Yahoo Tuning Groups Ultimate Backup tuning Sub-Mediants

Hi everyone,

I think I've found something interesting. I was looking for all of the ratios p/q between 5/4 and 81/64 with 5 < p < 81 and 4 < q < 64. My aim was to see if there is a way to get from the 'high-numbered' intervals to the 'low numbered' ones. Now, I started by taking all the mediants (5 + 81)/(4 + 64) = 43/34, then the mediants of the mediants, and so on. But the problem with this is that it goes on forever and doesn't approach 5/4. At any rate, after going on for some time I eventually got the following list: {5/4, 24/19, 43/34, 62/49, 81/64}.

What's interesting is that I found these match a formula. Given the higher numbered interval as say a/b, we use the Extended Euclidean Algorithm to solve Bezout's equation ax + by = 1. (Or you can use an applet at

http://wims.unice.fr/wims/wims.cgi?module=tool/arithmetic/bezout.en

Then all of the other intervals can be obtained as p = a (mod y) and q = b (mod x).

Eg: a/b = 81/64 in Bezout's equation gives x = -15, y = 19. Then 81 = (4 x 19) + 5, 62 = (3 x 19) + 5 etc...and 64 = (4 x 15) + 4, 49 = (3 x 15) + 4 etc...IOW all the numerators equal 5 (Mod 19) and all the denominators 4 (mod 15).

Eg2: a/b = 32/27 gives x = 11 and y = -13. Therefore 32/13 = 2 + 6/13 or 32 = 2*13 + 6 and 27 = 2*11 + 5. It is easy to calculate that the two intervals below are {6/5, 19/16}.

(Note also that each adjacent pair p/q and a/b satisfy the Diophantine equation aq - pb = 1 which suggests that they are a branch on the Stern-Brocot tree).

IOW this seems to give a direct method of getting from the higher intervals to the lower ones. But I don't quite understand why that is just yet. What I'd like to know is whether this method is well-known and if so where can I look it up?

Thanks

Rick

🔗Graham Breed <gbreed@...>

On 24 March 2010 06:57, rick <rick_ballan@...> wrote:

> IOW this seems to give a direct method of getting from the higher intervals to the lower ones. But I don't quite understand why that is just yet. What I'd like to know is whether this method is well-known and if so where can I look it up?

I don't think there's anything new but I don't know exactly what
you're suggesting might be new. The Stern-Brocot tree's well known,
sometimes from the perspective of regular temperaments. Adding twelve
to the number of notes in a meantone ET will give another meantone ET,
and the reason that works is the same property you're talking about.

It works better for schismatic temperaments: 5, 17, 29, 41, 53, ...

You can read Erv Wilson's articles about the scale tree:

http://www.anaphoria.com/wilson.html

Graham

🔗Mike Battaglia <battaglia01@...>

19 + 12 = 31

31 + 12 = 53... not meantone?

Confused here.

-Mike

On Wed, Mar 24, 2010 at 3:52 AM, Graham Breed <gbreed@...> wrote:

>
>
> On 24 March 2010 06:57, rick <rick_ballan@...<rick_ballan%40yahoo.com.au>>
> wrote:
>
> > IOW this seems to give a direct method of getting from the higher
> intervals to the lower ones. But I don't quite understand why that is just
> yet. What I'd like to know is whether this method is well-known and if so
> where can I look it up?
>
> I don't think there's anything new but I don't know exactly what
> you're suggesting might be new. The Stern-Brocot tree's well known,
> sometimes from the perspective of regular temperaments. Adding twelve
> to the number of notes in a meantone ET will give another meantone ET,
> and the reason that works is the same property you're talking about.
>
> It works better for schismatic temperaments: 5, 17, 29, 41, 53, ...
>
> You can read Erv Wilson's articles about the scale tree:
>
> http://www.anaphoria.com/wilson.html
>
> Graham
>
>

🔗Graham Breed <gbreed@...>

🔗Mike Battaglia <battaglia01@...>

LOL. I really need to stop staying up until 4 in the morning :)

Although one more question: how in this case do you rigorously define
"meantone?" 24-tet could be mapped in such a way so that the 350 cent
interval equates to 5/4 and the 400 cent interval equates to 81/64, or
something like that.

Or, put another way:

12-tet - commonly viewed as a "meantone", + 12 =
24-tet - also viewed as a "meantone" with 11-limit implications, + 12 =
...
...
...
72-tet - not viewed as a "meantone" anymore.

Is it just some kind of useful rule of thumb, or is there a deeper rigorous
mathematical background to it that I'm missing?

-Mike

On Wed, Mar 24, 2010 at 4:06 AM, Graham Breed <gbreed@...> wrote:

>
>
> On 24 March 2010 11:56, Mike Battaglia <battaglia01@...<battaglia01%40gmail.com>>
> wrote:
> >
> >
> > 19 + 12 = 31
> >
> > 31 + 12 = 53... not meantone?
>
> 31 + 12 = 43 ;-)
>
>

🔗Graham Breed <gbreed@...>

On 24 March 2010 12:14, Mike Battaglia <battaglia01@...> wrote:

> Although one more question: how in this case do you rigorously define "meantone?" 24-tet could be mapped in such a way so that the 350 cent interval equates to 5/4 and the 400 cent interval equates to 81/64, or something like that.

There are ways of defining meantones so that you exclude 24. They
could temperaments generated by a fifth and octave that temper out
81/64, for example. The 24-tet may appear to qualify, but you're
really generating only 12 notes per octave. You can also use the
jargon and say you have to temper out 81/64 with no contorsion.

Note there are meantone temperaments that don't fit the pattern. But
they're still the sum of other meantone temperaments. 19+31=50, for
example.

Note: the big temperament has a tuning in between the two little
temperaments, which you can show on the scale tree. Repeatedly adding
12 gets you progressively closer to 12-tet.

> Or, put another way:
>
> 12-tet - commonly viewed as a "meantone", + 12 =
> 24-tet - also viewed as a "meantone" with 11-limit implications, + 12 =

It would be a different temperament class in the 11-limit because the
generator isn't a fifth any more. There are different ways of doing
it, because you could split the fourth or the fifth, and the 7-limit
can be mapped differently if you include it. They may belong to the
"meantone family" but I forget how Gene defined that.

> 72-tet - not viewed as a "meantone" anymore.

It wouldn't normally be a meantone because it has a better 5:4 than
12-tet gives you. You could define a 72-tet mapping that belongs to
the meantone family (if I'm getting the word right) but that'd be
somewhat eccentric. And the result would still be 12-tet for 5-limit
harmony.

> Is it just some kind of useful rule of thumb, or is there a deeper rigorous mathematical background to it that I'm missing?

There certainly is a mathematical background, and it can be rigorous
if you like. The way Gene defined it, no temperament should have
contorsion. So 24-tet is misnamed in the 5-limit. It isn't a
temperament, only a contorted rendering of 12-tet.

Graham

🔗rick <rick_ballan@...>

Thanks Graham,

Of course I'm by no means claiming that the use of mediants or the S-B tree is unique. It's just that I've never seen these Mods (Moduli?) before or the fact that they can be obtained by Bezout's formula. They seem to be 'trimming' the S-B tree and I suspect that you guys do the same thing under another name (Eg, are these mods somehow related to MOS or something?). When you say "Adding twelve to the number of notes in a meantone ET will give another meantone ET, and the reason that works is the same property you're talking about", do you mean that this is a Mod 12? Excuse my ignorance but an example would be helpful if its not too much trouble.

-Rick

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 24 March 2010 06:57, rick <rick_ballan@...> wrote:
>
> > IOW this seems to give a direct method of getting from the higher intervals to the lower ones. But I don't quite understand why that is just yet. What I'd like to know is whether this method is well-known and if so where can I look it up?
>
> I don't think there's anything new but I don't know exactly what
> you're suggesting might be new. The Stern-Brocot tree's well known,
> sometimes from the perspective of regular temperaments. Adding twelve
> to the number of notes in a meantone ET will give another meantone ET,
> and the reason that works is the same property you're talking about.
>
> It works better for schismatic temperaments: 5, 17, 29, 41, 53, ...
>
> You can read Erv Wilson's articles about the scale tree:
>
> http://www.anaphoria.com/wilson.html
>
>
> Graham
>

🔗rick <rick_ballan@...>

31 + 12 = 53, yes definitely a 4 in the morning job. Are what you guys saying here about 11-limit related to Mod 12? If it is I've never made that connection.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> LOL. I really need to stop staying up until 4 in the morning :)
>
> Although one more question: how in this case do you rigorously define
> "meantone?" 24-tet could be mapped in such a way so that the 350 cent
> interval equates to 5/4 and the 400 cent interval equates to 81/64, or
> something like that.
>
> Or, put another way:
>
> 12-tet - commonly viewed as a "meantone", + 12 =
> 24-tet - also viewed as a "meantone" with 11-limit implications, + 12 =
> ...
> ...
> ...
> 72-tet - not viewed as a "meantone" anymore.
>
> Is it just some kind of useful rule of thumb, or is there a deeper rigorous
> mathematical background to it that I'm missing?
>
> -Mike
>
>
> On Wed, Mar 24, 2010 at 4:06 AM, Graham Breed <gbreed@...> wrote:
>
> >
> >
> > On 24 March 2010 11:56, Mike Battaglia <battaglia01@...<battaglia01%40gmail.com>>
> > wrote:
> > >
> > >
> > > 19 + 12 = 31
> > >
> > > 31 + 12 = 53... not meantone?
> >
> > 31 + 12 = 43 ;-)
> >
> >
>

🔗rick <rick_ballan@...>

Hi again Graham,

To answer my own earlier question concerning whether these mods relate to either limits or MOS, I've been revising these methods and I don't think they do. (At least I can't see it). Just to be clear, what I've found is that for any ratio a/b (where a and b are coprime) we can find the other approximate coprime ratios in three steps 1) first we solve ax + by = 1 to find x and y, 2) we take a/y = N + p/y, b/x = N + q/x to find the remainders p and q, and 3) all the intervals in this set will be ((n*y) + p)/((n*x) + q) where n = 0,1,2,3...

To give the eg again, a/b = 81/64,
1) solving 81x + 64y = 1 gives x = -15, y = 19,
2)taking 81/19 = 4 + 5/19 and 64/15 = 4 + 4/15 gives p = 5 and q = 4,and
3) gives ((n*19) + 5)/((n*15) + 4) = {5/4, 24/19, 43/34, 62/49, 81/64...}

Although this can be seen as the set of all mediants between n*19/n*15 and p/q, it is not exactly the same as taking the mediants between 81/64 and 5/4. Strictly speaking it is a sub-sequence of this mediant set.

-Rick

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 24 March 2010 12:14, Mike Battaglia <battaglia01@...> wrote:
>
> > Although one more question: how in this case do you rigorously
> define "meantone?" 24-tet could be mapped in such a way so that
> the 350 cent interval equates to 5/4 and the 400 cent interval
> equates to 81/64, or something like that.
>
> There are ways of defining meantones so that you exclude 24.

Here's how I think of it. "Meantone" is the 5-limit rank 2
temperament with 81/80 in its kernel.

24-ET, as a temperament, is rank 1. So it's not meantone.
And even if we made an exception, the meantone-like val you'd
use in 24 has torsion so it's not valid temperament anyway.

As a scale, however, there may be an acceptable meantone
tuning within 24-ET.

For a scale to support meantone temperament, it must satisfy
the map

< 1 2 4 ]
< 0 -1 -4 ]

in the 5-limit. A good scale will satisfy it so that the
most accurate approximation in the scale for each prime is
the one mapped.* In 24-ET, if we choose 24 steps & 10 steps
as generators, the above mapping reduces to

< 24 38 56 ]

that is, we get the 700 cents for 3:2 and 400 cents for 5:4,
which are the best approximations available. So as a scale,
24-ET supports meantone temperament.

-Carl

* For an ET, this is the same as saying the patent val must
be the result of the mapping. An acceptable variation on
this is that the val with least TOP damage must be the result
of the mapping.

Hm, maybe Igliashon did have a point on MMM...

🔗rick <rick_ballan@...>

Hi Carl,

Sorry, you guys have lost me here. Graham suggested something about adding 12 to meantone but what is 12 and to what are we adding, do you know? The numerator and denominator of each interval, or the generators? I also looked into Erv Wilson once again but apart from his using mediants to develop tunings I couldn't find anything specifically relating to my original problem which was the use of mods to develop interval matches.

-Rick

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > On 24 March 2010 12:14, Mike Battaglia <battaglia01@> wrote:
> >
> > > Although one more question: how in this case do you rigorously
> > define "meantone?" 24-tet could be mapped in such a way so that
> > the 350 cent interval equates to 5/4 and the 400 cent interval
> > equates to 81/64, or something like that.
> >
> > There are ways of defining meantones so that you exclude 24.
>
> Here's how I think of it. "Meantone" is the 5-limit rank 2
> temperament with 81/80 in its kernel.
>
> 24-ET, as a temperament, is rank 1. So it's not meantone.
> And even if we made an exception, the meantone-like val you'd
> use in 24 has torsion so it's not valid temperament anyway.
>
> As a scale, however, there may be an acceptable meantone
> tuning within 24-ET.
>
> For a scale to support meantone temperament, it must satisfy
> the map
>
> < 1 2 4 ]
> < 0 -1 -4 ]
>
> in the 5-limit. A good scale will satisfy it so that the
> most accurate approximation in the scale for each prime is
> the one mapped.* In 24-ET, if we choose 24 steps & 10 steps
> as generators, the above mapping reduces to
>
> < 24 38 56 ]
>
> that is, we get the 700 cents for 3:2 and 400 cents for 5:4,
> which are the best approximations available. So as a scale,
> 24-ET supports meantone temperament.
>
> -Carl
>
> * For an ET, this is the same as saying the patent val must
> be the result of the mapping. An acceptable variation on
> this is that the val with least TOP damage must be the result
> of the mapping.
>
> Hm, maybe Igliashon did have a point on MMM...
>

🔗Carl Lumma <carl@...>

Hi Rick,

Sorry, the thread has wandered a bit; I was replying to
Mike's question. I don't know what you mean by "interval
matches" so I can't help. Graham means adding 12 to the
number of notes/octave. If you look at this meantone
series

1 2 3 5 7 12 19 31 43 55

the successive differences are

1 1 2 2 5 7 12 12 12

Hope that helps,

-Carl

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:
>
> Hi Carl,
>
> Sorry, you guys have lost me here. Graham suggested something
> about adding 12 to meantone but what is 12 and to what are we
> adding, do you know? The numerator and denominator of each
> interval, or the generators? I also looked into Erv Wilson once
> again but apart from his using mediants to develop tunings I
> couldn't find anything specifically relating to my original
> problem which was the use of mods to develop interval matches.
>
> -Rick
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> > >
> > > On 24 March 2010 12:14, Mike Battaglia <battaglia01@> wrote:
> > >
> > > > Although one more question: how in this case do you rigorously
> > > define "meantone?" 24-tet could be mapped in such a way so that
> > > the 350 cent interval equates to 5/4 and the 400 cent interval
> > > equates to 81/64, or something like that.
> > >
> > > There are ways of defining meantones so that you exclude 24.
> >
> > Here's how I think of it. "Meantone" is the 5-limit rank 2
> > temperament with 81/80 in its kernel.
> >
> > 24-ET, as a temperament, is rank 1. So it's not meantone.
> > And even if we made an exception, the meantone-like val you'd
> > use in 24 has torsion so it's not valid temperament anyway.
> >
> > As a scale, however, there may be an acceptable meantone
> > tuning within 24-ET.
> >
> > For a scale to support meantone temperament, it must satisfy
> > the map
> >
> > < 1 2 4 ]
> > < 0 -1 -4 ]
> >
> > in the 5-limit. A good scale will satisfy it so that the
> > most accurate approximation in the scale for each prime is
> > the one mapped.* In 24-ET, if we choose 24 steps & 10 steps
> > as generators, the above mapping reduces to
> >
> > < 24 38 56 ]
> >
> > that is, we get the 700 cents for 3:2 and 400 cents for 5:4,
> > which are the best approximations available. So as a scale,
> > 24-ET supports meantone temperament.
> >
> > -Carl
> >
> > * For an ET, this is the same as saying the patent val must
> > be the result of the mapping. An acceptable variation on
> > this is that the val with least TOP damage must be the result
> > of the mapping.
> >
> > Hm, maybe Igliashon did have a point on MMM...
> >
>

🔗Graham Breed <gbreed@...>

On 25 March 2010 14:54, rick <rick_ballan@...> wrote:

> Sorry, you guys have lost me here. Graham suggested something about adding 12 to meantone but what is 12 and to what are we adding, do you know? The numerator and denominator of each interval, or the generators? I also looked into Erv Wilson once again but apart from his using mediants to develop tunings I couldn't find anything specifically relating to my original problem which was the use of mods to develop interval matches.

As Carl has said, 12 is a number of notes to the octave. Wilson
called this family "duodecimally negative" following Bosanquet's
"singly negative" because a Pythagorean comma is always 1 step in the
wrong direction.

When I said "any meantone" before, I should have said "except 12-equal
itself". Maybe that's why 24 and 72 crept into the discussion.

The reason is the MOS version of the scale tree. If you think of
meantones as having a fourth between 3/7 and 5/12 octaves, the tree is
(use mono-spacing for optimal satisfaction):

3 5
- --
7 12
8
--
19
11 13
-- --
26 31
18
--
43
23
--
55

Every step to the right is adding 12 to the denominator, which is the
number of steps to the octave. So adding 12 notes to the octave means
the meantone tuning gets a bit closer to 12-equal. The number of
notes to the octave is always congruent to 7 modulo 12 (or however
you're supposed to say that).

There are also valid meantones that aren't singly positive.

The original question was about tunings between 5/4 and 81/64, where
we could (but it looks like Rick didn't) take the scale tree between
5/4 and 19/15:

5 19
- --
4 15
24
--
19
29 43
-- --
23 34
34 53 67 62
-- -- -- --
27 42 53 49
81
--
64

Adding 5 to the numerator and 4 to the denominator will get you closer
to 5/4. So the numerator will always be the same modulo 5, and the
denominator modulo 4, for this progression. And it doesn't have
anything to do with odd limits.

Beyond that I forget what the question was.

Graham

🔗rick <rick_ballan@...>

Thanks Graham, this now makes perfect sense. I'll study this more fully (all those Wilson diagrams are so detailed!).

In the meantime I'll say that yes you're right, I didn't take the mediants between 5/4 and 19/15 in this example since 81x + 64y = 1 gives mod x = 15 and mod y = 19 to which I was adding 4 and 5 (respectively). However, I've since noticed that if we solve 19x + 15y = 1 then mod x = 4 and mod y = 5, which gives all the other numbers 29/23 etc...I now see that the 4 5 Mods follows the left branch of the tree and the 15 19 follows the right branch, which is all very clear now and exactly what I was looking for.

Thanks

Rick

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 25 March 2010 14:54, rick <rick_ballan@...> wrote:
>
> > Sorry, you guys have lost me here. Graham suggested something about adding 12 to meantone but what is 12 and to what are we adding, do you know? The numerator and denominator of each interval, or the generators? I also looked into Erv Wilson once again but apart from his using mediants to develop tunings I couldn't find anything specifically relating to my original problem which was the use of mods to develop interval matches.
>
> As Carl has said, 12 is a number of notes to the octave. Wilson
> called this family "duodecimally negative" following Bosanquet's
> "singly negative" because a Pythagorean comma is always 1 step in the
> wrong direction.
>
> When I said "any meantone" before, I should have said "except 12-equal
> itself". Maybe that's why 24 and 72 crept into the discussion.
>
> The reason is the MOS version of the scale tree. If you think of
> meantones as having a fourth between 3/7 and 5/12 octaves, the tree is
> (use mono-spacing for optimal satisfaction):
>
> 3 5
> - --
> 7 12
> 8
> --
> 19
> 11 13
> -- --
> 26 31
> 18
> --
> 43
> 23
> --
> 55
>
> Every step to the right is adding 12 to the denominator, which is the
> number of steps to the octave. So adding 12 notes to the octave means
> the meantone tuning gets a bit closer to 12-equal. The number of
> notes to the octave is always congruent to 7 modulo 12 (or however
> you're supposed to say that).
>
> There are also valid meantones that aren't singly positive.
>
> The original question was about tunings between 5/4 and 81/64, where
> we could (but it looks like Rick didn't) take the scale tree between
> 5/4 and 19/15:
>
> 5 19
> - --
> 4 15
> 24
> --
> 19
> 29 43
> -- --
> 23 34
> 34 53 67 62
> -- -- -- --
> 27 42 53 49
> 81
> --
> 64
>
> Adding 5 to the numerator and 4 to the denominator will get you closer
> to 5/4. So the numerator will always be the same modulo 5, and the
> denominator modulo 4, for this progression. And it doesn't have
> anything to do with odd limits.
>
> Beyond that I forget what the question was.
>
>
> Graham
>

🔗rick <rick_ballan@...>

Hi Carl,

Ah I think I see, thanks. Graham's talking about adding 12 to arrive at other steps of the tuning after a certain point? What I'm doing is adding multiples of 19 to 5 and 15 to 4 in 5/4 to arrive at different major thirds i.e. (0 x 19) + 5 = 5, (0 x 15) + 4 = 4 giving 5/4, (1 x 19) + 5 = 24, (1 x 15) + 4 = 19 giving 24/19, then 43/34, 62/49, 81/64 etc...As I said before, the 19 and 15 come from solving Bezout's equation. This is what I meant by "interval matching", finding different ratios that match the 'same' interval (here major thirds). Perhaps I should find a better way of putting it?

-Rick

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Rick,
>
> Sorry, the thread has wandered a bit; I was replying to
> Mike's question. I don't know what you mean by "interval
> matches" so I can't help. Graham means adding 12 to the
> number of notes/octave. If you look at this meantone
> series
>
> 1 2 3 5 7 12 19 31 43 55
>
> the successive differences are
>
> 1 1 2 2 5 7 12 12 12
>
> Hope that helps,
>
> -Carl
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> >
> > Hi Carl,
> >
> > Sorry, you guys have lost me here. Graham suggested something
> > about adding 12 to meantone but what is 12 and to what are we
> > adding, do you know? The numerator and denominator of each
> > interval, or the generators? I also looked into Erv Wilson once
> > again but apart from his using mediants to develop tunings I
> > couldn't find anything specifically relating to my original
> > problem which was the use of mods to develop interval matches.
> >
> > -Rick
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> > > >
> > > > On 24 March 2010 12:14, Mike Battaglia <battaglia01@> wrote:
> > > >
> > > > > Although one more question: how in this case do you rigorously
> > > > define "meantone?" 24-tet could be mapped in such a way so that
> > > > the 350 cent interval equates to 5/4 and the 400 cent interval
> > > > equates to 81/64, or something like that.
> > > >
> > > > There are ways of defining meantones so that you exclude 24.
> > >
> > > Here's how I think of it. "Meantone" is the 5-limit rank 2
> > > temperament with 81/80 in its kernel.
> > >
> > > 24-ET, as a temperament, is rank 1. So it's not meantone.
> > > And even if we made an exception, the meantone-like val you'd
> > > use in 24 has torsion so it's not valid temperament anyway.
> > >
> > > As a scale, however, there may be an acceptable meantone
> > > tuning within 24-ET.
> > >
> > > For a scale to support meantone temperament, it must satisfy
> > > the map
> > >
> > > < 1 2 4 ]
> > > < 0 -1 -4 ]
> > >
> > > in the 5-limit. A good scale will satisfy it so that the
> > > most accurate approximation in the scale for each prime is
> > > the one mapped.* In 24-ET, if we choose 24 steps & 10 steps
> > > as generators, the above mapping reduces to
> > >
> > > < 24 38 56 ]
> > >
> > > that is, we get the 700 cents for 3:2 and 400 cents for 5:4,
> > > which are the best approximations available. So as a scale,
> > > 24-ET supports meantone temperament.
> > >
> > > -Carl
> > >
> > > * For an ET, this is the same as saying the patent val must
> > > be the result of the mapping. An acceptable variation on
> > > this is that the val with least TOP damage must be the result
> > > of the mapping.
> > >
> > > Hm, maybe Igliashon did have a point on MMM...
> > >
> >
>

🔗rick <rick_ballan@...>

Actually, cancel what I just said Carl. Graham's post shows that there IS a relation between these Mods and MOS. I'll take some time to understand this and get back.