Yahoo Tuning Groups Ultimate Backup tuning Re: [tuning] Why not 31TET as basis for a cent-like logarithmic measure (was "so what about cents?")

>"3) 12-equal is a great coarse template, and not just because it's the
existing convention. It's a good coarse template because it's very low
in badness, and hence gets you very close to a lot of just intervals."

If we're talking low badness...why not a log system based on 31 equal? Beat 12 equal to death when it comes to that for virtually all limit ratios except 13-odd-limit and it's not that much larger... Of course, to get diatonic, one would have to memorize the steps involved to form the MOS, but the same problem occurs in 12TET. I'd say the dyadic ratios 31TET represents are pretty close to "inclusive of all scales"...unless you expect musicians to memorize the tons of steps in much higher TET's.

>"4) This leads to a handy encapsulating of information where,
heirarchically speaking, the coarse grid (hundreds-place) is enough to
get you decently close to where you want to go, and the tens-place is
enough to really dial it in, and then the ones-place is there to perfect the intonation and so on.
6) Hence in 12-equal, conformity to the coarse grid is roughly
inversely proportional to complexity, which is likely a very desirable
property if we're treating more complex intervals as less important
than less complex ones."

Using a 31TET logarithmic "31TET cent" measure also encapsulates these advantages...only better. It's pretty close to capturing an entire list of all the "normal limit" (read, under 11 odd limit or so) JI dyads and tons of the up to 25-limit or so triads.

So why not 31TET "cents"?

🔗Jake Freivald <jdfreivald@...>

> If we're talking low badness...why not a log system based on 31 equal?

Mike B said, "I'm absolutely not in favor of [moving off of cents], and not just because of convention," and then went on to point out a lot of good reasons to keep cents. Because he gave scant attention to convention, one might be tempted to move to a different, more effective "coarse template".

Don't be tempted. Convention is precisely why you should stick with cents, even if you can find a better coarse template.

Most musicians need a bridge between their well-traveled land and this alien backwater called microtonality. All musicians are familiar with 12-EDO, so the 12-EDO "bias" inherent in cents is a feature, not a bug, for communicating with them.

Put differently: 1200 cents may be "just convention", but convention is _everything_ in communication.

If you don't believe me, try to answer the following questions:

Q1. How good is a 588-millioctave fifth? Is it sharp or flat? By how much? And how much does it differ from 12-EDO? (And are you sure it's a fifth?)
Q2. Is 800 millioctaves a sixth or a seventh?
Q3. Is 800 millioctaves closer to a major sixth, to 7/4, or to two stacked 4/3s?

Non-cents indeed.

To be fair, creating a new system wouldn't make *everything* harder. Even with cents, newbies need to memorize the fact that a 7/4 is 968 cents. However, with cents, they can instantly see and easily remember that the harmonic seventh is well flat of their familiar 1000-cent minor seventh.

In sum: If people want to be more obscure, less able to explain themselves to 12-only folks, and make it harder to help people learn alternate tuning systems, they should remove their 12-EDO signposts and go to something less 12-centric. May their self-righteous objectivity compensate for their loneliness. :)

Regards,
Jake

P.S. If it matters:
A1: That's the 706-cent fifth from 17-EDO. And once I've said that, you don't need me to answer the rest.
A2: A seventh.
A3: The 7/4. A major sixth is about two half-steps higher than a P5, or ~900 cents. 4/3 is about 5 12-EDO steps, so two of them is 1000 cents. 7/4 is 968 cents. 800 millioctaves is 960 cents.

🔗ixlramp <ixlramp@...>

🔗cityoftheasleep <igliashon@...>

I agree with Jake. On all counts.

-Igs

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> > If we're talking low badness...why not a log system based on 31 equal?
>
> Mike B said, "I'm absolutely not in favor of [moving off of cents], and
> not just because of convention," and then went on to point out a lot of
> good reasons to keep cents. Because he gave scant attention to
> convention, one might be tempted to move to a different, more effective
> "coarse template".
>
> Don't be tempted. Convention is precisely why you should stick with
> cents, even if you can find a better coarse template.
>
> Most musicians need a bridge between their well-traveled land and this
> alien backwater called microtonality. All musicians are familiar with
> 12-EDO, so the 12-EDO "bias" inherent in cents is a feature, not a bug,
> for communicating with them.
>
> Put differently: 1200 cents may be "just convention", but convention is
> _everything_ in communication.
>
> If you don't believe me, try to answer the following questions:
>
> Q1. How good is a 588-millioctave fifth? Is it sharp or flat? By how
> much? And how much does it differ from 12-EDO? (And are you sure it's a
> fifth?)
> Q2. Is 800 millioctaves a sixth or a seventh?
> Q3. Is 800 millioctaves closer to a major sixth, to 7/4, or to two
> stacked 4/3s?
>
> Non-cents indeed.
>
> To be fair, creating a new system wouldn't make *everything* harder.
> Even with cents, newbies need to memorize the fact that a 7/4 is 968
> cents. However, with cents, they can instantly see and easily remember
> that the harmonic seventh is well flat of their familiar 1000-cent minor
> seventh.
>
> In sum: If people want to be more obscure, less able to explain
> themselves to 12-only folks, and make it harder to help people learn
> alternate tuning systems, they should remove their 12-EDO signposts and
> go to something less 12-centric. May their self-righteous objectivity
> compensate for their loneliness. :)
>
> Regards,
> Jake
>
> P.S. If it matters:
> A1: That's the 706-cent fifth from 17-EDO. And once I've said that, you
> don't need me to answer the rest.
> A2: A seventh.
> A3: The 7/4. A major sixth is about two half-steps higher than a P5, or
> ~900 cents. 4/3 is about 5 12-EDO steps, so two of them is 1000 cents.
> 7/4 is 968 cents. 800 millioctaves is 960 cents.
>

🔗Mike Battaglia <battaglia01@...>

Yikes, a lot of replies. I'll try to respond all at once:

Michael: it would be very interesting to investigate the use of
31-equal for this. When you get up to 31-equal, you end up with a
pretty fine coarse grid. The coarse grid basically has to do with
long-term memory - for cents, you have the sound of 12 different
signposts memorized, and then the "fine grid," which isn't really a
grid at all, is a formula to navigate you between signposts. For
31-equal, you'd have to memorize 31 different signposts. I'm not
ruling it out, but I wonder how hard that might be. It would be best
to pick a tuning that is easy to sing.

It's also noteworthy, again, that the memorization of the full
12-equal coarse grid is greatly facilitated by the fact that it is 7+5
in meantone - you can just think of the diatonic scale, and
intersperse one small step in between each of the large ones. So that
might go a long way in helping people memorize the layout for
31-equal, which they'll have to do whether people use it to replace
"cents" or not.

To Jake: We will likely always stick with cents simply because of
convention, but the reasons that I listed are, in my opinion, some of
the underlying "hidden conventions" that go with the use of cents. We
are, in fact, using it because it's based around 12-EDO, but the fact
that 12-EDO happens to be a pretty stellar tuning also leads to the
other properties mentioned. To pick another stellar tuning would
likely lessen the impact of switching the system up, because it'll be
intuitive enough that most of the vital features will still be met.
Kind of like how Esperanto is designed to be maximally easy to learn
because it builds on intuition.

Besides, this is all just for fun, anyway. It would be pretty useful
if we, as experts, suddenly discovered that an easier and more
intuitive system existed for describing the intonation of 13-limit
ratios than cents. But yeah, for pedagogical purposes, cents are
probably key.

To Mat Cooper: I say if you're going to use 72EDO you might as well
just use 12, since the point of this is to keep the coarse grid as
small as possible - ideally something you can sing. and round the
cents off. But the use of 72EDO is sort of implicit in using 12,
because you can just say that 11/8 is about 5 1/2 semitones or 550
cents, or that 7/4 is about 9 2/3 semitones or 966 cents. Likewise, it
might be useful to use 17-EDO as the coarse grid, since it easily
gives you 34-EDO and 68-EDO - 68 being kind of like the 17-EDO based
evil twin to 72-EDO.

-Mike

🔗Daniel Nielsen <nielsed@...>

🔗Michael <djtrancendance@...>

🔗Mike Battaglia <battaglia01@...>

🔗Jake Freivald <jdfreivald@...>

🔗Mike Battaglia <battaglia01@...>

On Wed, Jun 15, 2011 at 9:04 AM, Jake Freivald <jdfreivald@...> wrote:
>
> > Kind of like how Esperanto is designed to be maximally easy to learn
> > because it builds on intuition.
>
> And we can see how useful and successful that strategy is by noting the
> incredible growth of Esperanto speakers over the last few decades.

This is a strawman.

> I know it's "just for fun", but you have to decide what you want. You're
> under no obligation to expand the microtonal community, but it has
> always seemed like that something many people want. If you want to
> encourage growth of a community, you minimize the barriers to entry into
> that community.
>
> "We'll always use cents" will be true except for those places you use
> 19-EDO or 31-EDO non-cents, and those places will be more opaque to
> musicians who are new to microtonality.

I don't know where you see me saying not to use cents. I said that if
we wanted to find another coarse grid, this is the way to find one. If
we're dealing with novices getting into microtonality, we'd use cents.
If we're on tuning-math and talking amongst ourselves, maybe we'd find
it more useful to use 19-EDO or something. I'm sure the multilinear
algebra is going to put them off long before they even get to the
cents issue.

-Mike

🔗Michael <djtrancendance@...>

>"We'll always use cents" will be true except for those places you use

19-EDO or 31-EDO non-cents, and those places will be more opaque to

musicians who are new to microtonality."

I get the sense we're making a sweeping assumption musicians, even those not familiar with classical music, instinctively know cents. I highly doubt that. I asked my professional guitarist brother about them, for example, and he had no clue. After telling him "well, 100 cents is a semi-tone"...he could, for example, take a microtonal scale and tell me if it was 12TET-like or not and where to find chords he already knew, but that's about it. Certainly it seemed to tell him nothing about how JI works or how to make new chords.

In short, I think the main fault of cents is that it tempts people to simply find microtonal scales very like 12TET, rather than new ones...and the musical result is people tuning off 12TET to find more dissonance, rather than to find new chords. One key example of this: blues, where the quarter tones function for melodi/harmonic tension...but not much else. To me it's like adding sugar to ice cream and calling it gelato...not quite true.

--- On Wed, 6/15/11, Jake Freivald <jdfreivald@...> wrote:

From: Jake Freivald <jdfreivald@...>
Subject: Re: [tuning] Why not 31TET as basis for a cent-like logarithmic measure (was "so what about cents?")
To: tuning@yahoogroups.com
Date: Wednesday, June 15, 2011, 6:04 AM

> Kind of like how Esperanto is designed to be maximally easy to learn

> because it builds on intuition.

And we can see how useful and successful that strategy is by noting the

incredible growth of Esperanto speakers over the last few decades.

I know it's "just for fun", but you have to decide what you want. You're

under no obligation to expand the microtonal community, but it has

always seemed like that something many people want. If you want to

encourage growth of a community, you minimize the barriers to entry into

that community.

"We'll always use cents" will be true except for those places you use

19-EDO or 31-EDO non-cents, and those places will be more opaque to

musicians who are new to microtonality.

Regards,

Jake

🔗Michael <djtrancendance@...>

🔗Graham Breed <gbreed@...>

Michael <djtrancendance@...> wrote:

> Granted, 19EDO is not a bad alternative. I'd vouch
> for 19EDO as a good starter point...somewhere between
> 12EDO's "fair share of good ratios summarized in few
> tones" and 31EDO's "almost all important ratios with
> great accuracy, but requiring many tones to be
> memorized". Anyone else for 19EDO "cents"?

The trouble with 19 and 31 is that they're prime numbers.
Simple equal divisions of the octave won't be clear with
them. 12 does very well. Millioctaves have their
advantages as well, but only if you have a calculator, and
in that case multiplying by 1.2 isn't a big deal anyway.

Another thing about 19EDO is that it becomes
inconsistent/ambiguous in the 13-limit. Both mappings are
still good, but it's probably bad for a universal system
if there isn't a unique direction to get at each ratio from.

I'm quite a fan of 31-equal. My username might indicate
that. I believe Paul Erlich taught himself to recognize
all 31 intervals. That sounds like a good idea to me if
you want to learn a microtonal system. It's a framework
that'll get you close enough to a lot of ratios that you
can adjust by ear. (If you don't care about simple ratios,
it doesn't matter what framework you learn.) 31 also
supports Meantone, Miracle, Orwell, Mohajira, and so on.
(Of course, this follows from it having low badness as an
equal temperament.) But I still wouldn't propose a
cents-like system with 31.

You can take any equal division you like, and express
intervals in terms of it, with a few decimal places. If it
makes sense in the context, it'll be understood. There's
no need for another standard.

Graham

🔗Mike Battaglia <battaglia01@...>

On Wed, Jun 15, 2011 at 10:29 AM, Graham Breed <gbreed@...> wrote:
>
> Another thing about 19EDO is that it becomes
> inconsistent/ambiguous in the 13-limit. Both mappings are
> still good, but it's probably bad for a universal system
> if there isn't a unique direction to get at each ratio from.

12 is pretty ambiguous in the 13-limit, but we use it anyway. But just
how bad is 19 inconsistent in the 13-limit? C-Ab is close to 13/8,
C-D# is close to 13/11, C-E# is close to 13/10... I'm not on my main
computer to check now, I'll assume that it's something like 13/7 that
gets screwed up.

We could try comparing 19 and 26 and see what's more intuitive. 19 is
going to be more accurate, but 26 is consistent through the 13-limit.
Maybe it'll just boil down to which is larger.

> I'm quite a fan of 31-equal. My username might indicate
> that. I believe Paul Erlich taught himself to recognize
> all 31 intervals. That sounds like a good idea to me if
> you want to learn a microtonal system. It's a framework
> that'll get you close enough to a lot of ratios that you
> can adjust by ear. (If you don't care about simple ratios,
> it doesn't matter what framework you learn.) 31 also
> supports Meantone, Miracle, Orwell, Mohajira, and so on.
> (Of course, this follows from it having low badness as an
> equal temperament.) But I still wouldn't propose a
> cents-like system with 31.

31-equal is great. So is 34-equal, for that matter, although it never
gets any love anymore. I just feel like these are huge for a
cents-like system. Going all the way up to 31 for a coarse grid is
like going halfway to infinity. You might as well at that point
consider looking at 1200-EDO, which also has the added benefit of
representing all intervals within a cent.

> You can take any equal division you like, and express
> intervals in terms of it, with a few decimal places. If it
> makes sense in the context, it'll be understood. There's
> no need for another standard.

The fact that 12 subdivides into 72 might make it unbeatable.

-Mike

🔗Jake Freivald <jdfreivald@...>

I think this horse is well-beaten, so I'll make this my last post on the subject.

Mike B said:
> > > Kind of like how Esperanto is designed to be maximally easy to learn
> > > because it builds on intuition.
> >
> > And we can see how useful and successful that strategy is by noting the
> > incredible growth of Esperanto speakers over the last few decades.
>
> This is a strawman.

No, a strawman is where I put words in your mouth and then argue against the falsely constructed argument instead of what you actually think. I'm definitely not trying to do that. I'm just taking your analogy and running with it.

Also, my answer may have come off as flip, but I was just trying to be short. I don't mean to be disrespectful and I apologize if I came off that way.

In theory, Esperanto should be easy to learn and therefore should become a lingua franca; in practice, it is unrooted in any particular culture (except perhaps the subcultures of linguists and internationalists) and has, over the course of a century, barely gotten a foothold anywhere. By comparison, the incredibly unwieldy English language, grounded as it is in the particular culture and history of the Anglo-Saxons and their descendants, has several hundred million native speakers and (by some estimates) three times that number of non-native speakers.

This is not to say that some powerful people can't use Esperanto among themselves to great effect -- George Soros is a native speaker, since his father was a dedicated Esperantist -- but the language's theoretical benefits have not given rise to a significant population of speakers. It's likely that as many people speak Welsh as speak Esperanto, and Welsh is considered a threatened language that requires special support from the BBC and other cultural entities.

Abstract and theoretical arguments are great, as far as they go, but if you're talking about maximizing adoption rates, nothing beats sticking with a known entity and proven winner.

(This conversation reminds me of one of my favorite Yogi Berra quotes: "In theory, there's no difference between theory and practice. But in practice, there is!")

> I don't know where you see me saying not to use cents.

You didn't. But consider these two statements that are juxtaposed in your last post:

> If we're dealing with novices getting into microtonality, we'd use cents.
> If we're on tuning-math and talking amongst ourselves, maybe we'd find
> it more useful to use 19-EDO or something.

Now let's consider mailing lists and interaction mechanisms for a moment (i.e., ignoring websites). We have the tuning list, the tuning-math list, the Xenharmonic Alliance on Facebook, and a few other cats and dogs. The tuning-math list is clearly not for novices (unless they also have a high degree of mathematical competence). But which of the others is for "novices getting into microtonality" vs. "talking amongst ourselves"?

Michael said:

> I get the sense we're making a sweeping assumption musicians, even
> those not familiar with classical music, instinctively know cents.

Nope. I didn't know about cents until a year ago. I'm making a sweeping assumption that musicians are familiar with 12-EDO. Based on that assumption, I'm making an assertion (not an assumption) that the relation of cents to 12-EDO is a feature rather than a bug, because (for example) you can tell your brother that a semitone is 100 cents and he has an immediate understanding of how granular cents are.

Moreover, he has signposts related to what he already knows: If you told him that barbershop quartets use a 768-cent harmonic seventh a lot, he could, with little-to-no effort, see that this is significantly flat from the 12-EDO minor 7th he knows, but still closer to the m7 than to the M6.

Finally, if he looks around the Internet, almost everything he reads about tuning will be documented using ratios or cents.

> Certainly it seemed to tell him nothing about how JI works or how to make new chords.

I'll bet. But are you asserting that a 258.3-31-EDO-unit semitone, or even the 998-31-EDO-unit major 3rd, *would* tell him how JI works or how to make new chords?

There's a lot of explanation that has to accompany those units in order to teach him those things, whether those units are 31-EDO or 12-EDO based.

> In short, I think the main fault of cents is that it tempts people to
> simply find microtonal scales very like 12TET, rather than new ones...
> and the musical result is people tuning off 12TET to find more
> dissonance, rather than to find new chords.

Aren't you saying that people *should* be finding new chords instead of finding microtonal scales that are similar to 12TET? And isn't this imposing on people your view of what microtonality *should* be? ;)

I'm currently quite happy to tune off 12TET to find new *consonances*, not new dissonances. 7/4, 13/11, 14/11, and 7/6 are all beautifully consonant, and while I know you like chords with lots and lots of notes, I tend to like triads. And I can do things with these consonances that I wouldn't do in 12-EDO: I don't care for the sound of C-Eb-F, but if you replace the Eb with a 7/6, I think it sounds pretty good.

Nothing wrong with that, right? Even though I'm not deviating far from 12-EDO?

> One key example of this: blues, where the quarter tones function for
> melodi/harmonic tension...but not much else. To me it's like adding
> sugar to ice cream and calling it gelato...not quite true.

I love the blues. Its deviations from 12-EDO are significant enough that you can say "he plays a great blues guitar", and millions of musicians and non-musicians alike, all over the world, will instantly know what you mean.

So you can say that the blues aren't microtonal enough for you, but when a musical style goes beyond 12-EDO to add melodic and harmonic tension (and sometimes to reduce it -- lots of blues players go for "sweet" sounds), and millions of people enjoy it, I'd say microtonality is working.

Regards,
Jake

🔗Steve Parker <steve@...>

I disagree entirely. I work a lot with actual musicians in JI and the ones that struggle the most are the ones who reference everything to cents flat or sharp from 12ET.
The ones who take the intervals on their own merits get to actually play well much quicker.

Two quotes from a booklet I wrote a while ago:

'It quickly becomes necessary to learn how to move between ratios, notes, cents and sometimes frequencies. I try not to give descriptions in terms of cent 'deviation' (?!) from Equal Temperament as it is no more a part of my thinking than giving cent deviations from the pitch of the ping of my microwave. Inevitably though, at first there may be some use in understanding that 'a 5/4 E above C is about 13 cents flatter than 'piano E'.'

'Using intervals in cents is often a requirement of using synths, tuners or notation software - the implicit comparison with Equal Temperament should be battled!'

I stress that this is my experience with flesh and blood players - dividing the octave into 1200 is a barrier to new understanding.

Steve P.

On 15 Jun 2011, at 01:26, Jake Freivald wrote:

> In sum: If people want to be more obscure, less able to explain
> themselves to 12-only folks, and make it harder to help people learn
> alternate tuning systems, they should remove their 12-EDO signposts > and
> go to something less 12-centric. May their self-righteous objectivity
> compensate for their loneliness. :)

🔗gdsecor <gdsecor@...>

🔗cityoftheasleep <igliashon@...>

Without doing any calculations, Steve, can you put these intervals in order of largest to smallest? Please, to make this honest, don't do any calculations, don't factor them, don't give yourself more than a minute to think about it. Please note that these intervals, though complex-looking, occur between only two much much simpler ratios, no greater than 13-limit. So any of these could conceivably occur in a 7-to-13-limit JI system.

39/28
135/88
121/84
56/45
112/72
72/65

Now, order these cents values:
177.06862609662954
573.656756165573
764.91590473835
740.8607740962403
378.60219087351527
631.854977395001

Frankly, I understand that when working in JI, it's often easier to think in terms of ratios. 25/16 is obviously a 5/4 above a 5/4, for instance. The interval between 14/13 and 15/13 is obviously 15/14. Ratios can tell you about relationships: an 11/10 is the interval between 5/4 and 11/8, or 12/11 and 6/5. When the numbers are low, the relationships are obvious. But in every JI system outside the overtone series, one encounters intervals occurring between intervals where the ratios are more complex and less transparent, and these have to be internalized before they can make sense. A ratio that one has never encountered before, and whose factors are not immediately apparent, is obscure. It's been a minute since I calculated the above ratios and already I cannot order them in size of largest to smallest without converting them to cents. I'm not even immediately aware of the ratios they occur between, anymore. There are times when thinking in terms of ratios is very confusing. And when dealing with temperaments of lower accuracy, thinking in ratios quickly leads one into some very erroneous territory.

Cents, on the other hand, don't immediately tell you about harmonic relationships in terms of JI ratios, but they do immediately tell you how big the interval is, which is important. In dealing with temperaments, manipulation of cents values is often more sensible than dealing with approximate ratios, because you don't end up with error being multiplied. So while ratios clearly have a place in musical thinking, cents are often more transparent in some important ways.

-Igs
--- In tuning@yahoogroups.com, Steve Parker <steve@...> wrote:
>
> I disagree entirely. I work a lot with actual musicians in JI and the
> ones that struggle the most are the ones who reference everything to
> cents flat or sharp from 12ET.
> The ones who take the intervals on their own merits get to actually
> play well much quicker.
>
> Two quotes from a booklet I wrote a while ago:
>
> 'It quickly becomes necessary to learn how to move between ratios,
> notes, cents and sometimes frequencies. I try not to give descriptions
> in terms of cent 'deviation' (?!) from Equal Temperament as it is no
> more a part of my thinking than giving cent deviations from the pitch
> of the ping of my microwave. Inevitably though, at first there may be
> some use in understanding that 'a 5/4 E above C is about 13 cents
> flatter than 'piano E'.'
>
> 'Using intervals in cents is often a requirement of using synths,
> tuners or notation software - the implicit comparison with Equal
> Temperament should be battled!'
>
> I stress that this is my experience with flesh and blood players -
> dividing the octave into 1200 is a barrier to new understanding.
>
> Steve P.
>
>
> On 15 Jun 2011, at 01:26, Jake Freivald wrote:
>
> > In sum: If people want to be more obscure, less able to explain
> > themselves to 12-only folks, and make it harder to help people learn
> > alternate tuning systems, they should remove their 12-EDO signposts
> > and
> > go to something less 12-centric. May their self-righteous objectivity
> > compensate for their loneliness. :)
>

🔗Graham Breed <gbreed@...>

Mike Battaglia <battaglia01@...> wrote:

> 12 is pretty ambiguous in the 13-limit, but we use it
> anyway. But just how bad is 19 inconsistent in the
> 13-limit? C-Ab is close to 13/8, C-D# is close to 13/11,
> C-E# is close to 13/10... I'm not on my main computer to
> check now, I'll assume that it's something like 13/7 that
> gets screwed up.

12 isn't very ambiguous at all in the 13-limit. 12f is
clearly the best mapping. I think the only thing making it
ambiguous is that it isn't the patent val. But, yes,
12-equal isn't a very good 13-limit framework and 12-equal
ear training wouldn't get you very far. It might be useful
as a first step towards 72.

19 is extremely inconsistent in the 13-limit. 19p and 19e
have almost the same TE error (4.1 cents).

> We could try comparing 19 and 26 and see what's more
> intuitive. 19 is going to be more accurate, but 26 is
> consistent through the 13-limit. Maybe it'll just boil
> down to which is larger.

26 is unambiguous in the 13-limit. But it isn't that
good. This is the Cangwu parameter that makes it look best:

http://x31eq.com/cgi-bin/more.cgi?r=1&limit=13&error=5

31, 15, 9, 41, and 29 are also unambiguous and score better
than 26.

> The fact that 12 subdivides into 72 might make it
> unbeatable.

72 pretty much owns the 19-limit. See here:

http://x31eq.com/cgi-bin/pregular.cgi?limit=19&error=1

Unfortunately, steps in 72-equal don't work out as a whole
number of cents. But it's a good system for a universal
notation, as has been noted.

For ear training, you could start with 31 notes from
Miracle. It wouldn't matter if the tuning was 11-limit JI,
72-equal, or an optimized miracle.

Graham

🔗Daniel Nielsen <nielsed@...>

🔗genewardsmith <genewardsmith@...>

🔗Daniel Nielsen <nielsed@...>

🔗Steve Parker <steve@...>

Neither of these things bears much relation to how I work,
nor how I could conceivably work other than
players staring at a tuner.

On 15 Jun 2011, at 17:50, "cityoftheasleep" <igliashon@...> wrote:

> Without doing any calculations, Steve, can you put these intervals in order of largest to smallest? Please, to make this honest, don't do any calculations, don't factor them, don't give yourself more than a minute to think about it. Please note that these intervals, though complex-looking, occur between only two much much simpler ratios, no greater than 13-limit. So any of these could conceivably occur in a 7-to-13-limit JI system.
>
> 39/28
> 135/88
> 121/84
> 56/45
> 112/72
> 72/65
>
> Now, order these cents values:
> 177.06862609662954
> 573.656756165573
> 764.91590473835
> 740.8607740962403
> 378.60219087351527
> 631.854977395001
>
> Frankly, I understand that when working in JI, it's often easier to think in terms of ratios. 25/16 is obviously a 5/4 above a 5/4, for instance. The interval between 14/13 and 15/13 is obviously 15/14. Ratios can tell you about relationships: an 11/10 is the interval between 5/4 and 11/8, or 12/11 and 6/5. When the numbers are low, the relationships are obvious. But in every JI system outside the overtone series, one encounters intervals occurring between intervals where the ratios are more complex and less transparent, and these have to be internalized before they can make sense. A ratio that one has never encountered before, and whose factors are not immediately apparent, is obscure. It's been a minute since I calculated the above ratios and already I cannot order them in size of largest to smallest without converting them to cents. I'm not even immediately aware of the ratios they occur between, anymore. There are times when thinking in terms of ratios is very confusing. And when dealing with temperaments of lower accuracy, thinking in ratios quickly leads one into some very erroneous territory.
>
> Cents, on the other hand, don't immediately tell you about harmonic relationships in terms of JI ratios, but they do immediately tell you how big the interval is, which is important. In dealing with temperaments, manipulation of cents values is often more sensible than dealing with approximate ratios, because you don't end up with error being multiplied. So while ratios clearly have a place in musical thinking, cents are often more transparent in some important ways.
>
> -Igs
> --- In tuning@yahoogroups.com, Steve Parker <steve@...> wrote:
> >
> > I disagree entirely. I work a lot with actual musicians in JI and the
> > ones that struggle the most are the ones who reference everything to
> > cents flat or sharp from 12ET.
> > The ones who take the intervals on their own merits get to actually
> > play well much quicker.
> >
> > Two quotes from a booklet I wrote a while ago:
> >
> > 'It quickly becomes necessary to learn how to move between ratios,
> > notes, cents and sometimes frequencies. I try not to give descriptions
> > in terms of cent 'deviation' (?!) from Equal Temperament as it is no
> > more a part of my thinking than giving cent deviations from the pitch
> > of the ping of my microwave. Inevitably though, at first there may be
> > some use in understanding that 'a 5/4 E above C is about 13 cents
> > flatter than 'piano E'.'
> >
> > 'Using intervals in cents is often a requirement of using synths,
> > tuners or notation software - the implicit comparison with Equal
> > Temperament should be battled!'
> >
> > I stress that this is my experience with flesh and blood players -
> > dividing the octave into 1200 is a barrier to new understanding.
> >
> > Steve P.
> >
> >
> > On 15 Jun 2011, at 01:26, Jake Freivald wrote:
> >
> > > In sum: If people want to be more obscure, less able to explain
> > > themselves to 12-only folks, and make it harder to help people learn
> > > alternate tuning systems, they should remove their 12-EDO signposts
> > > and
> > > go to something less 12-centric. May their self-righteous objectivity
> > > compensate for their loneliness. :)
> >
>
>

🔗gdsecor <gdsecor@...>

🔗Daniel Nielsen <nielsed@...>

Um, have no idea why I typed that :/, wish this thing had an unsend button.
Sorry for the confusion.

Anyway, I agree 720 is definitely not ideal if one's going for exact
calculations of ratios up to a certain limit; of course, some folks might
not care about higher limits. It was just one notion that hadn't been
explicitly mentioned - simply using our typical degree convention.

44 is also not a great EDO for that, but it allows one to think in radians
well, if that's important to someone.

Combining 44 and 360 together might give something like 3960 (less than 1/3
of a cent), where
90\3960 is about 1/7 of a radian, and generates 44-EDO
180\3960 gives 22-EDO; 360\3960 gives 11-EDO
Of course, 11\3960 => 360-EDO; 22\3960 => 180-EDO; etc.
55\3960 => 72-EDO

Dan N

On Wed, Jun 15, 2011 at 1:37 PM, gdsecor <gdsecor@...> wrote:

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:
> >
> > What about 720°, since it agrees so well with 72 and 360 and is less than
> a
> > 1/2 cent? (Okay, quieting down now..)
>
> What's less than 1/2 cent? Not the unit of measure (1.666667 cents), nor
> the maximum error for 5-limit consonances (0.6413 cents). 720-EDO is only
> 5-limit consistent, and using its single degree as a unit of measure would
> require decimals for most calculations, since it would otherwise incur
> rounding errors.
>
> --George
>