Somebody requested that this file be edited so that the interval is presented in "ascending" instead of "descending" form. But I am not sure what that would mean, exactly, because this file is more complicated than a simple descending interval. It seems like the sequences of pitches in the current file is 1/1, 2/1, 1/1, 5/4, 25/16, 125/64, 2/1, 125/64, and then 2/1 and 125/64 at the same time. This is to illustrate not only the size of the 128/125 diesis interval that exists between the pitches 125/64 and 2/1, but also a way it can arise (as the difference between a stack of three 5/4's and an octave). What should be the sequence of pitches in the new version? —Keenan Pepper 19:28, 12 August 2010 (UTC)

In other words, the file shows:

• the ascending octave C-C' played melodically (i.e. successively)
• the three ascending 5:4's C-E-G#-B# played melodically
• the descending diesis C'-B# (125/128, from 2/1 to 125/64) played melodically
• the same diesis played harmonically (i.e. simultaneously)

As you wrote, the diesis is typically defined as a 128:125 interval. This means it is seen as an ascending inteval (ratio > 1, size in cents > 0). This audio file, however, shows a 125/128 interval (from 2/1 to 125/64), i.e a descending diesis. Thus, the descending diesis should be reversed. This means it should be B#-C' (128:125), rather than C'-B# (125/128). This may also imply, optionally, a different order in the previous sequences, i.e.:

• the three ascending 5:4's C-E-G#-B# played melodically
• the ascending octave C-C' played melodically
• the ascending diesis B#-C' (128:125, from 125/64 to 2:1) played melodically
• the same diesis played harmonically

− Paolo.dL (talk) 21:51, 12 August 2010 (UTC)

That sounds fine to me, and I'll make a new file when I have access to a machine with ZynAddSubFX on it. —Keenan Pepper 22:13, 12 August 2010 (UTC)

Thank you! By the way, you might know it better than I do, but I would like to remind you that defining all intervals as ascending in frequency (ratio > 1, size in cents > 0) is a widely accepted convention, valid even for commas which should be described as descending intervals, because when they ascend in pitch they "descend" in staff position! E.g., in Pythagorean tuning the interval from C' to B♯ is an ascending Pythagorean comma, but it goes down by one staff position. I discussed this topic with Glenn L in this talk page), and we agreed that there are valid reasons to define the Pythagorean comma as descending in pitch and hence ascending in staff position (e.g. from B♯ to C'). Clearly, this would make comparisons with other commas (such as the diesis) much more effective (see the table in comma (music), for instance), but as far as I know nobody in the literature does it.

− Paolo.dL (talk) 22:34, 12 August 2010 (UTC)

That's certainly an interesting idea, but if you're going to define intervals as having negative cent sizes, then phrases like "raise that pitch by a Pythagorean comma" become confusing. If the Pythagorean comma has a negative size (-23.4... cents), then "raising" a pitch by the Pythagorean comma would result in a pitch lower than the original. I prefer to define all intervals as positive.

(Aside: Something else you should be aware of is the existance of positive intervals which not only have negative "staff position" offsets, but also negative 12-equal representations. For example, consider the interval 78732/78125. It is certainly a positive interval in just intonation, because 78732 > 78125. However, if we let 1/1 be C, then 3/1 is G, 9/1 is D, and 3^9 = 19683/1 is a D#, and 78732/1 = 4*3^9 is also D#. Similarly, 78125/1 = 5^7 is enharmonically equivalent to an E (technically Cxx - C quadruple-sharp). So we're forced to conclude that even though 78732/78125 is positive in just intonation, it is negative if you're building it up by counting semitones of 12-equal. We say that 78732/78125 itself is positive, but its "mapping" into 12-equal is negative.

Something which might be surprising is that this is not an accident - you can prove that for any equal temperament and any mapping of prime numbers you can think of, there will always be intervals whose mapping has the wrong sign. In other words, no mapping preserves the sign of every interval.) —Keenan Pepper 00:08, 13 August 2010 (UTC)

"Raising by a comma" means going up in pitch, so in my opinion there's no ambiguity even if the comma is defined as negative. In other words, when you want to "descend by a comma", and the comma is defined as ascending (i.e. positive), you do understand that you need to change its sign, or invert its ratio. Similarly, when you want to "ascend by a comma", and the comma is defined as descending (i.e. negative), you do understand that you need to change its sign, or invert its ratio. In other words, "raising", "lowering", "flattening", sharpening", "ascending", "descending" is not equivalent to the neutral algebraic terminology, "adding", "subtracting", "plus", and "minus". And of course, if you define a comma as negative, an interval "plus" that comma would be flattened. What's the problem with that? On the contrary, in my opinion it is desirable: it is exactly what would make comparisons between Pythagorean comma and other commas more effective. In other words, the "difference" between Pythagorean comma and diesis would be 41.06- (-23.46) = exactly 3 syntonic commas, as it should be! Anyway, I am aware that it is impossible to change the sign of the Pythagorean comma, defined more than 2 millennia ago: it would be as unthinkable as changing the sign of the universal gravitational constant.

As for your definition of mapping into 12-equal, I am not sure I understand why 5^7 is enharmonic to E (although I trust you), and more importantly, I have studied 5-limit tuning, but I have never felt the need to use an interval such as 78732/78125, which cannot be found in a typical 12-tone scale (see table), nor to map it into 12-TET.

By the way, I am not even sure about the definition of accidentals in just intonation. Just intonation involves the use of 2-D construction tables (for 5-limit), or N-D for higher limits. However, User:Woodstone maintains that we need to use a unidimensional stack of fifths to define the type and number of accidentals in 5-limit. We briefly discussed this in Talk:Just intonation#Sharp or flat?. For instance, here is how I would use the accidentals in the 2-D construction table for 5-limit:

I marked in yellow all the notes with base ratio above C (so I am considering the sequence of notes obtained before multiplying them by 2^n, i.e. before bringing them within the same basic octave). This is the rule: all the non-diatonic notes are sharpened when their base ratio is above C, while the ones with base ratio below C are flattened. In my opinion, this is the correct "bidimensional thinking" to assign accidentals, but I am not sure if this method is used in the literature. Woodstone maintains that it is not. Thus, currently, the table in 5-limit tuning does not respect this simple rule and is built using the "unidimensional" strategy (stack of fifths). If you want to give your opinion about the strategy to assign accidentals, please do it in Talk:Just intonation#Sharp or flat?.

Paolo.dL (talk) 10:46, 13 August 2010 (UTC)