Terminology Crisis

Hello, first off, I'd like to introduce myself. You can call me Gotta Love Septimal Minor Thirds, I've been studying micro tonal music for over a year now, and I thought it was time to get started being more active in the microtonal community.

Anyway, delving into the huge lists of intervals named in scala, I found that many of the intervals had completely different names even when separated only by an interval of a schisma. (or smaller) Is it necessary to name such similar intervals with two distinct names? Does the schisma really have a musical function, or it is just theoretical? Worse yet, the information on these intervals on the internet is scarce, and you have to do a lot of digging around to find all the right stuff.

All of these things must be a huge turn off to anyone wanting to explore microtonal music past the basics. Why has no comprehensive dictionary of microtonal terms been made? Shouldn't we all be working on this? Is there something I've missed?

I say that we should all try to contribute as much missing information as possible to http://xenharmonic.wikispaces.com/ starting now.

I have a huge list, but I will list only a few of the intervals listed on scala with no explanation (and with no explanations found online):

Xenisma
Triaphonisma
Harmonisma
Ancient Chinese Tempering
Eratosthenes' Comma
Boethius Comma
Septendecimal Bridge Comma

--- In tuning@yahoogroups.com, "Gotta Love Septimal Minor Thirds" <microtonal76@...> wrote:

> Anyway, delving into the huge lists of intervals named in scala, I found that many of the intervals had completely different names even when separated only by an interval of a schisma. (or smaller) Is it necessary to name such similar intervals with two distinct names? Does the schisma really have a musical function, or it is just theoretical? Worse yet, the information on these intervals on the internet is scarce, and you have to do a lot of digging around to find all the right stuff.

The trouble is, two intervals separated by a schisma may have very different sizes and functions depending on temperament. If you decide to ignore the schisma, you are, more or less, tempering it out at least conceptually. And this is an idea with a lot of merit. Trouble is, once you've done that you've committed yourself. If you decide you want to do that in a meantone system, for instance, so that 81/80 is also tempered out, the only way to do that is 12 equal, and any idea you have of being xenharmonic has flown out the window. Typically, in a meantone tuning, the schisma is both larger much than it is in just intonation, and reversed in direction, so that it might for instance take you down a quarter-tone. On the other hand, if instead of 81/80 you decide you also want to temper out 250/243 you end up in 29 equal, and that may not be what you want at all. So you can't just blow things off because they are small, you must understand what tempering them out would entail.

> Xenisma

2058/2057, a comma of the 17-limit. If you do a search on the term in this group, you find a posting my Margo Schulter citing an earlier Xenharmonikon article by John Chalmers, and looking at that would presumably tell you why she was interested. Aside from that, I could expound at length just from what the math says, and you could put it into Graham's temperament finder and see what pops out. Probably some of the rest of your list will be similar, though 750/749, for instance, "ancient chinese tempering", is in the 107 limit and you'd need to find out how it was used and why they didn't just use the schisma for whatever purpose that was.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Gotta Love Septimal Minor Thirds" <microtonal76@> wrote:
>
> > Anyway, delving into the huge lists of intervals named in scala, I found that many of the intervals had completely different names even when separated only by an interval of a schisma. (or smaller) Is it necessary to name such similar intervals with two distinct names? Does the schisma really have a musical function, or it is just theoretical? Worse yet, the information on these intervals on the internet is scarce, and you have to do a lot of digging around to find all the right stuff.
>
> The trouble is, two intervals separated by a schisma may have very different sizes and functions depending on temperament. If you decide to ignore the schisma, you are, more or less, tempering it out at least conceptually. And this is an idea with a lot of merit. Trouble is, once you've done that you've committed yourself. If you decide you want to do that in a meantone system, for instance, so that 81/80 is also tempered out, the only way to do that is 12 equal, and any idea you have of being xenharmonic has flown out the window. Typically, in a meantone tuning, the schisma is both larger much than it is in just intonation, and reversed in direction, so that it might for instance take you down a quarter-tone. On the other hand, if instead of 81/80 you decide you also want to temper out 250/243 you end up in 29 equal, and that may not be what you want at all. So you can't just blow things off because they are small, you must understand what tempering them out would entail.

>
Ok, that is fine and all, but don't you find it kind of ridiculous how such small intervals can be represented by such large intervals? Wouldn't the significance of the intervals be null if the size is distorted by a certain degree?

I mean, what is the significance of this little tidbit from wikipedia on the breedsma?:

"In 12-tone equal temperament, the 50:49 interval is the unison, while the 49:48 interval is one step, so the breedsma is 100 cents (about 100 cents too wide); in 19-tone equal temperament the 50:49 interval is one step, while the 49:48 interval is the unison, so the breedsma is about b63 cents."

How can an interval that is about 100 cents wide exist in a temperament that has 100 cents as it's smallest interval?

> > --- In tuning@yahoogroups.com, "Gotta Love Septimal Minor Thirds" <microtonal76@> wrote:
> >
> I mean, what is the significance of this little tidbit from wikipedia on the breedsma?:
>
> "In 12-tone equal temperament, the 50:49 interval is the unison, while the 49:48 interval is one step, so the breedsma is 100 cents (about 100 cents too wide); in 19-tone equal temperament the 50:49 interval is one step, while the 49:48 interval is the unison, so the breedsma is about b63 cents."
>
> How can an interval that is about 100 cents wide exist in a temperament that has 100 cents as it's smallest interval?

Let me describe it this way:
In 12-equal, the prime 2 is represented by 12 steps, the prime 3 is represented by 19 steps, the prime 5 is represented by 28 steps, and the prime 7 is represented by 34 steps. -- So far so good.
Now, for example, 81/80 is equal to 3^4 / 2^4 / 5^1 (or 3*3*3*3/2/2/2/2/5), you can map this interval in 12-equal as "19*4 - 12*4 - 28*1" semitones, which is actually 0. You'll also get a result of 0 steps for intervals like the 5-limit schisma (32805/32768) or the Pyth. comma (531441/524288) or some others. But let's try something else -- for example, 2401/2400. This is 7^4 / 5^2 / 3^1 / 2^5, which you can express in terms of 12-equal steps as "34*4 - 28*2 - 19*1 - 12*5". This gives you a result of 1 step, not 0 steps. So although the original untempered interval is smaller than a single cent, it is, nevertheless, mapped to 100 cents in 12-equal -- and the fact that the original size is closer to 0 steps is irrelevant.
Petr

> You can call me Gotta Love Septimal Minor Thirds

That's funny -- the sm7 is the first non-twelve interval (triad, really, with the root and perfect 5th) that made me say, "Wow! I have to know more!"

I see you've been given an answer by Gene, but I find that I sometimes have to unpack his answers over the course of several days. Let me see if I can help, as a relative newbie from a newbie perspective; Gene and others, I'd be very interested if you think that my explanations are correct, even though they elide some of the details.

> Anyway, delving into the huge lists of intervals named in scala,
> I found that many of the intervals had completely different
> names even when separated only by an interval of a schisma.
> (or smaller) Is it necessary to name such similar intervals with
> two distinct names?

Short answer: Yes.

One reason is historical, cultural, or scholarly accuracy.

The other requires a longer answer, which I will get to after I explain a few other things.

> Does the schisma really have a musical function, or it is just
> theoretical?

Answer: Both, sort of. It depends on what you mean by "just theoretical".

I doubt anyone plays the schisma as a tone: That is, nobody plays a root C and, concurrently or consecutively, plays a 32805/32768 interval. If they do, it's part of some trickery known as a "comma pump", and for the most part you don't have to care.

That said, the schisma does have a musical function, or, perhaps more properly, the schisma is part of tonal reality. It's the difference between:

* 8 justly tuned perfect fifths (3/2 raised to the 8th power) plus a justly tuned major third and 5 octaves (2 raised to the 5th power times 5/4);

* the major limma and Pythagorean limma (I'd have to look them up);

* the syntonic comma (81/80) and the diaschisma (2048/2025).

Those are tiny differences, of course. You can wave them away if you like. That's what we mean when we talk about "tempering out a comma" -- you're pretending that a very small interval (a comma) is actually 1/1, a unison, which is zero cents. And tempering a comma changes your tuning somewhat. The more commas you temper, and the larger each comma is, the more your tuning changes.

That's not "just theoretical" at all. It has major implications for real-life scales.

Let's look at the just Canton scale, which you can find here:
http://xenharmonic.wikispaces.com/Canton

In it, you see a 13/11 major third, a 14/11 minor third, and a 3/2 perfect fifth. That mimics the traditional 5/4 major third, 6/5 minor third, and 3/2 perfect fifth.

But there's a difference.

* With traditional thirds, you multiply (5/4)*(6/5) and the 5s cancel out, so you get 6/4, which reduces to 3/2: A major third with a minor third stacked on top equals a perfect fifth.

* When you multiply (13/11)*(14/11), though, you get 182/121. That's close to a perfect fifth, but it's a hair sharp: 1.504132231 instead of 1.5 exactly. A 13/11 major third with a 14/11 minor third on top does *not* equal a perfect fifth.

That has all kinds of implications for Canton. Wouldn't it be nice if you could pretend (13/11)*(14/11) = 3/2? Hell yes. Well, what's the difference between 181/121 and 3/2? (181/121)/(3/2) = 364/363. That's 4.76 cents, practically undetectable. You'll never play it as a tone -- so can you just wave it away, pretend it doesn't exist, make it equal to zero?

Yes, you can. This is exactly the time when we want to "temper out a comma". We want to pretend that 364/363 = 1/1 = 0 cents.

Of course, saying that 364/363 = 1/1 implies that 364 = 363, which is ridiculous.

Or is it? We're talking about tempering the scale, which means we're making *approximations* in it. So what if we took the prime factors of 364 and 363, and approximated them in such a way that when you multiply them together, the approximate value for 364 = the approximate value for 363? We can make it so "364" and "363" are approximate, but the equality is exact.

To see what I mean, let's do it. Take the prime factors of 364 and 363: 364 = 2*2*7*13, and 363 = 3*11*11.

Now there are a bunch of ways you can make "364" = "363".

* You can replace the 11 with 11.01513 to make both the 13/11 and 14/11 slightly smaller: Instead of 289.21 and 417.51 cents, they're 286.83 and 415.13 cents. (Notice that you didn't change either 3 or 2, so the perfect fifth, 3/2, remains unchanged at 701.96 cents.)

* Maybe you'd prefer the sound of a pure 13/11, and you're worried that mucking about with 11 will mess that sound up. You could replace 7 with 6.9808. That will change your 14/11 (which is (2*7)/11) to 412.75. (Notice that the 3/2 is still unchanged, because you still didn't mess with 3 or 2.) Notice the 14/11 is even flatter than it was in the last example. That's because you have to squeeze the 14/11 a little harder, because you're not squeezing the 13/11 at all anymore, and you still want them to stack into a pure 3/2.

* Maybe you still want pure 13/11s, but you want less damage done to your 14/11. You could reduce the 7 a little bit, but not so much -- say, to 6.9900 -- and then *increase* the size of your 3/2 by replacing 3 with 3.00396. Now your 14/11 is 415.03 cents (vs. a 417.51 pure value), and your fifth is 704.24 cents (vs. a 701.96 pure value). You did less damage to your 14/11, but you had to stretch the 3/2 out a little bit to fit a (13/11)*(14/11) stack into it.

* Maybe you prefer the sound of a pure 14/11. You could keep the 7 pure and replace the 13 with 12.9643. Only the 13/11 is impure: 284.45 instead of 289.21.

You can see how each of these tempering methods does different amounts of "damage" to different intervals.

Generally speaking, however, you're going to be tempering multiple commas to make your scale function as a whole, so the easiest thing is to either (a) tell Scala to temper out your commas, or (b) pick an EDO that tempers all the commas you want, and then do the mapping of your values to the EDOs.

I believe Gene did the latter when he created the Cantonpenta scale, which is just a tempered version of the Canton scale. Because he tempered out 364/363 (and a few others), he created a more functional and useful scale with values practically indistinguishable from the original just version. You can find it here:
http://xenharmonic.wikispaces.com/cantonpenta

So, after all that talk, here's the point: That barely detectable less-than-5-cent comma isn't "just theoretical". Depending on context, how you handle it can be very important.

=====

Now let's get back to that question about whether almost-identical intervals need different names.

As you can see, different commas represent different powers of different primes. Also, since the way you tweak primes affects how much damage you do to the various intervals in your scales, you can't ignore the fact that these very close intervals are made of different primes.

(Well, really, you can, but you're giving up any consistent approach to developing a scale. Arbitrary changes seem less good to me than algorithms.)

Let's use a different example. consider 243/242, which Scala calls the "neutral third comma", and 225/224, the "septimal kleisma". 243/242 is 7.14 cents, and 225/242 is 7.71 cents. I doubt anyone can hear the difference between them.

I don't know what they're good for, honestly, because I'm still figuring out a lot of things. I'm guessing that 243 is something that involves neutral thirds, and the just neutral third is 11/9... Well, I'll take a guess and stack two 11/9s on top of each other and see how close I get to the 3/2. Turns out that 121/81 is 243/242 less than 3/2: The "neutral third comma" is the difference between two stacked neutral thirds and a perfect fifth.

As an aside: I think this is the first time I've guessed the use of a comma from its name.

243/242 uses primes 2, 3, and 11: it's 2^-1 * 3^5 * 11^-2. (I'll assume you understand that notation, but let me know if you don't.) The "monzo" for that (a notation named after Joe Monzo, who promoted its use) is | -1 5 0 0 -2 >. (The place for each number determines which prime you're talking about, and these are just in order: 2, 3, 5, 7, 11, etc. The number itself is the power that you're raising the prime to. The -1 is in the 2s place, so it represents 2^-1 = 1/2. The 5 is in the 3s place, so it represents 3^5 = 243. The -2 is in the 11s place, so it represents 11^-2 = 1/(11^2) = 1/121. Multiply them all together and you get 243/242.)

Meanwhile, 225/224 uses primes 2, 3, 5, and 7: it's 2^-5 * 3^2 * 5^2 * 7^-1, the monzo for which is | -5 2 2 -1 >.

So notice: If you temper out 243/242, you may affect your 3s and 11s. (Let's assume that we are keeping pure octaves, which is pretty common.) If you temper out 225/224, you may affect your 3s, 5s, and 7s. Totally different operations and implications.

Now, what if I want to stack two 11/9 neutral thirds to add up to a 3/2?

Tempering out the Septimal Kleisma won't work. 5 and 7 don't matter because they're not factors of the 11/9. I can shrink the 3 to 2.99333 to temper out 225/224, but while that shrinks the 3/2, it increases the 11/9: Now my neutral third is 355.11 cents, and my 11/9 is only 698 cents, and 2*355.11 does not equal 698. So yeah, I could mess around with 3 until I get a solution (it's about 2.9975), and you could further mess around with 5 and 7 to temper out 225/224 (if you don't change 5 or 7, you get 4.82 cents, and if you do change them, you're altering other notes for no good reason), so the whole idea of tempering commas isn't worth approaching in this way. In other words, you lose the conceptual framework of "tempering commas", you're left with methods are a lot less flexible than simply tempering commas, and everything has to be hand-crafted.

Instead, I can tackle the problem directly by tempering out 243/242: *By definition* I am getting rid of the difference between two 11/9s and a 3/2. *Any* method for tempering out 243/242 will give me what I want: TOP, Frobenius, Minimax, and a whole bunch of other methods I don't understand and really don't care about. (Hit ctrl-T in Scala to see them.) I could also pick an EDO that tempers out 243/242 and perform a mapping of my intervals in a spreadsheet. There are a bunch: 7, 10, 14, 17, 24, 31, 34, 38, 41, 48, 55, 58... -- yes, I have a spreadsheet for that, too -- and I could choose whatever EDO I like based on other intervals that I want to map, other commas I want to temper out, or other characteristics of that EDO.

Tempering out 243/242 isn't even close to tempering out 225/224. It's not about how close together they are, it's about a) what their prime factors are and b) what stacks of intervals they equate when they're tempered out. Can you see how they definitely need different names?

There are other implications of this, but I'll stop here to see if you get what I've said so far. Does that help?

=====

If you do get that, then maybe I can explain this part, too.

> I mean, what is the significance of this little tidbit from wikipedia
> on the breedsma?:
>
> "In 12-tone equal temperament, the 50:49 interval is the unison, while
> the 49:48 interval is one step, so the breedsma is 100 cents (about
> 100 cents too wide); in 19-tone equal temperament the 50:49 interval
> is one step, while the 49:48 interval is the unison, so the breedsma
> is about −63 cents."
>
> How can an interval that is about 100 cents wide exist in a
> temperament that has 100 cents as it's smallest interval?

Hopefully, after the discussion above, you can see that the Breedsma is the difference between a variety of different pitches or commas; whether that's important or not depends on what you're trying to do.

I think you meant, "How can an interval that's LESS THAN 100 cents wide exist in a temperament that has 100 cents as its smallest interval?"

You might consider that a major 3rd is 5/4 = 386 cents. How can it exist in 12-tone equal temperament? It doesn't "exist in" 12-tET -- but 12-tET *maps* that interval to the fourth step, or 400 cents. The minor 3rd, 6/5, is 316 cents, and it maps to 3 steps or 300 cents.

Every ratio, no matter how big or small, can get mapped to some number of steps within any EDO. Small ratios sometimes map to zero steps, and sometimes they don't. How it gets done seems a little complicated at first, but if you trust the mathematicians and just do the arithmetic, it's easy enough to deal with.

Using what's called the "patent val" for 12-EDO ("val" is a vector algebra construct, and "patent" means obvious: a "patent val for 12-EDO" is the obvious way to map ratios to EDO steps), you can see that 5/4 gets mapped to 4 steps, 6/5 (316 cents) gets mapped to 3 steps, 7/6 (266 cents) gets mapped to 3 steps, 7/4 (969 cents) maps to 10 steps (1000 cents) and so on. Trust me on that for a minute -- I intend to write this up for the wiki someday. (By the way: note that the patent val isn't always the *best* mapping, although it often is.)

Meanwhile, in 12-EDO:

* 50/49 gets mapped to 0 steps: That's what it means for it "to be the unison", and also what it means to "temper out a comma".

* 49/48 gets mapped to 1 step -- and since 1 step is 100 cents, that's a pretty jump from its pure value, which is about 36 cents. You can also see that this mapping process doesn't just mean simple rounding. Seems crazy, I know, but there's reason for it.

* The difference between 50/49 (as mapped) and 49/48 (as mapped) is 100 cents.

Similarly for 19-EDO, where the two are flipped:

* 50/49 gets mapped to 1 step.

* 49/48 gets mapped to zero steps (i.e., gets tempered out, becomes a unison).

* The step size for 19-EDO is about 63 cents, so the difference between 50/49 (as mapped) and 49/48 (as mapped) = 0 steps - 1 step = -63 cents.

So now, the Breedsma, which is the difference between 50/49 and 49/48, is 100 cents in 12 EDO and -63 cents in 19-EDO. In pure terms, it should be .72 cents. So 12-EDO and 19-EDO not only don't temper out the Breedsma, they exaggerate it wildly.

That's what the paragraph you quoted meant.

If you look at the next paragraph on Wikipedia, you see this: "In both 31-tone equal temperament and 41-tone equal temperament, the 49:48 and 50:49 intervals are each one step, so the breedsma is tempered out".

Since 49/48 and 50/49 both map to one step in 31-EDO and 41-EDO, they are the same; the Breedsma, which is the difference between 49/48 and 50/49, is therefore zero steps (a unison). Therefore, 31-EDO and 41-EDO both temper out the Breedsma.

Again, whether that's useful or not depends on what you're trying to do.

=====

On another note, I'd like to point out that you were fairly rude.

A lot of people have spent a great deal of time figuring this stuff out. (I'm not among them; I'm trying to keep up with stuff they knew years ago.) When you are asking questions, you should avoid phrasing them like "don't you find it kind of ridiculous how..." Because talking like that makes those who *have* contributed think that you are essentially saying, "Isn't this aspect of your life's work ridiculous?" to which the proper answer is, of course, a variant of "Pound sand, jerk."

And no, it's not ridiculous, and the fact that you think it is shows much more about your ignorance than anything else. Humble ignorance is fine, but condescending ignorance is not.

It's also quite annoying when the people who are demanding that it be updated are currently so incapable of updating it.

If you want to prevent turning off the people who have already done such deep work to make this stuff work out -- the giants on whose shoulders you are currently standing -- then you should listen and ask questions first, and be less demanding in your demands.

So I suggest you adopt a more polite tone, ask questions, absorb the answers, and then, when you are competent to do so (and only then!), start giving back to the community by updating the wiki.

Regards,
Jake
Newbie

Hey guys, thanks for all the information. I think that the main thing that was keeping my understanding back was that I have always thought of the schisma as being fundamentally the difference between the syntonic and Pythagorean comma. I didn't realize that it was also the difference between 8 perfect fifths plus a major third and 5 octaves. With that in mind, I can see how it would make a difference in temperaments. I still think the breedsma thing is kind of ridiculous though, especially considering there is nothing really septimal at all in 12-tet. But anyway, I think we can all agree that in just intonation the ear "physiologically tempers out" the schisma. For example, the pythagorean diminished fourth dosen't really act like a diminished fourth, it acts like a 5/4 interval. (thus going along with mathieu's claim that the pythagorean comma itself isn't really functional in just intonation, because it really acts as a syntonic comma)

Besides that, I still think that there should be more comprehensive descriptions of these commas on the xenharmonic wiki. You guys found info on the xenisma, but that is just the tip of the iceburg, I tried searching other comma names here with no results. The xenharmonic wiki seems to have a great deal of description on various temperament schemes, and what commas they temper out, but no information on the various commas themselves. (as far as I can tell)

--- In tuning@yahoogroups.com, "Gotta Love Septimal Minor Thirds" <microtonal76@...> wrote:

The xenharmonic wiki seems to have a great deal of description on various temperament schemes, and what commas they temper out, but no information on the various commas themselves. (as far as I can tell)
>

The 5 and 7 limit commas have a lot of information on them in the guise of discussion of the associated temperaments. What kind of information do you think is needed? Do you think commas need articles on them?

> I still think the breedsma thing is kind of ridiculous though, especially considering
> there is nothing really septimal at all in 12-tet.

Actually, the fact that there's nothing really septimal at all in
12-tET seems to be directly related to "the Breedsma thing" --
specifically, to the fact that 12-tET wildly exaggerates the Breedsma.

I'm not sure why your standard for something being ridiculous or not
is 12-tET, though. Maybe you could ask Graham Breed, who is on this
list, why this comma is interesting and worthy of his name. He seems
to be a fan of 31-tET -- maybe it's related to that.

But ask him politely. I think you calling things you don't understand
"ridiculous" is kind of ridiculous, and likely to highlight how
ignorant you are. You should learn some manners.

Regards,
Jake

On 6/19/11, Gotta Love Septimal Minor Thirds <microtonal76@...> wrote:
> Hey guys, thanks for all the information. I think that the main thing that
> was keeping my understanding back was that I have always thought of the
> schisma as being fundamentally the difference between the syntonic and
> Pythagorean comma. I didn't realize that it was also the difference between
> 8 perfect fifths plus a major third and 5 octaves. With that in mind, I can
> see how it would make a difference in temperaments. I still think the
> breedsma thing is kind of ridiculous though, especially considering there is
> nothing really septimal at all in 12-tet. But anyway, I think we can all
> agree that in just intonation the ear "physiologically tempers out" the
> schisma. For example, the pythagorean diminished fourth dosen't really act
> like a diminished fourth, it acts like a 5/4 interval. (thus going along
> with mathieu's claim that the pythagorean comma itself isn't really
> functional in just intonation, because it really acts as a syntonic comma)
>
> Besides that, I still think that there should be more comprehensive
> descriptions of these commas on the xenharmonic wiki. You guys found info on
> the xenisma, but that is just the tip of the iceburg, I tried searching
> other comma names here with no results. The xenharmonic wiki seems to have a
> great deal of description on various temperament schemes, and what commas
> they temper out, but no information on the various commas themselves. (as
> far as I can tell)
>
>
>
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--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> > I still think the breedsma thing is kind of ridiculous though, especially considering
> > there is nothing really septimal at all in 12-tet.
>
> Actually, the fact that there's nothing really septimal at all in
> 12-tET seems to be directly related to "the Breedsma thing" --
> specifically, to the fact that 12-tET wildly exaggerates the Breedsma.

If 2401/2400 is not tempered out, then 49/48 and 50/49 must come to a different number of scale steps. If the equal division is not too complex, one of them is therefore going to have to vanish. If it's 50/49, you get 12, 22, 26. If it's 49/48, you get 15, 19, 24, 29 and (by one mapping) 34.

> I'm not sure why your standard for something being ridiculous or not
> is 12-tET, though. Maybe you could ask Graham Breed, who is on this
> list, why this comma is interesting and worthy of his name. He seems
> to be a fan of 31-tET -- maybe it's related to that.

Some observations: 2401/2400 has a fourth power for a numerator; for n^4/(n^4-1) below the 13 limit there's only 16/15 and 81/80, both 5-limit, and 2401/2400, 7-limit. This may sound like numerology but actually it relates to some important approximations; just as 81/80 = (9/8)/(10/9), we have 2401/2400 = (49/48)/(50/49). Also, it is the gap between the two different septimal neutral thirds, 60/49 and 49/40, so that if you temper it out these are the same and you split the fifth in half. 7-limit pitch classes can be reduced from three to two dimensions by tempering out 2401/2400, where you can place classes on a grid with one direction given by the 60/49~49/40 neutral third, and the other by 10/7 up or 7/5 down. This is actually quite efficient, and since 2401/2400 is so small, you effectively get just intonation (this is called "microtempering".)

2401/2400 with its 7^4 numerator is strongly septimal in terms of the comma pumps or "puns" associated to it: if you have root movements by 7/6, 7/5 or 7/4 you are likely to end with a root shifted 2401/2400 up, which you are welcome to try to hear. Or 8/7, 10/7, 12/7 movements are likely to lead to 2400/2401 movements down. So long as you are willing to plunge into full-blown septimal harmony, it's not that complex a relationship, yet it's a very small comma.

> But ask him politely. I think you calling things you don't understand
> "ridiculous" is kind of ridiculous, and likely to highlight how
> ignorant you are. You should learn some manners.

I doubt Graham's been offended by that.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@> wrote:
> >
> > > I still think the breedsma thing is kind of ridiculous though, especially considering
> > > there is nothing really septimal at all in 12-tet.
> >
> > Actually, the fact that there's nothing really septimal at all in
> > 12-tET seems to be directly related to "the Breedsma thing" --
> > specifically, to the fact that 12-tET wildly exaggerates the Breedsma.
>
> If 2401/2400 is not tempered out, then 49/48 and 50/49 must come to a different number of scale steps. If the equal division is not too complex, one of them is therefore going to have to vanish. If it's 50/49, you get 12, 22, 26. If it's 49/48, you get 15, 19, 24, 29 and (by one mapping) 34.
>
> > I'm not sure why your standard for something being ridiculous or not
> > is 12-tET, though. Maybe you could ask Graham Breed, who is on this
> > list, why this comma is interesting and worthy of his name. He seems
> > to be a fan of 31-tET -- maybe it's related to that.

OK, that seems to make more sense. The only thing I am reacting to is the general lack of competent explanations in one location in the microtonal community for explanations. No specific person. :) (and if I have missed something, please, just point it out to me, a book, an article, anything.)

As to why the xenharmonic wiki doesn't explain the commas themselves enough, this is why:

A. Clues as to how the commas are constructed; what intervals are used to create them are needed.

B. Historical background; When the comma was discovered, when/why it was named, etc... is needed.

C. More extensive examples of usage are needed.

And yes, I do think the commas deserve their own articles, especially the ones that aren't specifically related to any one temperament scheme.

--- In tuning@yahoogroups.com, "Gotta Love Septimal Minor Thirds" <microtonal76@...> wrote:
>
> As to why the xenharmonic wiki doesn't explain the commas themselves enough, this is why:
>
> A. Clues as to how the commas are constructed; what intervals are used to create them are needed.
this can be taken from the monzo notation, for example this |-7 3 1>
means 2^(-7) * 3^3 * 5^1 or 3*3*3*5/(2*2*2*2*2*2*2) - there are many way to get a comma, but the prime factors in use are the same. Maybe, if you combine 3 just fifths (with ratio 3:2) - say c->g->d1->a1 - and a just major third (5:4) a1->cis2 and lower this tone by two octaves, than you get the pelogic comma (135:128)
http://xenharmonic.wikispaces.com/135_128

>
> B. Historical background; When the comma was discovered, when/why it was named, etc... is needed.
absolutely.
>
> C. More extensive examples of usage are needed.
dito.
>
> And yes, I do think the commas deserve their own articles, especially the ones that aren't specifically related to any one temperament scheme.
>
the first pages dedicated to individual commas just appeared :)

Best,
Wolf

--- In tuning@yahoogroups.com, "wolfpeuker" <wolfpeuker@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Gotta Love Septimal Minor Thirds" <microtonal76@> wrote:
> >
> > As to why the xenharmonic wiki doesn't explain the commas themselves enough, this is why:
> >
> > A. Clues as to how the commas are constructed; what intervals are used to create them are needed.
> this can be taken from the monzo notation, for example this |-7 3 1>
> means 2^(-7) * 3^3 * 5^1 or 3*3*3*5/(2*2*2*2*2*2*2) - there are many way to get a comma, but the prime factors in use are the same. Maybe, if you combine 3 just fifths (with ratio 3:2) - say c->g->d1->a1 - and a just major third (5:4) a1->cis2 and lower this tone by two octaves, than you get the pelogic comma (135:128)
> http://xenharmonic.wikispaces.com/135_128
>
> >
> > B. Historical background; When the comma was discovered, when/why it was named, etc... is needed.
> absolutely.
> >
> > C. More extensive examples of usage are needed.
> dito.
> >
> > And yes, I do think the commas deserve their own articles, especially the ones that aren't specifically related to any one temperament scheme.
> >
> the first pages dedicated to individual commas just appeared :)
>
> Best,
> Wolf
>
Great. Well, I do believe that my rant is over (for now). Thanks for dealing me, and I hope I didn't make the wrong first impression; just take all this as constructive criticism from an outsider looking in. Now I plan to contribute to the community and try and help with some of the issues where possible. :)