) because it did not make sense (it was a senseless translation from German; it should have been "Extended algebra"!). I was absolutely right (in a way), but of course, unfortunately mathematicians accepted that bad translation for centuries, and it is now almost unthinkable to change it, even in the literature! I had been fighting so passionately in that discussion, that an administrator even decided to block me (see my talk page).

So, I am on your side, but I am bound to respect Wikipedia policies, not only because I am forced to, but also because they make a lot of sense. And even if you find one book that defines Pythagorean comma as negative, you perfectly know that the Pythagorean comma was known as a positive number even before the meantone and just intonation were invented.

Think about it: how can you change history? It would be the same as defining the universal gravitational constant as a negative number! It woulk make a lot of sense (because gravity is downward, and typically positive is upward), but unfortunately Newton decided to define it as a positive number, and EVERY book of physics gives it as a positive number. NOBODY will ever accept such a revolution (unfortunately), neither in the academic world (where original contributions are highly valued), nor in the much more limited world of Wikipedia, where original contributions are forbidden.

So, if you want to fight to change the definition in Pythagorean tuning, do it. I will even give you a (conceptual) weapon: in order to be positive, the Pythagorean comma must be (e.g.) C-B♯, instead of B♯-C. That shows the inconsistency better than everything else we wrote before, doesn't it? Even a baby would understand. But even with this weapon, you are bound to lose, because you are fighting against one of the strongest Wikipedia policies.

− Paolo.dL (talk) 21:31, 10 August 2010 (UTC)

Intro mistake?

For instance, if the base note for tuning the scale is C, this interval is from G♯ to E♭.

Correct me if I'm wrong, but it appears to me that G#-Eb is wolf fith for D-based pythagorian tune. For C we obtain fifths-circle sequence Gb Db Eb Bb F C G D A E B F# so Gb is not enharmonical to F#. So it appears to me that wolf fifts in this case are F#-Db, B-Gb. --Veprbl (talk) 00:35, 23 November 2011 (UTC)

Partially true. The sentence is at the same time unnecessarily detailed (in this context), and insufficiently detailed to be completely understood. Therefore, it creates a legitimate doubt, about a topic which is not relevant in this context. Specifying the base note is not enough to determine the endpoints of the stack, unless we say that the stack is "symmetrical", and that the note on the left of the 13-tone stack is removed to form the 12-tone scale (see 1/4-comma meantone#C- based construction tables). But in this context we don't need (and I don't want) to be so picky. See my edit. Paolo.dL (talk) 16:50, 24 November 2011 (UTC)

Midi example

Can we not use midi files for examples like this? Almost all popular media players and browsers don't support the format anymore; I had to import it into a new garageband project to hear it, and I'm pretty sure it didn't even play it correctly 71.225.210.10 (talk) 20:28, 2 December 2014 (UTC)

Title, introduction and redirects

Though the title of the article is Wolf interval, and Wolf fourth redirects here, the introduction is written as if the title were Wolf fifth. It looks like we need a good generic definition of wolf interval that is broad enough to encompass anything that could be called a wolf interval.—Theodore Kloba (☎) 13:58, 18 June 2019 (UTC)

plus-or-minus versus minus-or-plus

In this article I find this:

Similarly, we obtain nine minor thirds of 300 ± 3ε cents and three minor thirds (or augmented seconds) of 300 ∓ 9ε cents.

The "±" and the "∓" symbols are in two different fonts. Can anything be done about that? Michael Hardy (talk) 14:31, 29 July 2023 (UTC)