Yahoo Tuning Groups Ultimate Backup tuning-math A generalized approach to units of interval measure, part 3: the case for cents, minutes and degrees

This is the last post in the series about generalized units of
interval measure. This was intended to be a simple and fun thing to
get me back into the swing of it after my huge hiatus, but now that
we're talking about topology on Facebook, I gotta finish this up and
move onto the next thing. I just wrote part 2 here:
/tuning-math/message/21341. Part 1
talked about what makes for a good coarse division, part 2 talked
about what makes for a good fine division.

In this section, I'll talk about what makes for a good
one-size-fits-all fine division for any coarse division, and introduce
a few notations to use that I thought were nice and compact.

A fine division is a good candidate for a "one-size fits all" division
of a coarse reference if it meets as many of the following properties
as possible:
1) It's something people are already used to dividing things into
2) It has lots of factors, so it divides steps into a lot of different
simple ways
3) It's sufficiently fine
4) It's more likely, compared to some other random fine division, to
lead to a zeta multiple of an arbitrarily chosen coarse reference zeta
EDO

There are three divisions that stand out as handling a good amount of
these properties: 100, 60, 360, and 100, which for some coarse
reference N-EDO, I will term a division into "N-cents," "N-minutes,"
and "N-degrees."

100 cents is useful especially because it meets criteria #1: it is
much more common for people to divide things into 100 than into other
non-power of 10 numbers, so no learning curve is needed (obviously).
It also clearly meets criterion #3, and to some extent meets criteria
#2 and 4 as well, although not as well as 60 and 360.

60 and 360 outperform 100 on criteria #2 and #4 by far, on average
meet #3 about as well, and meet #1 enough to still be quite useful.
Since we already understand the various ways to subdivide 60 minutes
(15 min = 1/4, 30 min = 1/2, etc) and 360 degrees (90° = 1/4, 30° =
1/12, 45° = 1/8, etc), much like with cents, no learning curve is
needed. This is fairly significant, since the main reason to stick
with cents is that a significant learning curve is typically assumed
to use anything else.

60 and 360 perform so well on #2 and #4 because they're highly
composite numbers, which have more factors than anything smaller than
them. Furthermore, they're "superior" highly composite numbers, which
in general outperform all other types of numbers at this task if
weighted by size. A fine division into a (superior) highly composite
number enables one to split the step in as many ways as possible for
its size, meeting criterion #2. Furthermore, since highly composite
numbers have so many factors, the chance that one of them makes your
coarse division a zeta multiple is higher than for other numbers
chosen at random in its size vicinity, meeting criterion #4.

The benefits of using a highly composite number to divide something
into parts are well-known, as some traditional units of measure retain
their divisions into highly composite numbers as a result (1 hour = 60
minutes, 1 turn = 360 degrees, 1 foot = 12 inches, 1 day = 24 hours,
etc). This can be thought of as an application of this concept to
music; there are some people suggesting mathematics would have been
more elegant if we used base-12 as well.

Finally, we need some notation. To notate something in N-minutes, we
can write steps:minutes, to notate it in N-degrees, we can write
steps, degrees°, and to notate it in N-cents, we can write it either
as steps.cents, or as a single number 100*steps+cents. Both cents,
minutes, and degrees can be any real number less than the number of
the fine division they specify; steps must always be an integer.

Some examples:

12-minutes, rounded to the nearest minute
3/2: 7:01
5/4: 3:52
6/5: 3:09
7/4: 9:41
9/7: 4:21
11/8: 5:31
13/8: 8:24

This should make sense. Since 12-EDO is very accurate, 3/2 is just a
hair past 7:00. 5/4 is about 3:50, which is roughly 1/6 of a step less
than 4:00. 6/5 is roughly 1/6 of a step greater than 3:00. 7/4 is
about 1/3 of a step less than 10:00, and likewise, 9/7 is about 1/3 of
a step past 4:00. 11/8 is about 5 1/2 steps, and 13/8 is about 5 2/5
steps. Seeing that 5/4 is about 1/6 of a step less than 4:00 makes
sense, and it's something which isn't apparent when you look at cents:

12-cents, rounded to the nearest cent (notating them as a real number of steps):
3/2: 7.02
5/4: 3.86
6/5: 3.16
7/4: 9.69
9/7: 4.35
11/8: 5.51
13/8: 8.41

That 386 cents is about 1/6 of a step less than 400 cents isn't
something I ever realized before. Now degrees:

12-degrees, rounded to the nearest degree:
3/2: 7, 7°
5/4: 3, 311°
6/5: 3, 56°
7/4: 9, 248°
9/7: 4, 126°
11/8: 5, 185°
13/8: 8, 146°

This should also make sense: 3/2 is just a few degrees past 7 steps.
5/4 makes sense if thought of as about 300° or 5/6 of the way past 3
steps, 6/5 is about 60° or 1/6 of a step past 3, 7/4 is about 240° or
2/3's of a step past 9, 9/7 is about 120° or 1/3 of the way past 4,
11/8 is about 180° or 1/2 of the way past 5. 13/8 is a bit tricker,
but makes sense if thought about as 150°, or 180°-30°, or about 5/12,
past 8 steps.

I personally find that I actually rather enjoy using minutes: the
notation is very compact and they're simple to use, and since I
already know how to tell time, I happen to think they're very
intuitive. Cents may always remain dominant, but this might be another
nice choice as well.

Here's 19-cents, 19-minutes, and 19-degrees for completeness:

19-cents, notated as a real number of steps, rounded
3/2: 11.11
5/4: 6.12
6/5: 5.00
7/4: 15.34
9/7: 6.89
11/8: 8.73
13/8: 13.31

19-minutes, rounded
3/2: 11:07
5/4: 6:07
6/5: 5:00
7/4: 15:20
9/7: 6:53
11/8: 8:44
13/8: 13:19

Interesting to see that 5/4 is a bit less than 10 min or 1/6 of the
way past 6 19-steps, and that 7/2 is about 1/3 past 15 steps, and 9/7
is about a sixth less than 7 steps. Now degrees:

19-degrees, rounded
3/2: 11, 41°
5/4: 6, 42°
6/5: 4, 359°
7/4: 15, 122°
9/7: 6, 320°
11/8: 8, 263°
13/8: 13, 111°

From this perspective, you can see that 5/4 is about 45° or 1/8 past 6
steps, that 6/5 is just a hair less than 5 steps, that 7/4 is about
1/3 of the way past 15 steps, that 9/7 is about 1/9 less than 7 steps,
that 1/8 is a bit more than 240° or 2/3 past 11/8, and 13/8 is about
1/3 of the way past 13 steps.

Degrees work much better than minutes for 19, since lots of nice
intervals seem to be very close to exactly 1/8, 1/3, 1/9, etc of the
way past various steps, probably because 19-EDO has zeta multiples at
19*8 = 152 and 19*9 = 171.

Mike

🔗gdsecor <gdsecor@yahoo.com>

So what about my suggestion to divide 41-EDO into 60 minutes? This works far better than anything you've mentioned. It's a unit of measure that allows you to round intervals to the nearest minute *before* making interval calculations, because rounding errors will occur only rarely in the result.

--George

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> This is the last post in the series about generalized units of
> interval measure. This was intended to be a simple and fun thing to
> get me back into the swing of it after my huge hiatus, but now that
> we're talking about topology on Facebook, I gotta finish this up and
> move onto the next thing. I just wrote part 2 here:
> /tuning-math/message/21341. Part 1
> talked about what makes for a good coarse division, part 2 talked
> about what makes for a good fine division.
>
> In this section, I'll talk about what makes for a good
> one-size-fits-all fine division for any coarse division, and introduce
> a few notations to use that I thought were nice and compact.
>
> A fine division is a good candidate for a "one-size fits all" division
> of a coarse reference if it meets as many of the following properties
> as possible:
> 1) It's something people are already used to dividing things into
> 2) It has lots of factors, so it divides steps into a lot of different
> simple ways
> 3) It's sufficiently fine
> 4) It's more likely, compared to some other random fine division, to
> lead to a zeta multiple of an arbitrarily chosen coarse reference zeta
> EDO
>
> There are three divisions that stand out as handling a good amount of
> these properties: 100, 60, 360, and 100, which for some coarse
> reference N-EDO, I will term a division into "N-cents," "N-minutes,"
> and "N-degrees."
>
> 100 cents is useful especially because it meets criteria #1: it is
> much more common for people to divide things into 100 than into other
> non-power of 10 numbers, so no learning curve is needed (obviously).
> It also clearly meets criterion #3, and to some extent meets criteria
> #2 and 4 as well, although not as well as 60 and 360.
>
> 60 and 360 outperform 100 on criteria #2 and #4 by far, on average
> meet #3 about as well, and meet #1 enough to still be quite useful.
> Since we already understand the various ways to subdivide 60 minutes
> (15 min = 1/4, 30 min = 1/2, etc) and 360 degrees (90° = 1/4, 30° =
> 1/12, 45° = 1/8, etc), much like with cents, no learning curve is
> needed. This is fairly significant, since the main reason to stick
> with cents is that a significant learning curve is typically assumed
> to use anything else.
>
> 60 and 360 perform so well on #2 and #4 because they're highly
> composite numbers, which have more factors than anything smaller than
> them. Furthermore, they're "superior" highly composite numbers, which
> in general outperform all other types of numbers at this task if
> weighted by size. A fine division into a (superior) highly composite
> number enables one to split the step in as many ways as possible for
> its size, meeting criterion #2. Furthermore, since highly composite
> numbers have so many factors, the chance that one of them makes your
> coarse division a zeta multiple is higher than for other numbers
> chosen at random in its size vicinity, meeting criterion #4.
>
> The benefits of using a highly composite number to divide something
> into parts are well-known, as some traditional units of measure retain
> their divisions into highly composite numbers as a result (1 hour = 60
> minutes, 1 turn = 360 degrees, 1 foot = 12 inches, 1 day = 24 hours,
> etc). This can be thought of as an application of this concept to
> music; there are some people suggesting mathematics would have been
> more elegant if we used base-12 as well.
>
> Finally, we need some notation. To notate something in N-minutes, we
> can write steps:minutes, to notate it in N-degrees, we can write
> steps, degrees°, and to notate it in N-cents, we can write it either
> as steps.cents, or as a single number 100*steps+cents. Both cents,
> minutes, and degrees can be any real number less than the number of
> the fine division they specify; steps must always be an integer.
>
> Some examples:
>
> 12-minutes, rounded to the nearest minute
> 3/2: 7:01
> 5/4: 3:52
> 6/5: 3:09
> 7/4: 9:41
> 9/7: 4:21
> 11/8: 5:31
> 13/8: 8:24
>
> This should make sense. Since 12-EDO is very accurate, 3/2 is just a
> hair past 7:00. 5/4 is about 3:50, which is roughly 1/6 of a step less
> than 4:00. 6/5 is roughly 1/6 of a step greater than 3:00. 7/4 is
> about 1/3 of a step less than 10:00, and likewise, 9/7 is about 1/3 of
> a step past 4:00. 11/8 is about 5 1/2 steps, and 13/8 is about 5 2/5
> steps. Seeing that 5/4 is about 1/6 of a step less than 4:00 makes
> sense, and it's something which isn't apparent when you look at cents:
>
> 12-cents, rounded to the nearest cent (notating them as a real number of steps):
> 3/2: 7.02
> 5/4: 3.86
> 6/5: 3.16
> 7/4: 9.69
> 9/7: 4.35
> 11/8: 5.51
> 13/8: 8.41
>
> That 386 cents is about 1/6 of a step less than 400 cents isn't
> something I ever realized before. Now degrees:
>
> 12-degrees, rounded to the nearest degree:
> 3/2: 7, 7°
> 5/4: 3, 311°
> 6/5: 3, 56°
> 7/4: 9, 248°
> 9/7: 4, 126°
> 11/8: 5, 185°
> 13/8: 8, 146°
>
> This should also make sense: 3/2 is just a few degrees past 7 steps.
> 5/4 makes sense if thought of as about 300° or 5/6 of the way past 3
> steps, 6/5 is about 60° or 1/6 of a step past 3, 7/4 is about 240° or
> 2/3's of a step past 9, 9/7 is about 120° or 1/3 of the way past 4,
> 11/8 is about 180° or 1/2 of the way past 5. 13/8 is a bit tricker,
> but makes sense if thought about as 150°, or 180°-30°, or about 5/12,
> past 8 steps.
>
> I personally find that I actually rather enjoy using minutes: the
> notation is very compact and they're simple to use, and since I
> already know how to tell time, I happen to think they're very
> intuitive. Cents may always remain dominant, but this might be another
> nice choice as well.
>
> Here's 19-cents, 19-minutes, and 19-degrees for completeness:
>
> 19-cents, notated as a real number of steps, rounded
> 3/2: 11.11
> 5/4: 6.12
> 6/5: 5.00
> 7/4: 15.34
> 9/7: 6.89
> 11/8: 8.73
> 13/8: 13.31
>
> 19-minutes, rounded
> 3/2: 11:07
> 5/4: 6:07
> 6/5: 5:00
> 7/4: 15:20
> 9/7: 6:53
> 11/8: 8:44
> 13/8: 13:19
>
> Interesting to see that 5/4 is a bit less than 10 min or 1/6 of the
> way past 6 19-steps, and that 7/2 is about 1/3 past 15 steps, and 9/7
> is about a sixth less than 7 steps. Now degrees:
>
> 19-degrees, rounded
> 3/2: 11, 41°
> 5/4: 6, 42°
> 6/5: 4, 359°
> 7/4: 15, 122°
> 9/7: 6, 320°
> 11/8: 8, 263°
> 13/8: 13, 111°
>
> From this perspective, you can see that 5/4 is about 45° or 1/8 past 6
> steps, that 6/5 is just a hair less than 5 steps, that 7/4 is about
> 1/3 of the way past 15 steps, that 9/7 is about 1/9 less than 7 steps,
> that 1/8 is a bit more than 240° or 2/3 past 11/8, and 13/8 is about
> 1/3 of the way past 13 steps.
>
> Degrees work much better than minutes for 19, since lots of nice
> intervals seem to be very close to exactly 1/8, 1/3, 1/9, etc of the
> way past various steps, probably because 19-EDO has zeta multiples at
> 19*8 = 152 and 19*9 = 171.
>
> Mike
>

🔗Mike Battaglia <battaglia01@gmail.com>

🔗gdsecor@yahoo.com

Mike, I don't see any text in your message #21344, even though the list of
messages seems to indicate that it begins with "Not sure why you think this is
out of the ...". ???

--George

--- In tuning-math@yahoogroups.com, <battaglia01@...> wrote:

On Wed, Aug 28, 2013 at 2:56 PM, gdsecor <gdsecor@...> wrote:
>
> So what about my suggestion to divide 41-EDO into 60 minutes? This works far
better than anything you've mentioned. It's a unit of measure that allows you to
round intervals to the nearest minute *before* making interval calculations,
because rounding errors will occur only rarely in the result.

Not sure why you think this is out of the scope of what I mentioned,
since my approach is
1) Find a coarse reference that's a small zeta EDO
2) Find a zeta EDO that's a multiple of this
3) Pick a fine division that gives you the zeta multiple as a subset
4) If you want a one-size-fits all solution that works as much as
possible, choose a (superior) highly composite number for your fine
division, particularly 60 or 360

Dividing 41-EDO into 60 minutes is compatible with all four of these
ideas, even #4, so it's right in the same vein of all those ideas,
except that it's kind of large for a "coarse" reference.

Mike

🔗gdsecor@yahoo.com

This interface is really goofy! Mike, I'm now able to read your reply in msg.
#21344, but only *after* I replied to what looked like an empty message.

You wrote:

< Not sure why you think this is out of the scope of what I mentioned, ...

I didn't think my suggestion was out of the scope of the 4 points you mentioned.
But your candidates (coarse 12 & 19) don't result in "good" (low-error)
divisions when subdivided into 60 "minutes" or 360 "degrees". How do you figure
that 720-, 1140-, and 6840-EDO are zeta EDOs?

I don't think that 41-EDO is "kind of large" for a coarse reference. I think
that 31-EDO would be about the right amount for a coarse reference, since
consecutive coarse intervals would be subminor 3rd (6:7), minor 3rd (5:6),
neutral 3rd (9:11), major 3rd (4:5), and supermajor 3rd (7:9) -- all readily
distinguishable by ear; however, 31 doesn't subdivide into anything fine enough
(217 being too coarse). 41-EDO throws 11:13 and 11:14 into the coarse interval
sequence, but subdivides very nicely into 2460-EDO, which is 27-limit
consistent. Note that 2460-EDO also allows simple division of the syntonic comma
into 4 parts, for calculation of the historic meantone temperament.

--George

--- In tuning-math@yahoogroups.com, <gdsecor@...> wrote:

Mike, I don't see any text in your message #21344, even though the list of
messages seems to indicate that it begins with "Not sure why you think this is
out of the ...". ???

--George

--- In tuning-math@yahoogroups.com, <battaglia01@...> wrote:

Dividing 41-EDO into 60 minutes is compatible with all four of these
ideas, even #4, so it's right in the same vein of all those ideas,
except that it's kind of large for a "coarse" reference.

Mike

--- In tuning-math@yahoogroups.com, <gdsecor@...> wrote:

Mike, I don't see any text in your message #21344, even though the list of
messages seems to indicate that it begins with "Not sure why you think this is
out of the ...". ???

--George

--- In tuning-math@yahoogroups.com, <battaglia01@...> wrote:

Dividing 41-EDO into 60 minutes is compatible with all four of these
ideas, even #4, so it's right in the same vein of all those ideas,
except that it's kind of large for a "coarse" reference.

Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Aug 29, 2013, at 1:39 PM, "gdsecor@yahoo.com" <gdsecor@yahoo.com> wrote:

This interface is really goofy! Mike, I'm now able to read your reply in
msg. #21344, but only *after* I replied to what looked like an empty message

Yikes! Yahoo's screwing everything up here...

You wrote:

< Not sure why you think this is out of the scope of what I mentioned, ...

I didn't think my suggestion was out of the scope of the 4 points you
mentioned. But your candidates (coarse 12 & 19) don't result in "good"
(low-error) divisions when subdivided into 60 "minutes" or 360 "degrees".
How do you figure that 720-, 1140-, and 6840-EDO are zeta EDOs?

They result in *multiples* of something that is, itself, a zeta EDO that is
a multiple > 1 of another zeta EDO, which was one of my criteria in part 2.
They have 72 as a factor, which has 12 as a factor.

I don't think that 41-EDO is "kind of large" for a coarse reference. I
think that 31-EDO would be about the right amount for a coarse reference,
since consecutive coarse intervals would be subminor 3rd (6:7), minor 3rd
(5:6), neutral 3rd (9:11), major 3rd (4:5), and supermajor 3rd (7:9) -- all
readily distinguishable by ear; however, 31 doesn't subdivide into anything
fine enough (217 being too coarse). 41-EDO throws 11:13 and 11:14 into the
coarse interval sequence, but subdivides very nicely into 2460-EDO, which
is 27-limit consistent. Note that 2460-EDO also allows simple division of
the syntonic comma into 4 parts, for calculation of the historic meantone
temperament.

If you can handle 41-EDO as a coarse reference, I say go for it then! It
isn't a bad choice.

Mike

A generalized approach to units of interval measure, part 3: the case for cents, minutes and degrees

🔗Mike Battaglia <battaglia01@gmail.com>

🔗gdsecor <gdsecor@yahoo.com>

🔗Mike Battaglia <battaglia01@gmail.com>

🔗gdsecor@yahoo.com

🔗gdsecor@yahoo.com

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@gmail.com>

🔗Mike Battaglia <battaglia01@gmail.com>