The uniqueness of the septimal comma?

Let's say for the purposes of definition below, a comma is any interval less
than or equal to a diesis of 128/125 in size.

Let's say "comma distance", or c-distance on the lattice, is the sum of the
absolute values of the powers of a comma's monzo, not including the octave
powers of two.

Then the "easiness" of a given prime-limit comma is its ranking within a list
of commas sorted by increasing c-distance, and then by size, at or below a
given prime-limit in JI space.

For example:

1.01587, 64/63, c-distance = 3, monzo is [ 6, -2, 0, -1 > , 7-limit easiness=1
1.02083, 49/48, c-distance = 3, monzo is [ -4, -1, 0, 2 >, 7-limit easiness=2
1.02400, 128/125, c-distance = 3, monzo is [ 7, 0, -3, 0 >, 7-limit easiness=3
1.00488, 1029/1024, c-distance = 4, monzo is [ -10, 1, 0, 3 >, " easiness=4
1.02041, 50/49, c-distance = 4, monzo is [ 1, 0, 2, -2 > etc.
1.00446, 225/224, c-distance = 5, monzo is [ -5, 2, 2, -1 > etc,
1.01250, 81/80, c-distance = 5, monzo is [ -4, 4, -1, 0 >
1.00310, 6144/6125, c-distance = 6, monzo is [ 11, 1, -3, -2 >
1.00800, 126/125, c-distance = 6, monzo is [ 1, 2, -3, 1 >
1.01136, 2048/2025, c-distance = 6, monzo is [ 11, -4, -2, 0 >
1.01725, 3125/3072, c-distance = 6, monzo is [ -10, -1, 5, 0 >
1.02357, 12288/12005, c-distance = 6, monzo is [ 12, 1, -1, -4 >

and so on....

It seems that no comma at any prime-limit with an easiness equal to 64/63's
easiness of 1 in the 7-limit will have a numerator as low as 64, which I
think makes the septimal comma (64/63) quite special in this regard.

One could of course devise any arbitrary such measure for ranking the relative
simplicity of given commas, for instance, by asking, how low is the prime
limit? Which would make the Pythagorean comma most "important", followed by
the syntonic comma, and then septimal comma. And, obviously, where you place
the cutoff point of what makes a comma exactly, changes things. We could make
the syntonic comma "win" by not considering any limit higher than 5, and
making the diesis too large to be a 'comma'....

Anyway, just blabbing...maybe I want to prove something about why I'm
attracted to the septimal comma. Anyone have any related thoughts or
insights? Perhaps someone can find a more general formulation that accounts
for something more universal, and we can find a "most special comma", then
worship that?

-Aaron.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> Let's say "comma distance", or c-distance on the lattice, is the sum
of the
> absolute values of the powers of a comma's monzo, not including the
octave
> powers of two.

I'd say a better way to do this would be Hahn distance or symmetric
Euclidean distance:

http://66.98.148.43/~xenharmo/hahn.htm

http://66.98.148.43/~xenharmo/sevlat.htm

The reason I say "better" is that it seems to me to be a problem that
5/3 and 15/8 are given the same distance; it makes more sense to me to
give 5/3 the same distance as the other 7-limit consonances, and 15/8
a greater distance.

> It seems that no comma at any prime-limit with an easiness equal to
64/63's
> easiness of 1 in the 7-limit will have a numerator as low as 64,
which I
> think makes the septimal comma (64/63) quite special in this regard.

Only if c-distance is really what is wanted; moreover, you are relying
on an arbitary cutoff in defining what a comma is. If we use Hahn
distance and the 128/125 bound, we get 49/48 and 50/49 coming out as
special, so 50/49 edges out 49/48. If however we use symmetrical
Euclidean distance, 49/48 is a bit closer, and now is the winner.

I think one can argue that the size of the comma ought to factor in to
the equation, and when you do that, 2401/2400 is pretty certain to
look awfully special. Foe example, we can take the fourth power of
distance times size of commas, and 2401/2400 comes out on top.

> Anyway, just blabbing...maybe I want to prove something about why I'm
> attracted to the septimal comma. Anyone have any related thoughts or
> insights? Perhaps someone can find a more general formulation that
accounts
> for something more universal, and we can find a "most special
comma", then
> worship that?

This kind of thing has been discussed from various angles on
tuning-math, if you read that group.

On Thursday 21 July 2005 2:49 am, Gene Ward Smith wrote:

> I think one can argue that the size of the comma ought to factor in to
> the equation, and when you do that, 2401/2400 is pretty certain to
> look awfully special. Foe example, we can take the fourth power of
> distance times size of commas, and 2401/2400 comes out on top.

Gene,

I intuitively want to give "extra points" to comma ratios that have smaller
numerators, in light that they are more elegant. This is why the 'c-distance'
worked for me, but I can see the argument for hahn distance, and even more so
for not having an arbitrary cutoff for the comma, and using the 4th power of
distance times the comma size. Incidentally, why the 4th power?

A thought would be to take the final product of the 4th power of distance
(pending your explanation) times comma size times numerator, to "punish" a
fraction for being larger

or

easiness=(d^4)*c*n

Then the best comma to worship will have the smallest easiness....

-A

I like the 64/63 and the full Dallesandro has a pentatonic doubled at such an animal that when my brass tubes were up and running i used quite a bit.

>Message: 9 > Date: Wed, 20 Jul 2005 20:53:36 -0500
> From: Aaron Krister Johnson <aaron@akjmusic.com>
>Subject: The uniqueness of the septimal comma?
>
>and so on....
>
>It seems that no comma at any prime-limit with an easiness equal to 64/63's >easiness of 1 in the 7-limit will have a numerator as low as 64, which I >think makes the septimal comma (64/63) quite special in this regard.
>
>
> >
>
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> Let's say for the purposes of definition below, a comma is any
interval less
> than or equal to a diesis of 128/125 in size.

Aaron, if you're interested in some *non-arbitrary* boundaries for
small rational intervals, then you should read this:

/tuning/topicId_56202.html#56261

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> > Let's say for the purposes of definition below, a comma is any
> interval less
> > than or equal to a diesis of 128/125 in size.
>
> Aaron, if you're interested in some *non-arbitrary* boundaries for
> small rational intervals, then you should read this:
>
> /tuning/topicId_56202.html#56261

I think these boundries are clearly arbitrary. This reminds me of the
discussion of "moats" for temperaments, and whether a non-arbitary
boundry can be drawn there. The only non-arbitary boundry I can see is
that in any prime limit, there is always a smallest superparticular
comma. However, these rapidly become very small for increasing prime
limits.

le 21/07/05 3:53, Aaron Krister Johnson à aaron@akjmusic.com a écrit :

> ( The uniqueness of the septimal comma ? )...

> ... maybe I want to prove something about why I'm
> attracted to the septimal comma. Anyone have any related thoughts or
> insights? Perhaps someone can find a more general formulation that accounts
> for something more universal, and we can find a "most special comma", then
> worship that ?
>
> -Aaron.

Find all the divisors of the numerator and the denominator of a comma's
ratio.
Multiply them by the right powers of 2 in order to bring them inside one
octave.

Example : 63 has 3, 7, 9, 21 and 64 has only powers of 2
Final scale is 32 36 42 48 56 63-64
(and contains two symmetrical slendros, M with 63 and N with 64)

Then play music with that, at different octaves.
Worship only the comma that generates the scales you are resonating with.
(You may also worship several commas ;)

----------------------------------------------------------------
Jacques Dudon
Atelier d'Exploration Harmonique - Les Camails 83340 LE THORONET
tel & fax 04 94 73 87 78 - tel & répondeur 04 94 73 80 25
fotosonix@wanadoo.fr
http://aeh.free.fr

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> Find all the divisors of the numerator and the denominator of a comma's
> ratio.
> Multiply them by the right powers of 2 in order to bring them inside one
> octave.

For some reason which bears further examination, this proceedure seems
to usually result in a scale whose smallest step size is the very same
comma which was the starting point, and hence one which is a logical
candidate for tempering by that comma, though on the small size for
such a tempering. For example, 81/80 leads to the meantone pentatonic.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> I intuitively want to give "extra points" to comma ratios that have
smaller
> numerators, in light that they are more elegant. This is why the
'c-distance'
> worked for me, but I can see the argument for hahn distance, and
even more so
> for not having an arbitrary cutoff for the comma, and using the 4th
power of
> distance times the comma size. Incidentally, why the 4th power?

A related function we have been using a lot is the Tenney height,
which is the product of the numerator and denominator, or the Tenney
distance, which is log base two of the Tenney height.

The 4th power is still low enough that we can expect an infinity of
intervals beneath some upper bound. Of course, you may not want an
infinity of intervals, but this requirement at least gives a ground
for a particular choice of exponent.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

The numerator is a divisor of the numerator, and the denominator of
the denominator, so there is always an interval of
numerator/denominator = comma in a dudon scale. It is not always the
smallest interval, however.
In such cases the temperament combining the dudon comma with smaller
scale steps seems an obvious choice. For example, dudon(16875/16807) has
25 intervals. Tempering out the two smallest, 2401/2400 and
16875/16807, leads to miracle, and a 22-note miracle scale. In the
same way, dudon(250/243) leads to tempering out 250/243 and 81/80, and
so 7-equal, etc.

le 22/07/05 0:00, Gene Ward Smith à gwsmith@svpal.org a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>
>> Find all the divisors of the numerator and the denominator of a comma's
>> ratio.
>> Multiply them by the right powers of 2 in order to bring them inside one
>> octave.
>
> For some reason which bears further examination, this proceedure seems
> to usually result in a scale whose smallest step size is the very same
> comma which was the starting point, and hence one which is a logical
> candidate for tempering by that comma, though on the small size for
> such a tempering. For example, 81/80 leads to the meantone pentatonic.

All right, but my procedure didn't mention any tempering !
The scale 32 36 42 48 56 63-64 has six tones to the octave, not five.
Suppose we hear 64/63 as a "semitone" in this scale, this is a different
thing than simply 64 72 84 112 127.

I found these "comma-divisors" scales usually musical to my ears.

Since each note is by definition at least divisor of one of the two comma
frequencies, these scales have the property to be "spectral-coherent" - it
means that any note of the scale has among its overtones at least another
note of the scale (abstraction done of octave position and exception made
for the comma notes : 63 has none here).

You can also apply the procedure to several commas, ex :

128/125 & 81/80 generate the 8 notes scale

1 9 5-81 3 25 27 125-1

I use generally the term "harmonic coïncidences" rather than commas, because
the procedure can apply to schismas as well and anyway I don't have a
definitive definition of the boundaries between "schismas", "commas" and
even quartertones and so on, it depends on the context.

le 22/07/05 0:39, Gene Ward Smith à gwsmith@svpal.org a écrit :

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> The numerator is a divisor of the numerator, and the denominator of
> the denominator, so there is always an interval of
> numerator/denominator = comma in a dudon scale. It is not always the
> smallest interval, however.
> In such cases the temperament combining the dudon comma with smaller
> scale steps seems an obvious choice. For example, dudon(16875/16807) has
> 25 intervals. Tempering out the two smallest, 2401/2400 and
> 16875/16807, leads to miracle, and a 22-note miracle scale. In the
> same way, dudon(250/243) leads to tempering out 250/243 and 81/80, and
> so 7-equal, etc.

I am glad I invented the dudon scales, but you're too fast for me and
besides I am missing the precedent chapters.
Can you give an example where it is not the smaller interval ?
Is 16875/16807 an example of it ?
Do you find 2401/2400 in dudon(16875/16807) ?
And can somebody explain me what Gene means by "tempering out" ?
(Is Gene an temperament worshipper ? ;)
And how do you "temper out" for example 250/243 and 81/80 ?

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> All right, but my procedure didn't mention any tempering !

My point was that the scales thus produced seem to want to be natural
candidates for tempering.

> I found these "comma-divisors" scales usually musical to my ears.

Consider dudon(2401/2400). This is a ten-note scale

1, 75/64, 2401/2048, 5/4, 343/256, 3/2, 49/32, 25/16, 7/4, 15/8

It can be "rotated" to the following mode:

1, 50/49, 8/7, 60/49, 64/49, 75/49, 49/32, 80/49, 7/4, 96/49

The interval between 75/64 and 2401/2048 in the first version, or
75/49 and 49/32 in the second, is 2401/2400, which is less than a
cent. This is hardly a convincingly musical interval to have in a
ten-note scale. Removing a note, or tempering via 2401/2400, now gives
a nine-note scale which *does* make at least some sense musically.

> Since each note is by definition at least divisor of one of the two
comma
> frequencies, these scales have the property to be
"spectral-coherent" - it
> means that any note of the scale has among its overtones at least
another
> note of the scale (abstraction done of octave position and exception
made
> for the comma notes : 63 has none here).

This property really derives from the fact that it is the union of two
Euler genera, doesn't it? This whole business is a little reminiscent
of the "corner clipper comma" stuff for Euler genera I was talking
about a while back.

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> Can you give an example where it is not the smaller interval ?
> Is 16875/16807 an example of it ?

Right. 16875/16807 is 6.99 cents, but 2401/2400 is 0.72 cents, far
snaller.

> Do you find 2401/2400 in dudon(16875/16807) ?

It's the interval between 75/64 and 2401/2048.

> And can somebody explain me what Gene means by "tempering out" ?

Choosing a tuning so that the commas in question vanish. For instance,
in 175 equal, both 16875/16807 and 2401/2400 vanish. If both these
commas vanish, you are in miracle temperament territory. In 175,
344373768/341796875 also vanishes, but we don't care about that.

> (Is Gene an temperament worshipper ? ;)

Gene uses both temperaments and JI. Gene thinks the theory these are
opposing forces is silly.

> And how do you "temper out" for example 250/243 and 81/80 ?

The only way to do that is 7-et, or at least some irregular form of it.

le 22/07/05 9:00, Gene Ward Smith à gwsmith@svpal.org a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>
>> Can you give an example where it is not the smaller interval ?
>> Is 16875/16807 an example of it ?
>
> Right. 16875/16807 is 6.99 cents, but 2401/2400 is 0.72 cents, far
> smaller.
>
>> Do you find 2401/2400 in dudon(16875/16807) ?
>
> It's the interval between 75/64 and 2401/2048.

I understand better - I thought you meant we could find 2401/2400
between the actual divisors of 16875 and 16807, which is impossible.
You find it when you compare different intervals of the scale. That's
interesting...
And it is an incredibly rare thing.

We can find many examples if we admit complementing two commas, such as
(56/55 & 55/54) (or 56/55/54) : in that case,
55/54 has divisors 1 3 5 9 11 27
56/55 has divisors 1 5 7 11 55 and both compiled : 1 3 5 7 9 11 27 55
where you find 55/48 and 8/7 that differ from the smaller comma 385/384

The only second example I found of a single, non simplifiable comma hiding
such smaller skisma is
dudon(351/350) that has divisors 1 3 5 7 9 13 25 27 35 39 117 175-351
(12 steps if 351/350 considered as a double one)
where 351/350 is not the smaller comma, but 4096/4095
I remind seeing this oriental beauty between 128/117 and 35/32
I used it in a disk I called "Yantra"

Do you have a division of the octave for 351/350 ?
Or 4096/4095 ?
It has 23 steps with 4096/4095 considered as double
and a few commas such as 64/63, 65/64, 91/45 and 105/104

Gene and all,

I think your suggestion of the Hahn distance is not what I'm looking for for
the 'easiness' function.

If you define 'easiness' of production of a comma to be how many step of JI
tuning you'd have to take on a keyboard to tune up a given comma by ear, that
is definately 'taxicab distance' on the lattice, which is, for a 7-limit
comma

abs(x)+abs(y)+abs(z)

where

3^x*5^y*7^z are the x,y,z components of the comma's position on the lattice
(it's 'monzo' as you all like to say for short)

given these parameters, for intervals we commonly consider to be small enough
to be considered commas, the septimal comma is special, because it is the
smallest comma available in 3 'taxicab steps' on a 7-limit lattice-space. (if
we look at the intervals derivable within 2 taxicab steps, we get sizes
larger than the typical comma)

perhaps we might derive a function that would return a rating based on the
tradeoff between taxicab-steps and smallness of the comma, and find a more
'efficiently derived' comma, but my hypothesis would be that the septimal
comma is the most efficient one out there, at least in the 7-limit.

-Aaron.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> > --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
> > > Let's say for the purposes of definition below, a comma is any
> > interval less
> > > than or equal to a diesis of 128/125 in size.
> >
> > Aaron, if you're interested in some *non-arbitrary* boundaries for
> > small rational intervals, then you should read this:
> >
> > /tuning/topicId_56202.html#56261
>
> I think these boundries are clearly arbitrary.

For what reason?

Expressing commas as ratios between a rational pitch and nominals in a
Pythagorean sequence (extended indefinitely) results in sets of commas
(the word being used in the "generic" sense) that have the same prime
factors >3 as the rational pitch, thereby allowing those commas to be
grouped (and named) according to those factors and distinguished from
one another on the basis of size. If you consider the selection of
nominals to be arbitrary, then I should point out that the practice of
constructing chains of tones by a 2:3 or 3:4 seems to have been trans-
culturally popular.

If you know of an alternative method for finding the boundary between
what we might call a "diesis" vs. a "(categorical) comma" or
a "(categorical) comma" vs. a "kleisma" (or whatever vs. whatever
else), then let's hear it. Otherwise, I think Dave has come up with a
very useful classification of commas (word used in the generic sense)
that ought not to be dismissed as "arbitrary".

> This reminds me of the
> discussion of "moats" for temperaments, and whether a non-arbitary
> boundry can be drawn there.

Do you have a link to that discussion?

> The only non-arbitary boundry I can see is
> that in any prime limit, there is always a smallest superparticular
> comma. However, these rapidly become very small for increasing prime
> limits.

and therefore not very useful for distinguishing categories of commas
by size.

--George

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> The only second example I found of a single, non simplifiable comma
hiding
> such smaller skisma is
> dudon(351/350) that has divisors 1 3 5 7 9 13 25 27 35 39 117 175-351
> (12 steps if 351/350 considered as a double one)
> where 351/350 is not the smaller comma, but 4096/4095

I'm not checking you here. I get the folowing 13-note scale:

1 35/32 9/8 39/32 5/4 175/128 351/256 3/2 25/16 13/8 27/16 7/4 117/64

This has a step of 351/350 between 175/128 and 351/350, but no smaller
one that I can find. To get 128/117, you can add the inverse intervals
to the scale, producing an inversely symmetrical scale. That certainly
makes sense, but now we have a different setup--one more likely to
produce smaller intervals than the defining comma. The resulting 25
note scale in this instance has two intervals of 4096/4095, and two of
351/350, with the remaining intervals much larger. Hence it would
certainly make sense to "temper out" 4096/4095 and 351/350, for
instance by using 130 equal. It is possible to temper out only these
two commas, producing what is called a planar temperament, since the
octave equivalent note classes lie in a plane, like 5-limit just
intonation.

> I remind seeing this oriental beauty between 128/117 and 35/32
> I used it in a disk I called "Yantra"

Have you been reading Ernest McClain? Are any samples of your music
available on-line?

> Do you have a division of the octave for 351/350 ?

130 works for both 351/350 and 4096/4095. Other nice 351/350 divisions
are 58, 72, 111, 183 and 241.

> Or 4096/4095 ?

1506 is very nice for extreme accuracy, but 270, 311 or 494 are likely
to prove sufficient.

> It has 23 steps with 4096/4095 considered as double
> and a few commas such as 64/63, 65/64, 91/45 and 105/104

Indeed. In 1506-et, 4096/4095 is 0 steps, 351/350 is 6 steps, 512/507
21 steps, 64/63 and 65/64 both 34 steps (the difference being
4096/4095.) I don't see 105/104 in there, which however would also be
21 steps. The largest intervals are 35/32 and 128/117, 195 steps,
which is much larger than 6 steps, so this scale is still quite
irregular even with 4096/4095 removed.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> Gene and all,
>
> I think your suggestion of the Hahn distance is not what I'm looking
for for
> the 'easiness' function.
>
> If you define 'easiness' of production of a comma to be how many
step of JI
> tuning you'd have to take on a keyboard to tune up a given comma by
ear, that
> is definately 'taxicab distance' on the lattice, which is, for a
7-limit
> comma

> abs(x)+abs(y)+abs(z)

Actually, it isn't, which is the point of Hahn distance. You get
instead max(|x|, |y|, |z|, |x+y|, |y+z|, |z+x|, |x+y+z|). You are
assuming a particular kind of path, but if you allow *any* path using
7-limit consonances, which I think is what you can assume can be
"tuned up", you can beat it. Of course, if a 5/3 has to be tuned as a
5 and a 3 separately, you get your metric, but I don't see why primes
are privileged.

This can make for a difference of a factor of two: 50/49 is obtained
as two 10/7 intervals, a 7-limit consonance, with a Hahn distance of
2. Your measure gives a 2 for the power of 5, and another 2 for the
power of 7, totaling 4. Your path would replace 10/7 10/7 with 5/4 8/7
5/4 8/7.

> given these parameters, for intervals we commonly consider to be
small enough
> to be considered commas, the septimal comma is special, because it
is the
> smallest comma available in 3 'taxicab steps' on a 7-limit
lattice-space.

But it isn't the smallest availble in 3 consonant steps, which in fact
is 126/125. We get

Smallest interval for a given number of lattice steps

1: 8/7
2: 50/49
3: 126/125
4: 2401/2400
5: 2401/2400
6: 2401/2400
7: 4375/4374

The one looking rather stupifyingly special to me is 2401/2400.

> perhaps we might derive a function that would return a rating based
on the
> tradeoff between taxicab-steps and smallness of the comma, and find
a more
> 'efficiently derived' comma, but my hypothesis would be that the
septimal
> comma is the most efficient one out there, at least in the 7-limit.

That's the sort of thing we've done on tuning-math, but not with your
taxicab metric. I don't see a convincing rationale for that yet.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
> Let's say for the purposes of definition below, a comma is any
interval less
> than or equal to a diesis of 128/125 in size.
>
> Let's say "comma distance", or c-distance on the lattice, is the
sum of the
> absolute values of the powers of a comma's monzo, not including the
octave
> powers of two.

As you probably know, I consider this wanting as a distance measure
in certain respects. For one thing, if you're getting rid of the
powers of two, a ratio having exponents of opposite sign can indicate
a much closer relationship than a ratio having exponents of the same
sign, all other things equal. For example, compare 5/3 against 16/15 -
- the latter clearly represents a much further distance harmonically
and on any reasonable sort of lattice. Other problems with this
measure include the fact that higher primes should represent a longer
distance than lower ones. The Tenney harmonic distance seems a much
better measure for several reasons, some of which I tried to get into
in the 'Middle Path' paper I sent you. And it's easy to calculate --
just multiply the numerator by the denominator (and take the log if
you're after more than just ranking).

You might be interested in looking at this diagram of 7-limit commas:

/tuning/files/Erlich/planar.gif

where the horizontal axis shows Tenney harmonic distance and the
vertical axis is proportional to the cents size of the ratio divided
by its Tenney harmonic distance. So the "most important" commas will
be found to the lower left of the diagram; depending on how you draw
your "indifference curves", 64/63 might indeed show up as "best for
you", though 81/80, 50/49, 49/48, and 126/125 would all probably fare
quite well too.

On Friday 22 July 2005 1:15 pm, Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> > Gene and all,
> >
> > I think your suggestion of the Hahn distance is not what I'm looking
>
> for for
>
> > the 'easiness' function.
> >
> > If you define 'easiness' of production of a comma to be how many
>
> step of JI
>
> > tuning you'd have to take on a keyboard to tune up a given comma by
>
> ear, that
>
> > is definately 'taxicab distance' on the lattice, which is, for a
>
> 7-limit
>
> > comma
> >
> > abs(x)+abs(y)+abs(z)
>
> Actually, it isn't, which is the point of Hahn distance. You get
> instead max(|x|, |y|, |z|, |x+y|, |y+z|, |z+x|, |x+y+z|). You are
> assuming a particular kind of path, but if you allow *any* path using
> 7-limit consonances, which I think is what you can assume can be
> "tuned up", you can beat it. Of course, if a 5/3 has to be tuned as a
> 5 and a 3 separately, you get your metric, but I don't see why primes
> are privileged.
>
> This can make for a difference of a factor of two: 50/49 is obtained
> as two 10/7 intervals, a 7-limit consonance, with a Hahn distance of
> 2. Your measure gives a 2 for the power of 5, and another 2 for the
> power of 7, totaling 4. Your path would replace 10/7 10/7 with 5/4 8/7
> 5/4 8/7.

Tuning a 10/7 by ear is too hard. That's why the taxicab approach works
better.

Granted, 5/3 is tunable easily in one step, but as the prime limit goes up, so
does the nightmare of trying to tune various dyads alone in one step. Try to
tune 14/11 by ear in one step, and you'll see what I mean.

Hence the taxicab distance as making more sense from a practical consistency
point-of-view.

BTW, my script made an error---here's the corrected list of positive valued
7-limit commas, in order of decreasing taxicab efficiency:

decimal ratio monzo taxi-d efficiency

1.01587 64/63 [ 6, -2, 0, -1 > 3 0.04233
1.02083 49/48 [ -4, -1, 0, 2 > 3 0.04067
1.02400 128/125 [ 7, 0, -3, 0 > 3 0.03962
1.06667 16/15 [ 4, -1, -1, 0 > 2 0.03810
1.00488 1029/1024 [ -10, 1, 0, 3 > 4 0.03449
1.04167 25/24 [ -3, -1, 2, 0 > 3 0.03373
1.04490 256/245 [ 8, 0, -1, -2 > 3 0.03265
1.05000 21/20 [ -2, 1, -1, 1 > 3 0.03095
1.02041 50/49 [ 1, 0, 2, -2 > 4 0.03061
1.02539 525/512 [ -9, 1, 2, 1 > 4 0.02937
1.02857 36/35 [ 2, 2, -1, -1 > 4 0.02857
1.00446 225/224 [ -5, 2, 2, -1 > 5 0.02768
1.03704 28/27 [ 2, -3, 0, 1 > 4 0.02646
1.01250 81/80 [ -4, 4, -1, 0 > 5 0.02607
1.09375 35/32 [ -5, 0, 1, 1 > 2 0.02455
1.07143 15/14 [ -1, 1, 1, -1 > 3 0.02381
1.02582 16807/16384 [ -14, 0, 0, 5 > 5 0.02341
1.00310 6144/6125 [ 11, 1, -3, -2 > 6 0.02329
1.00800 126/125 [ 1, 2, -3, 1 > 6 0.02248
1.05469 135/128 [ -7, 3, 1, 0 > 4 0.02204
1.01136 2048/2025 [ 11, -4, -2, 0 > 6 0.02192
1.01725 3125/3072 [ -10, -1, 5, 0 > 6 0.02093
1.00042 2401/2400 [ -5, -1, -2, 4 > 7 0.02035
1.00136 65625/65536 [ -16, 1, 5, 1 > 7 0.02021
1.00352 3136/3125 [ 6, 0, -5, 2 > 7 0.01991
1.02357 12288/12005 [ 12, 1, -1, -4 > 6 0.01988
1.00758 1728/1715 [ 6, 3, -1, -3 > 7 0.01933
1.04675 8575/8192 [ -13, 0, 2, 3 > 5 0.01922
1.00937 33075/32768 [ -15, 3, 2, 2 > 7 0.01907
1.01094 65536/64827 [ 16, -3, 0, -4 > 7 0.01885
1.01273 875/864 [ -5, -3, 3, 1 > 7 0.01859
1.03200 4096/3969 [ 12, -4, 0, -2 > 6 0.01848
1.03359 1323/1280 [ -8, 3, -1, 2 > 6 0.01821
1.05350 256/243 [ 8, -5, 0, 0 > 5 0.01787
1.07187 343/320 [ -6, 0, -1, 3 > 4 0.01775
1.01902 131072/128625 [ 17, -1, -3, -3 > 7 0.01769
1.00178 2100875/2097152 [ -21, 0, 3, 5 > 8 0.01764
1.00333 5120/5103 [ 10, -6, 1, -1 > 8 0.01744
1.04025 8192/7875 [ 13, -2, -3, -1 > 6 0.01710
1.00823 245/243 [ 0, -5, 1, 2 > 8 0.01683
1.04210 2401/2304 [ -8, -2, 0, 4 > 6 0.01679
1.00979 1058841/1048576 [ -20, 2, 0, 6 > 8 0.01663
1.02900 1029/1000 [ -3, 1, -3, 3 > 7 0.01627
1.04533 392/375 [ 3, -1, -3, 2 > 6 0.01625
1.02997 16875/16384 [ -14, 3, 4, 0 > 7 0.01613
1.04632 1875/1792 [ -8, 1, 4, -1 > 6 0.01609
1.01545 51200/50421 [ 11, -1, 2, -5 > 8 0.01593
1.01630 686/675 [ 1, -3, -2, 3 > 8 0.01582
1.01702 3645/3584 [ -9, 6, 1, -1 > 8 0.01573
1.04858 16384/15625 [ 14, 0, -6, 0 > 6 0.01571
1.03316 405/392 [ -3, 4, 1, -2 > 7 0.01567
1.04956 360/343 [ 3, 2, 1, -3 > 6 0.01555
1.01860 4194304/4117715 [ 22, 0, -1, -7 > 8 0.01553
1.01945 28672/28125 [ 12, -2, -5, 1 > 8 0.01543
1.06622 2560/2401 [ 9, 0, 1, -4 > 5 0.01533
1.03661 3200/3087 [ 7, -2, 2, -3 > 7 0.01518
1.02222 1071875/1048576 [ -20, 0, 5, 3 > 8 0.01508
1.06812 4375/4096 [ -12, 0, 4, 1 > 5 0.01495
1.03821 8505/8192 [ -13, 5, 1, 1 > 7 0.01495
1.03982 262144/252105 [ 18, -1, -1, -5 > 7 0.01472
1.02881 250/243 [ 1, -5, 3, 0 > 8 0.01426
1.05820 200/189 [ 3, -3, 2, -1 > 6 0.01411
1.03040 540225/524288 [ -19, 2, 2, 4 > 8 0.01406
1.08844 160/147 [ 5, -1, 1, -2 > 4 0.01361
1.06148 16384/15435 [ 14, -2, -1, -3 > 6 0.01356
1.03680 648/625 [ 3, 4, -4, 0 > 8 0.01326
1.06337 1225/1152 [ -7, -2, 2, 2 > 6 0.01325
1.07666 2205/2048 [ -11, 2, 1, 2 > 5 0.01324
1.05326 4608/4375 [ 9, 2, -4, -1 > 7 0.01280
1.04123 2500/2401 [ 2, 0, 4, -4 > 8 0.01270
1.08000 27/25 [ 0, 3, -2, 0 > 5 0.01257
1.05513 21609/20480 [ -12, 2, -1, 4 > 7 0.01253
1.04308 546875/524288 [ -19, 0, 7, 1 > 8 0.01247
1.04446 122880/117649 [ 13, 1, 1, -6 > 8 0.01230
1.06998 32768/30625 [ 15, 0, -4, -2 > 6 0.01215
1.05864 343/324 [ -2, -4, 0, 3 > 7 0.01203
1.04719 823543/786432 [ -18, -1, 0, 7 > 8 0.01196
1.08360 1024/945 [ 10, -3, -1, -1 > 5 0.01185
1.06193 3584/3375 [ 9, -3, -3, 1 > 7 0.01156
1.05044 16807/16000 [ -7, 0, -3, 5 > 8 0.01155
1.09714 192/175 [ 6, 1, -2, -1 > 4 0.01143
1.05143 275625/262144 [ -18, 2, 4, 2 > 8 0.01143
1.06293 625/588 [ -2, -1, 4, -2 > 7 0.01142
1.07520 672/625 [ 5, 1, -4, 1 > 6 0.01128
1.08889 49/45 [ 0, -2, -1, 2 > 5 0.01079
1.06787 2187/2048 [ -11, 7, 0, 0 > 7 0.01071
1.05796 1296/1225 [ 4, 4, -2, -2 > 8 0.01061
1.11111 10/9 [ 1, -2, 1, 0 > 3 0.01058
1.05984 138915/131072 [ -17, 4, 1, 3 > 8 0.01038
1.09227 2048/1875 [ 11, -1, -4, 0 > 5 0.01012
1.06313 1701/1600 [ -6, 5, -2, 1 > 8 0.00997
1.07475 9216/8575 [ 10, 2, -2, -3 > 7 0.00973
1.08482 243/224 [ -5, 5, 0, -1 > 6 0.00967
1.08507 625/576 [ -6, -2, 4, 0 > 6 0.00963
1.06856 420175/393216 [ -17, -1, 2, 5 > 8 0.00929
1.07022 16384/15309 [ 14, -7, 0, -1 > 8 0.00908
1.08025 175/162 [ -1, -4, 2, 1 > 7 0.00894
1.12500 9/8 [ -3, 2, 0, 0 > 2 0.00893
1.09863 1125/1024 [ -10, 2, 3, 0 > 5 0.00884
1.07288 140625/131072 [ -17, 2, 6, 0 > 8 0.00875
1.09181 65536/60025 [ 16, 0, -2, -4 > 6 0.00851
1.07545 784/729 [ 4, -6, 0, 2 > 8 0.00843
1.07621 3375/3136 [ -6, 3, 3, -2 > 8 0.00833
1.07711 352947/327680 [ -16, 1, -1, 6 > 8 0.00822
1.10204 54/49 [ 1, 3, 0, -2 > 5 0.00816
1.07878 32768/30375 [ 15, -5, -3, 0 > 8 0.00801
1.07955 2592/2401 [ 5, 4, 0, -4 > 8 0.00791
1.08147 70875/65536 [ -16, 4, 3, 1 > 8 0.00767
1.12000 28/25 [ 2, 0, -2, 1 > 3 0.00762
1.08991 15625/14336 [ -11, 0, 6, -1 > 7 0.00756
1.08315 163840/151263 [ 15, -2, 1, -5 > 8 0.00746
1.09909 36015/32768 [ -15, 1, 1, 4 > 6 0.00729
1.10742 567/512 [ -9, 4, 0, 1 > 5 0.00709
1.09329 375/343 [ 0, 1, 3, -3 > 7 0.00708
1.09421 16807/15360 [ -10, -1, -1, 5 > 7 0.00695
1.10250 441/400 [ -4, 2, -2, 2 > 6 0.00673
1.11607 125/112 [ -4, 0, 3, -1 > 4 0.00670
1.09669 18432/16807 [ 11, 2, 0, -5 > 7 0.00660
1.09012 35721/32768 [ -15, 6, 0, 2 > 8 0.00659
1.09037 214375/196608 [ -16, -1, 4, 3 > 8 0.00656
1.09760 686/625 [ 1, 0, -4, 3 > 7 0.00647
1.10617 448/405 [ 6, -4, -1, 1 > 6 0.00611
1.11953 384/343 [ 7, 1, 0, -3 > 4 0.00583
1.09739 800/729 [ 5, -6, 2, 0 > 8 0.00568
1.11456 4096/3675 [ 12, -1, -2, -2 > 5 0.00566
1.10571 10240/9261 [ 11, -3, 1, -3 > 7 0.00531
1.11654 1715/1536 [ -9, -1, 1, 3 > 5 0.00526
1.10080 65536/59535 [ 16, -5, -1, -2 > 8 0.00526
1.11409 131072/117649 [ 17, 0, 0, -6 > 6 0.00479
1.10592 3456/3125 [ 7, 3, -5, 0 > 8 0.00462
1.10960 131072/118125 [ 17, -3, -4, -1 > 8 0.00416
1.11065 8000/7203 [ 6, -1, 3, -4 > 8 0.00403
1.11157 2401/2160 [ -4, -3, -1, 4 > 8 0.00391
1.11237 18225/16384 [ -14, 6, 2, 0 > 8 0.00381
1.11262 109375/98304 [ -15, -1, 6, 1 > 8 0.00378
1.12152 18375/16384 [ -14, 1, 3, 2 > 6 0.00356
1.11502 6272/5625 [ 7, -2, -4, 2 > 8 0.00348
1.11848 262144/234375 [ 18, -1, -7, 0 > 8 0.00305
1.12199 588245/524288 [ -19, 0, 1, 6 > 7 0.00298
1.12347 24576/21875 [ 13, 1, -5, -1 > 7 0.00277
1.12373 4096/3645 [ 12, -6, -1, 0 > 7 0.00273
1.12547 7203/6400 [ -8, 1, -2, 4 > 7 0.00248
1.12875 640/567 [ 7, -4, 1, -1 > 6 0.00235
1.13049 9261/8192 [ -13, 3, 0, 3 > 6 0.00206
1.13002 2025/1792 [ -8, 4, 2, -1 > 7 0.00183
1.12849 6912/6125 [ 8, 3, -3, -2 > 8 0.00180
1.13426 245/216 [ -3, -3, 1, 2 > 6 0.00143
1.13225 262144/231525 [ 18, -3, -2, -3 > 8 0.00133
1.13379 500/441 [ 2, -2, 3, -2 > 7 0.00130
1.13778 256/225 [ 8, -2, -2, 0 > 4 0.00127
1.13730 8192/7203 [ 13, -1, 0, -4 > 5 0.00111
1.13400 567/500 [ -2, 4, -3, 1 > 8 0.00111
1.13932 875/768 [ -8, -1, 3, 1 > 5 0.00071
1.13906 729/640 [ -7, 6, -1, 0 > 7 0.00054
1.13885 3125/2744 [ -3, 0, 5, -3 > 8 0.00050
1.13980 84035/73728 [ -13, -2, 1, 5 > 8 0.00038
1.14131 524288/459375 [ 19, -1, -5, -2 > 8 0.00019
1.14238 19200/16807 [ 8, 1, 2, -5 > 8 0.00006
1.14286 8/7 [ 3, 0, 0, -1 > 1 0.00000

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
>
> > Let's say "comma distance", or c-distance on the lattice, is the
sum
> of the
> > absolute values of the powers of a comma's monzo, not including
the
> octave
> > powers of two.
>
> I'd say a better way to do this would be Hahn distance or symmetric
> Euclidean distance:
>
> http://66.98.148.43/~xenharmo/hahn.htm
>
> http://66.98.148.43/~xenharmo/sevlat.htm
>
> The reason I say "better" is that it seems to me to be a problem
that
> 5/3 and 15/8 are given the same distance; it makes more sense to me
to
> give 5/3 the same distance as the other 7-limit consonances, and
15/8
> a greater distance.

I'd go further and say that, since 7-prime-limit sometimes can mean 9-
odd-limit in practice, a weighting such as Tenney's, which gives
shorter distance to the lower primes, would be even more appropriate
in general. If octave-equivalence is to be enforced, I strongly
recommend using the Kees van Prooijen lattice, where the relevant
distance measure (using a hexagonal or rhombic-dodecahedral norm, as
I've come to realize) is the van Prooijen 'expressibility' or
colloquially the 'odd limit' -- the largest odd factor of either the
numerator or denominator. (You'd take the log if you're interested in
more than ranking -- I'll use log base 2). For example:

5/3: expressibility = log(5) (=2.3219)
15/8: expressibility = log(15) (=3.9069)
49/48: expressibility = log(49) (=5.6147)
50/49: expressibility = log(49) (=5.6147)
64/63: expressibility = log(63) (=5.9773)
81/80: expressibility = log(81) (=6.3399)
126/125: expressibility = log(125) (=6.9658)
2401/2400: expressibility = log(2401) (=11.229)

These numbers are real easy to derive, and they make a whole lot of
sense when you're looking at the lattice.

But really, isn't this a topic for the tuning-math list? I hope you
join if you haven't yet, Aaron . . .

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:
> On Thursday 21 July 2005 2:49 am, Gene Ward Smith wrote:
>
> > I think one can argue that the size of the comma ought to factor in
to
> > the equation, and when you do that, 2401/2400 is pretty certain to
> > look awfully special. Foe example, we can take the fourth power of
> > distance times size of commas, and 2401/2400 comes out on top.
>
> Gene,
>
> I intuitively want to give "extra points" to comma ratios that have
smaller
> numerators, in light that they are more elegant. This is why the 'c-
distance'
> worked for me,

It seems Tenney harmonic distance or van Prooijen expressibility would
work even better for you, then. Wouldn't you consider them even more
elegant? You can express either of them in terms of the exponents in
the factorization, but you can also immediately see how they rank
ratios just by looking at the size of the numbers in the ratios. Isn't
this closer to what you're after, Aaron?

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> ...
> > This reminds me of the
> > discussion of "moats" for temperaments, and whether a non-arbitary
> > boundry can be drawn there.
>
> Do you have a link to that discussion?

Never mind -- I did a search in tuning-math and found a definition
of "moat" here:
/tuning-math/message/9199
from which I suspect that your concern is not with the Pythagorean-
nominal comparison at all, but rather with how many places the chain of
nominals should be extended in order to make the comparisons (which
would, in turn, determine how many size boundaries will result).

But even so, the *locations* of the boundaries would not be arbitrary.

--George

Hi Aaron,

The Tenney Harmonic Distance, as shown in the 'Middle Path' paper I
mailed you, is also a taxicab metric. It's the taxicab distance in
the Tenney lattice.

For reasons I tried to explain in a post I wrote today, it seems far
superior as a distance measure to the metric you're proposing.

It's also a lot easier to see how the Tenney Harmonic Distances
compare if you're just given the ratios as fractions. Smaller numbers
in the comma's ratio implies shorter harmonic distance.

Best,
Paul

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
> Gene and all,
>
> I think your suggestion of the Hahn distance is not what I'm
looking for for
> the 'easiness' function.
>
> If you define 'easiness' of production of a comma to be how many
step of JI
> tuning you'd have to take on a keyboard to tune up a given comma by
ear, that
> is definately 'taxicab distance' on the lattice, which is, for a 7-
limit
> comma
>
> abs(x)+abs(y)+abs(z)
>
> where
>
> 3^x*5^y*7^z are the x,y,z components of the comma's position on the
lattice
> (it's 'monzo' as you all like to say for short)
>
> given these parameters, for intervals we commonly consider to be
small enough
> to be considered commas, the septimal comma is special, because it
is the
> smallest comma available in 3 'taxicab steps' on a 7-limit lattice-
space. (if
> we look at the intervals derivable within 2 taxicab steps, we get
sizes
> larger than the typical comma)
>
> perhaps we might derive a function that would return a rating based
on the
> tradeoff between taxicab-steps and smallness of the comma, and find
a more
> 'efficiently derived' comma, but my hypothesis would be that the
septimal
> comma is the most efficient one out there, at least in the 7-limit.
>
> -Aaron.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
> On Friday 22 July 2005 1:15 pm, Gene Ward Smith wrote:

> > This can make for a difference of a factor of two: 50/49 is
obtained
> > as two 10/7 intervals, a 7-limit consonance, with a Hahn distance
of
> > 2. Your measure gives a 2 for the power of 5, and another 2 for
the
> > power of 7, totaling 4. Your path would replace 10/7 10/7 with
5/4 8/7
> > 5/4 8/7.
>
> Tuning a 10/7 by ear is too hard.

You can use 7/5 instead and it amounts to the same thing as far as
what Gene was saying.

>That's why the taxicab approach works
> better.

This argument clearly can't be right if you're happy tuning 7/5 by
ear (I am). Meanwhile, Gene and I have both given you an example
where what you're calling 'taxicab' is clearly deficient. Aaron,
yours is not the only taxicab approach, and Hahn distance is a type
of taxicab measure too (though not on a rectangular lattice). We've
discussed different lattice geometries and different metrics
extensively on this list and then the tuning-math list (where some of
your posts on this clearly belong). I hope you'll have the patience
to listen to the other ideas and arguments here, and also to read
the 'Middle Path' paper that I snail-mailed you.

> Granted, 5/3 is tunable easily in one step, but as the prime limit
goes up, so
> does the nightmare of trying to tune various dyads alone in one
step. Try to
> tune 14/11 by ear in one step, and you'll see what I mean.

Partch considered this interval mildly consonant. Using sawtooth
waves, I was able to tune 17/13 and all simpler (lower-numbers)
ratios by ear (I tuned them first just by listening and then
determined the exact intervals later). But for the purposes of
evaluating Hahn distance, which assumes octave-equivalence, 14/11 is
the same thing as 11/7, which is not really all that hard to tune by
ear. Meanwhile, your particular 'taxicab' metric would assign the
very same distance to 77/64 or 128/77, which are clearly far more
difficult to tune by ear, and I'd say impossible to tune by ear
without tuning up additional, auxillary notes along the way.

On Friday 22 July 2005 2:13 pm, wallyesterpaulrus wrote:

> As you probably know, I consider this wanting as a distance measure
> in certain respects. For one thing, if you're getting rid of the
> powers of two, a ratio having exponents of opposite sign can indicate
> a much closer relationship than a ratio having exponents of the same
> sign, all other things equal. For example, compare 5/3 against 16/15 -
> - the latter clearly represents a much further distance harmonically
> and on any reasonable sort of lattice. Other problems with this
> measure include the fact that higher primes should represent a longer
> distance than lower ones. The Tenney harmonic distance seems a much
> better measure for several reasons, some of which I tried to get into
> in the 'Middle Path' paper I sent you. And it's easy to calculate --
> just multiply the numerator by the denominator (and take the log if
> you're after more than just ranking).
>
> You might be interested in looking at this diagram of 7-limit commas:
>
> /tuning/files/Erlich/planar.gif
>
> where the horizontal axis shows Tenney harmonic distance and the
> vertical axis is proportional to the cents size of the ratio divided
> by its Tenney harmonic distance. So the "most important" commas will
> be found to the lower left of the diagram; depending on how you draw
> your "indifference curves", 64/63 might indeed show up as "best for
> you", though 81/80, 50/49, 49/48, and 126/125 would all probably fare
> quite well too.

Paul,

As I mentioned to Gene, I wasn't neccessarily interested in the harmonic
distance, per se, but the 'tunable' distance in terms of primes. What you say
about Tenney makes sense, too, though. When I get a chance, I will rank the
commas by the various means you suggested....did you see the posting I did
earlier today, where I ranked them by a simple measure of the reference ratio
of the given limit mius the comma, divided by the taxicab distance. Indeed,
64/63 proves to be the most efficient way to get a decently small comma.

So you see, it has nothing to do with consonance or dissonance whatsoever, and
not doing so, has nothing to do with the otherwise true and excellent points
you bring up.

-Aaron.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:

> Paul,
>
> As I mentioned to Gene, I wasn't neccessarily interested in the
harmonic
> distance, per se, but the 'tunable' distance in terms of primes.

OK. But I agree with Gene that there's no reason to privelege primes
here. If you think about 'tunable' distance in terms of any
intervals, not just primes, I think you'll end up with something more
like Tenney or van Prooijen distnace. The Partch Tonality Diamond,
for example, consists of all ratios within a van Prooijen distance of
log(11) from the tonic 1/1, and tunability was certainly a big issue
for Partch, who tuned his Chromelodeon by ear and his other
instruments to that.

>What you say
> about Tenney makes sense, too, though. When I get a chance, I will
rank the
> commas by the various means you suggested....did you see the
posting I did
> earlier today, where I ranked them by a simple measure of the
reference ratio
> of the given limit mius the comma, divided by the taxicab distance.
Indeed,
> 64/63 proves to be the most efficient way to get a decently small
comma.

I find these kinds of investigations interesting (as Gene mentioned,
a whole lot of work along these lines has taken place on the tuning-
math list) but I think it would be a good idea to re-direct them to
the tuning-math list, since long lists of numbers tend to result in
mass unsubscriptions on this list :) I'd love to engage with you more
deeply on these matters over there.

> So you see, it has nothing to do with consonance or dissonance
>whatsoever,

I think 'tunability' has an intimate relationship with consonance, so
I'm not sure I can offer you unqualified agreement on this.

But sorry for being so critical of your remarks. As usual, my zeal
oversteps the boundaries of civil communication. I apologize.

Best,
Paul

On Friday 22 July 2005 4:15 pm, wallyesterpaulrus wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
>
> wrote:
> > On Friday 22 July 2005 1:15 pm, Gene Ward Smith wrote:
> > > This can make for a difference of a factor of two: 50/49 is
>
> obtained
>
> > > as two 10/7 intervals, a 7-limit consonance, with a Hahn distance
>
> of
>
> > > 2. Your measure gives a 2 for the power of 5, and another 2 for
>
> the
>
> > > power of 7, totaling 4. Your path would replace 10/7 10/7 with
>
> 5/4 8/7
>
> > > 5/4 8/7.
> >
> > Tuning a 10/7 by ear is too hard.
>
> You can use 7/5 instead and it amounts to the same thing as far as
> what Gene was saying.
>
> >That's why the taxicab approach works
> > better.
>
> This argument clearly can't be right if you're happy tuning 7/5 by
> ear (I am). Meanwhile, Gene and I have both given you an example
> where what you're calling 'taxicab' is clearly deficient. Aaron,
> yours is not the only taxicab approach, and Hahn distance is a type
> of taxicab measure too (though not on a rectangular lattice). We've
> discussed different lattice geometries and different metrics
> extensively on this list and then the tuning-math list (where some of
> your posts on this clearly belong). I hope you'll have the patience
> to listen to the other ideas and arguments here, and also to read
> the 'Middle Path' paper that I snail-mailed you.

I have read your paper.

-A.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> > Aaron, if you're interested in some *non-arbitrary* boundaries for
> > small rational intervals, then you should read this:
> >
> > /tuning/topicId_56202.html#56261
>
> I think these boundries are clearly arbitrary. This reminds me of the
> discussion of "moats" for temperaments, and whether a non-arbitary
> boundry can be drawn there. The only non-arbitary boundry I can see is
> that in any prime limit, there is always a smallest superparticular
> comma. However, these rapidly become very small for increasing prime
> limits.

From a purely mathematical point of view, yes there is some
arbitrariness. But this is only in the choice of the schismina/schisma
and schisma/kleisma boundaries. However, in a broader sense they are
not arbitrary either, due to the need to be consistent with current
and historical usage of the terms schisma and kleisma (as given in
Scala's intnam.par).

Here's how all the other boundaries can be derived purely mathematically:

Find all the 3-prime-limit rational intervals no greater than an
apotome ([-11 7> ~113.685 c) and ordering them by the absolute value
of their 3-exponent (or their 2-exponent, it makes no difference).

You can do this easily by trying all 3-exponents up to say 53 (or 200
if you want to see the somewhat arbitrary schisma/kleisma boundary)
and finding the 2-exponent for each that makes the resulting ratio as
close to 1/1 as possible,
i.e. 2_exponent = -Round(3_exponent * ln(3)/ln(2))
Then eliminate any whose absolute value is greater than an apotome.

Also eliminate any whose 2-exponent and 3-exponent are both divisible
by 2. This is because the boundaries are at the irrational square
roots of ratios. We don't want any of the boundaries to be rational or
there will be a comma sitting on the boundary.

Now take the square roots of these ratios and convert them to cents,
or equivalently convert them to cents and divide by 2 (and take the
absolute values). Also calculate the apotome-complement of each of
these cents values. We want the boundaries to be symmetrical about the
half-apotome so there is a simple correspondence between comma
categories for alternate spellings of a ratio.

You should then have a table that looks like this. (Hit the Reply
button to see it correctly formatted if viewing on the web).

apotome-complement
2-exp 3-exp cents cents
------------------------------------------
8 -5 45.112 68.573 used
-11 7 56.843 56.843 used
-19 12 11.730 101.955 used
27 -17 33.382 80.303 used
46 -29 21.652 92.033
-57 36 35.190 78.495
65 -41 9.922 103.763
73 -46 55.035 58.650
-84 53 1.808 111.877 used for historical reasons
.
.
.
317 -200 4.500 109.185 4.5 only used for historical reasons

Then you just work your way down the list until you've got enough
boundaries (enough categories). For most practical purposes this
happens after only the first four lines, and I don't think it is any
accident that boundaries and the categories so produced are quite
consistent with historical and current usage.

Then we've added the mathematically-more-arbitrary boundaries near
1.808, 4.500 and 111.877 cents to deal with the historical categories
of schisma, kleisma and apotome, and made the boundaries of the
apotome category symmetrical about the actual 3-limit apotome.

-- Dave Keenan

Gene and Paul,

All of your theoretical considerations appear to be just that: theoretical.

I've implemented your suggestions and sorted the lists of the output, using:

a) Hahn distance
b) Tenney distance
c) log Tenney distance

Each time, something on the list was unsatisfactory in terms of: how easy is
this comma tuned?

That is to say, what I'm talking about is far removed from the pure paper
world of '10/7' is as easy to tune as '5/3'.

Let me clearly rephrase my problem, so that you don't go resuggesting that my
problem has anything whatsoever to do with harmony on the lattice: what is
the maximully efficient comma, i.e. given the tradeoff between small size and
number of steps.

I already think that any formula that would, for instance, give the septimal
comma after 50/49 on the list is flawed: given the ease of tuning 2 fifths,
and then a harmonic 7th, and how relatively small this comma is, I think it
ought top 50/49 on any list, since two 10/7's are more difficult (at least
they would take me a significantly greater time to tune) than two 5ths and a
single 7th, and 50/49 is not as small as 64/63.

I have yet to find a technique that you suggested that jives with how easily
these comma are tuned: save the simple square lattice taxicab distance.

d) square lattice taxicab distance

....remains the most accurate

Why this works you have every theoretical reason to dislike, however, your
metrics fail when considering 'ease'.

If you disagree, I challenge you to come up with a 7-limit comma smaller than
the septimal comma in three or less steps in *any* metric of your choosing.
When you fail, you will see why it should be the top of the list. And you
will see why your metrics fail when they *don't* put it there. And if you
succeed, well, I will gracefully stand corrected.

Or, perhaps you can illustrate a formula that would take into account the
difference in tunability between two 10/7's and two 3/2's plus a 7/4...

-Aaron.

On Friday 22 July 2005 4:43 pm, wallyesterpaulrus wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
>
> wrote:
> > Paul,
> >
> > As I mentioned to Gene, I wasn't neccessarily interested in the
>
> harmonic
>
> > distance, per se, but the 'tunable' distance in terms of primes.
>
> OK. But I agree with Gene that there's no reason to privelege primes
> here. If you think about 'tunable' distance in terms of any
> intervals, not just primes, I think you'll end up with something more
> like Tenney or van Prooijen distnace. The Partch Tonality Diamond,
> for example, consists of all ratios within a van Prooijen distance of
> log(11) from the tonic 1/1, and tunability was certainly a big issue
> for Partch, who tuned his Chromelodeon by ear and his other
> instruments to that.
>
> >What you say
> > about Tenney makes sense, too, though. When I get a chance, I will
>
> rank the
>
> > commas by the various means you suggested....did you see the
>
> posting I did
>
> > earlier today, where I ranked them by a simple measure of the
>
> reference ratio
>
> > of the given limit mius the comma, divided by the taxicab distance.
>
> Indeed,
>
> > 64/63 proves to be the most efficient way to get a decently small
>
> comma.
>
> I find these kinds of investigations interesting (as Gene mentioned,
> a whole lot of work along these lines has taken place on the tuning-
> math list) but I think it would be a good idea to re-direct them to
> the tuning-math list, since long lists of numbers tend to result in
> mass unsubscriptions on this list :) I'd love to engage with you more
> deeply on these matters over there.
>
> > So you see, it has nothing to do with consonance or dissonance
> >whatsoever,
>
> I think 'tunability' has an intimate relationship with consonance, so
> I'm not sure I can offer you unqualified agreement on this.
>
> But sorry for being so critical of your remarks. As usual, my zeal
> oversteps the boundaries of civil communication. I apologize.
>
> Best,
> Paul
>
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
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>
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>
>
>

le 22/07/05 19:53, Gene Ward Smith à gwsmith@svpal.org a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:

>> The only second example I found of a single, non simplifiable comma
>> hiding such smaller skisma is
>> dudon(351/350) that has divisors 1 3 5 7 9 13 25 27 35 39 117 175-351
>> (12 steps if 351/350 considered as a double one)
>> where 351/350 is not the smaller comma, but 4096/4095
>
> I'm not checking you here. I get the folowing 13-note scale:
>
> 1 35/32 9/8 39/32 5/4 175/128 351/256 3/2 25/16 13/8 27/16 7/4 117/64
>
> This has a step of 351/350 between 175/128 and 351/350, but no smaller
> one that I can find.

Certainly, with my excuses,
I thought you were just COMPARING different intervals close in size
in the 16875/16807 scale of divisors, because I thought it was impossible,
this is why I was asking you - but I was wrong -. Yes, you do have indeed a
2401/2400 at octaves between the two divisors 75 (2400) and 2401.

Then this is even more rare than I thought !

Of course, the scale of 351/350 cannot contain 4096/4095, just as
the scale of ANY comma cannot contain a smaller comma that would be
expressed with a ratio of higher numbers than himself.

Are they MANY examples like 16875/16807 and 2401/2400, I wonder.
Most famous skismas are not easily found that way in the scales of larger
intervals when combining the same factors.

> To get 128/117, you can add the inverse intervals
> to the scale, producing an inversely symmetrical scale. That certainly
> makes sense, but now we have a different setup--one more likely to
> produce smaller intervals than the defining comma. The resulting 25
> note scale in this instance has two intervals of 4096/4095, and two of
> 351/350, with the remaining intervals much larger. Hence it would
> certainly make sense to "temper out" 4096/4095 and 351/350, for
> instance by using 130 equal. It is possible to temper out only these
> two commas, producing what is called a planar temperament, since the
> octave equivalent note classes lie in a plane, like 5-limit just
> intonation.
>
>> I remind seeing this oriental beauty between 128/117 and 35/32
>> I used it in a disk I called "Yantra"
>
> Have you been reading Ernest McClain?
No, why ?

> Are any samples of your music available on-line ?
Not much, only 3 very short samples of my CD "Lumières audibles" (Sounding
lights at http://aeh.free.fr (page .mp3) - I am on the way to several new
albums, but slowly.
The reason I called 4096/4095 Yantra is related with the famous indian
mandala called "Sri Yantra" - when you place the divisors of 4095 on a
logarithmic representation of the octave, with 4095 on the summit,
you have a perfect symmetry between the divisors and many horizontal lines
between them (such as 105-395 or 13-315) out of which you may find some kind
of possible construction of the "Sri Yantra" triangles. I did this many,
many years ago, and I can't swear it works perfectly - but then certainly
one (n.m.o.p.q.. / power of 2) or power of 2 / n.m.o.p.q..) skisma should
do it then !
Less anecdotic, out of 4095 (and related) I was able to make one interesting
"Yantra-Rast" :
13-105 117 1 35 39-315 175-351-11 3
Where the tonic is strangely ambiguous between 13, 105 and 35...
It is not an example of an arab model, just a photosonic experience
testing its special subharmonic possibilities appearing with bass tones
11, 12, 13, 16, 35, 39 (defective scale)

>> Do you have a division of the octave for 351/350 ?
>
> 130 works for both 351/350 and 4096/4095. Other nice 351/350 divisions
> are 58, 72, 111, 183 and 241.
>
>> Or 4096/4095 ?
>
> 1506 is very nice for extreme accuracy, but 270, 311 or 494 are likely
> to prove sufficient.
>
>> It has 23 steps with 4096/4095 considered as double
>> and a few commas such as 64/63, 65/64, 91/45 and 105/104
>
> Indeed. In 1506-et, 4096/4095 is 0 steps, 351/350 is 6 steps, 512/507
> 21 steps, 64/63 and 65/64 both 34 steps (the difference being
> 4096/4095.) I don't see 105/104 in there, which however would also be
> 21 steps.
How come ? both 105 and 13 are divisors of 4095,
and 104 is the 3rd octave of 13.

> The largest intervals are 35/32 and 128/117, 195 steps,
> which is much larger than 6 steps, so this scale is still quite
> irregular even with 4096/4095 removed.

Amazing, I don't know your method, but it's effective.

...

Now I think I found another comma whose scale contains a smaller one
(to see if I understood well) :
dudon(8192/8085) - or 13th power of 2 / 11.5.7.7.3 (22,76 cents)
It has 24 divisors : 1 3 5 7 11 15 21 33 35 49 55 77 105 147 165 231 245 385
539 735 1155 1617 2695 8085
now by ascendent order in the octave :
1 33 539 35 1155 147 77 5 165 21-2695 11 735 3-385 49 1617 105 55 7 231 15
245 8085
where 385/384 (4,5 cents) occurs 2 times (2695/2688, 385/384)

then I guess you may be able to find a temperament where 8192/8085 is five
times bigger than 385/384 ??
or, ignore 385/384, in a 53 edo may be ? (reminds me something that
contained also 22 something else)

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> Tuning a 10/7 by ear is too hard. That's why the taxicab approach works
> better.

But you are saying 10/7 is too hard, but 13/8 is tunable. What's the
rationale for that? I think it would make more sense to come up with,
and justify, a list of tunable consonances, and then see what the
implications are. We've been using n-odd-limit for standard consonance
sets, but clearly one doesn't have to.

> Granted, 5/3 is tunable easily in one step, but as the prime limit
goes up, so
> does the nightmare of trying to tune various dyads alone in one
step. Try to
> tune 14/11 by ear in one step, and you'll see what I mean.
>
> Hence the taxicab distance as making more sense from a practical
consistency
> point-of-view.

If the proof is in the pudding, I'm not finding the pudding very
palatable, by which I mean the ordering below does not strike me as
intuitively plausible. We've got dubious commas like 256/245 and
525/512 appearing high on the list; in fact both come out better than
225/224. That's pretty hard to justify, and it isn't what you'd get
from, for example, Hahn distance to the fourth power times comma size.
There 256/245 and 525/512 appear well down on the list, where I think
they clearly belong, and 225/224 higher. We get something more like
2401/2400, 50/49, 49/48, 36/35, 4375/4374, 126/125, 225/224,
250047/250000, 1029/1024, 64/63, 128/125, 1728/1715, 6144/6125, ...
This ranking makes far more sense to me, and is clearly closer to what
you'd get by using a goodness ranking on the resulting rank 3
temperaments, which would be another way to sort commas out.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I'd go further and say that, since 7-prime-limit sometimes can mean 9-
> odd-limit in practice, a weighting such as Tenney's, which gives
> shorter distance to the lower primes, would be even more appropriate
> in general.

It might indeed, but Aaron was assuming symmetry between 3,5, and 7.
Using Kees height as a measure means 4375/4374 and 225/224, as 7-limit
bridge commas, do better, which one might favor. 5-limit commas such
as 81/80 also do better. All of this arguably makes good sense.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> apotome-complement
> 2-exp 3-exp cents cents
> ------------------------------------------
> 8 -5 45.112 68.573 used
> -11 7 56.843 56.843 used
> -19 12 11.730 101.955 used
> 27 -17 33.382 80.303 used
> 46 -29 21.652 92.033
> -57 36 35.190 78.495
> 65 -41 9.922 103.763
> 73 -46 55.035 58.650
> -84 53 1.808 111.877 used for historical reasons

If you look at the ones you use here, you'll find that except for
|27 -17> they are derived from the convergents of log2(3). These are
just then sucessively smaller 3-limit intervals, involving the powers
of 3 of 2, 5, 7, 12, 41, 53, 306, 665, ... . The problem with this is
that the boundries are very irregular, since log2(3) behaves like a
typical transcendental number. Aside from that, you toss in one
semiconvergent , 17, but not the rest of them. Using all of the
semiconvergents would certainly lead to more regularity in the end
result.

If we look at cents values for the square roots of semiconvergents, we
get this:

|3 -2> 101.955
|-11 7> 56.843
|8 -5> 45.112
|27 -17> 33.382
|46 -29> 21.652
|-19 12> 11.730
|65 -41> 9.922
|149 -94> 8.115
|233 -147> 6.370
|401 -253> 2.692
|-84 53> 1.808
|-569 359> 0.9227
|485 -386> 0.8849

It's still somewhat irregular, but less so.

Anyway, the method you describe strikes me as ad hoc, and I don't see
how you can support a claim it isn't pretty arbitrary, even if you
assume to start out with that you should base your boundries on the
3-limit, and should take square roots when you do. If only there was a
consistent way to refine an equal division to a higher one, it would
be grand, but even lacking that it seems to me that categorizing
commas with respect to selected equal divisions might make more sense
than this.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> I already think that any formula that would, for instance, give the
septimal
> comma after 50/49 on the list is flawed: given the ease of tuning 2
fifths,
> and then a harmonic 7th, and how relatively small this comma is, I
think it
> ought top 50/49 on any list, since two 10/7's are more difficult (at
least
> they would take me a significantly greater time to tune) than two
5ths and a
> single 7th, and 50/49 is not as small as 64/63.

Have you tried tuning two 7/5s up to 49/25 instead?

> I have yet to find a technique that you suggested that jives with
how easily
> these comma are tuned: save the simple square lattice taxicab distance.

Which assumes a 7/4 is just as easy to tune as a 3/2, and twice as
easy to tune as a 5/3. Is it?

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> >> I remind seeing this oriental beauty between 128/117 and 35/32
> >> I used it in a disk I called "Yantra"
> >
> > Have you been reading Ernest McClain?
> No, why ?

He defines a musical "yantra" in his book "Myth of Invariance". The
book itself doesn't make much sense, but his yantras--scales derived
from taking all p-limit intervals below some bound--are interesting.

> > Are any samples of your music available on-line ?
> Not much, only 3 very short samples of my CD "Lumières audibles"
(Sounding
> lights at http://aeh.free.fr (page .mp3) - I am on the way to
several new
> albums, but slowly.

Great! I'll check it out.

> The reason I called 4096/4095 Yantra is related with the famous indian
> mandala called "Sri Yantra" - when you place the divisors of 4095 on a
> logarithmic representation of the octave, with 4095 on the summit,
> you have a perfect symmetry between the divisors and many horizontal
lines
> between them (such as 105-395 or 13-315) out of which you may find
some kind
> of possible construction of the "Sri Yantra" triangles.

This is very much in the spirit of Mcclain. You might find his book
interesting.

> How come ? both 105 and 13 are divisors of 4095,

It's in dudon(4096/4095).

> Now I think I found another comma whose scale contains a smaller one
> (to see if I understood well) :
> dudon(8192/8085) - or 13th power of 2 / 11.5.7.7.3 (22,76 cents)
> It has 24 divisors : 1 3 5 7 11 15 21 33 35 49 55 77 105 147 165 231
245 385
> 539 735 1155 1617 2695 8085
> now by ascendent order in the octave :
> 1 33 539 35 1155 147 77 5 165 21-2695 11 735 3-385 49 1617 105 55 7
231 15
> 245 8085
> where 385/384 (4,5 cents) occurs 2 times (2695/2688, 385/384)
>
> then I guess you may be able to find a temperament where 8192/8085
is five
> times bigger than 385/384 ??

270. However tuning the above scale in 72 or 118 might be more
interesting, because then the 385/384s vanish.

le 23/07/05 23:27, Gene Ward Smith à gwsmith@svpal.org a écrit :

> --- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
> wrote:
>
>>>> I remind seeing this oriental beauty between 128/117 and 35/32
>>>> I used it in a disk I called "Yantra"
>>>
>>> Have you been reading Ernest McClain?
>> No, why ?
>
> He defines a musical "yantra" in his book "Myth of Invariance". The
> book itself doesn't make much sense, but his yantras--scales derived
> from taking all p-limit intervals below some bound--are interesting.
>
>> The reason I called 4096/4095 Yantra is related with the famous indian
>> mandala called "Sri Yantra" - when you place the divisors of 4095 on a
>> logarithmic representation of the octave, with 4095 on the summit,
>> you have a perfect symmetry between the divisors and many horizontal
>> lines
>> between them (such as 105-395 or 13-315) out of which you may find
>> some kind
>> of possible construction of the "Sri Yantra" triangles.
>
> This is very much in the spirit of McClain. You might find his book
> interesting.

Mmmmm...? because it doesn't make much sense ? I don't know then ! ;)

le 24/07/05 23:55, Jacques Dudon (AEH) à fotosonix@wanadoo.fr a écrit :

Knowing my interest for harmonics, somebody told me recently of a book
called "The Harmonic Experience", by W. A. Mathieu -

The editor (Inner Traditions) writes on internet :

"His theory of music reconciles the ancient harmonic system of just
intonation with the modern system of twelve-tone temperament"...

... which leaves me highly dubious, but does anyone knows this book and is
it worthy of the 50$, or should be avoided ?

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:

> ... which leaves me highly dubious, but does anyone knows this book
and is
> it worthy of the 50$, or should be avoided ?

It's definately worth reading. I've corresponded with the author in an
attempt to expand his comma horizons, but it's got some basic,
important material.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> If we look at cents values for the square roots of semiconvergents, we
> get this:
>
> |3 -2> 101.955
> |-11 7> 56.843
> |8 -5> 45.112
> |27 -17> 33.382
> |46 -29> 21.652
> |-19 12> 11.730
> |65 -41> 9.922
> |149 -94> 8.115
> |233 -147> 6.370
> |401 -253> 2.692
> |-84 53> 1.808
> |-569 359> 0.9227
> |485 -386> 0.8849
>
> It's still somewhat irregular, but less so.
>
> Anyway, the method you describe strikes me as ad hoc,

Your method above, is indeed far more elegant. So if we take the first
four semi-convergents of log2(3) and their apotome complements we have
the same result (excluding the "historicals").

I note that although a boundary at the 5th semiconvergent (21.652 c)
would be logical, there is little practical use and no historical
precedent for a boundary that would put the Pythagorean and Didymus
commas in different categories.

> and I don't see
> how you can support a claim it isn't pretty arbitrary,

Well at least we agree that there are degrees of arbitraryness and it
is a long way from being _completely_ arbitrary.

> even if you
> assume to start out with that you should base your boundries on the
> 3-limit, and should take square roots when you do. If only there was a
> consistent way to refine an equal division to a higher one, it would
> be grand, but even lacking that it seems to me that categorizing
> commas with respect to selected equal divisions might make more sense
> than this.

It all depends on your purpose, or the use you have for these size
categories. It turns out that these
square-roots-of-semiconvergents-of-log2(3) boundaries satisfy two
purposes simultaneously.

1. To give precise boundaries to existing historical categories.

2. For most ratios used in tuning, to put each of the commas used to
notate a given ratio, into a different size category.

This lets us unambiguously refer to the septimal comma versus the
septimal diesis or the septimal large diesis or the septimal kleisma
etc. Although, as I mentioned in a previous post, the distinction
between schisma and kleisma is not necessary for this purpose.

It seems fine to have categories approximately 12 cents wide until we
get to the smallest such category, after which we seem to want
categories where each is a third to a half the size of the previous,
hence the historically motivated, schisma/kleisma boundary near 4.5
cents and the schismina/schisma boundary near 1.8 cents.

These boundaries were originally found empirically, for purpose 2
above, by taking all the ratios that occurred in the Scala archive,
sorted according to commonness of ocurrence, and generating all the
commas that could be used to notate them relative to a chain of +-12
fifths.

Initial guesses for boundaries were made on the basis of common usage
of the terms schisma, kleisma, comma, minor diesis etc. Some pairs of
commas for notating the same ratio (relative to different
nominals-plus-sharps-or-flats), are extremely close together and when
boundaries were nudged over so as to fall between them it was found
that these boundaries served to disambiguate alternate-spelling commas
for many other ratios as well. The boundaries were successively
refined in this manner to make the first ambiguity occur as far down
the popularity list as possible. Only then was it noticed that they
all occurred at the irrational square roots of 3-commas.

Thanks for showing a more elegant derivation of these boundaries.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Well at least we agree that there are degrees of arbitraryness and it
> is a long way from being _completely_ arbitrary.

I was objecting to the claim that it was more or less forced by the
objective nature of the problem; I wasn't trying to claim the
boundries were unreasonable.

> It all depends on your purpose, or the use you have for these size
> categories. It turns out that these
> square-roots-of-semiconvergents-of-log2(3) boundaries satisfy two
> purposes simultaneously.
>
> 1. To give precise boundaries to existing historical categories.
>
> 2. For most ratios used in tuning, to put each of the commas used to
> notate a given ratio, into a different size category.

Why can't you do this by a simple loglog scale? An example would be
1 cent, 2 cents, 4 cents, etc.

> Thanks for showing a more elegant derivation of these boundaries.

You're welcome. Actually I thought I was deriving an alternative
system, in an attempt to show what you gave wasn't forced. I think it
would make more sense to have a systematic proceedure such as using
3-limit semiconvergents, but the problem of the unwanted comma boundry
clearly doesn't help matters.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > Well at least we agree that there are degrees of arbitraryness and it
> > is a long way from being _completely_ arbitrary.
>
> I was objecting to the claim that it was more or less forced by the
> objective nature of the problem;

I don't recall claiming that exactly.

But I claim that to the degree that the problem is objective, the
solution is more or less forced -- the problem of putting
alternate-spelling commas for the most popular ratios (and "most
popular" can be interpreted _very_ broadly) into different categories
that agree closely with historical usage.

>
> > It all depends on your purpose, or the use you have for these size
> > categories. It turns out that these
> > square-roots-of-semiconvergents-of-log2(3) boundaries satisfy two
> > purposes simultaneously.
> >
> > 1. To give precise boundaries to existing historical categories.
> >
> > 2. For most ratios used in tuning, to put each of the commas used to
> > notate a given ratio, into a different size category.
>
> Why can't you do this by a simple loglog scale? An example would be
> 1 cent, 2 cents, 4 cents, etc.

Because you soon come across ratios like 35/11. Determine the possible
commas for notating that, and you should get the idea. They are both
extremely close together. A boundary at half the pythagorean comma
makes one a comma and the other a kleisma. Almost anything else will
give us either two 11:35 kleismas or two 11:35 commas. This would be
inconvenient.

There are other such examples.

-- Dave Keenan

>-- Original Message --
>To: <tuning@yahoogroups.com>
>From: "Jacques Dudon (AEH)" <fotosonix@wanadoo.fr>
>Date: Mon, 25 Jul 2005 00:26:24 +0200
>Subject: [tuning] Are you experienced ?
>Reply-To: tuning@yahoogroups.com
>
>
>le 24/07/05 23:55, Jacques Dudon (AEH) à fotosonix@wanadoo.fr a écrit :
>
>Knowing my interest for harmonics, somebody told me recently of a book
>called "The Harmonic Experience", by W. A. Mathieu -
>
>The editor (Inner Traditions) writes on internet :
>
>"His theory of music reconciles the ancient harmonic system of just
>intonation with the modern system of twelve-tone temperament"...
>
>... which leaves me highly dubious, but does anyone knows this book and
is
>it worthy of the 50$, or should be avoided ?
>

I know the book and thoroughly enjoyed it , both for the information and
inspiration.

mopani

___________________________________________________________

Book yourself something to look forward to in 2005.
Cheap flights - http://www.tiscali.co.uk/travel/flights/
Bargain holidays - http://www.tiscali.co.uk/travel/holidays/

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:
>
> Gene and Paul,
>
> All of your theoretical considerations appear to be just that:
>theoretical.

I believe that's totally unfair and untrue, at least in my case. But
I'm happy to turn this claim around and use it against you. I hope
you'll see this in the spirit of fair play and good humor.

> I've implemented your suggestions and sorted the lists of the
output, using:
>
> a) Hahn distance
> b) Tenney distance
> c) log Tenney distance
>
> Each time, something on the list was unsatisfactory in terms of:
how easy is
> this comma tuned?
>
> That is to say, what I'm talking about is far removed from the pure
paper
> world of '10/7' is as easy to tune as '5/3'.

I never made that claim or anything like it, so I have no idea what
this "pure paper" world you're talking about is.

> Let me clearly rephrase my problem, so that you don't go
resuggesting that my
> problem has anything whatsoever to do with harmony on the lattice:
what is
> the maximully efficient comma, i.e. given the tradeoff between
small size and
> number of steps.
>
> I already think that any formula that would, for instance, give the
septimal
> comma after 50/49 on the list is flawed: given the ease of tuning 2
fifths,
> and then a harmonic 7th, and how relatively small this comma is, I
think it
> ought top 50/49 on any list, since two 10/7's are more difficult
(at least
> they would take me a significantly greater time to tune) than two
5ths and a
> single 7th, and 50/49 is not as small as 64/63.

I think *you're* the one living in a pure paper, theoretical
world :). How do you tune 64/63 using two 5ths and a single 7th?
Maybe you mean two 4ths? Meanwhile, I can tune 50/49 easily using two
7:5s and a 2:1. The 7:5s are harder than 4ths (4:3), but the 2:1 is
much easier than the 7th (7:4), so the two intervals are pretty close
in terms of 'tunability'. I can think of other ways of tuning 50/49
that may appear slightly easier still, for example a 7:5, a 7:2, and
a 1:5.

> I have yet to find a technique that you suggested that jives with
how easily
> these comma are tuned: save the simple square lattice taxicab
distance.
>
> d) square lattice taxicab distance
>
> ....remains the most accurate

It doesn't look like you've even thought about the alternatives, let
alone really evaluated them. Meanwhile, you're simply ignoring the
clear problems with your 'square lattice taxicab distance' (I'd
add "with stripped 2s" or something for a mor accurate description)
that have been brought up in this thread, and many others that
haven't been brought up yet here. I brought up the examples of 11/7
vs. 77/64 or 128/77. You seem to be just ignoring those points.

> Why this works you have every theoretical reason to dislike,
however, your
> metrics fail when considering 'ease'.
> If you disagree, I challenge you to come up with a 7-limit comma
smaller than
> the septimal comma in three or less steps in *any* metric of your
choosing.

Which of the metrics claim there should be one? And how do you
define "steps" in the case of Tenney or van Prooijen distance?

> When you fail, you will see why it should be the top of the list.
And you
> will see why your metrics fail when they *don't* put it there.

They don't fail. Perhaps you expected them to fail, but since you
didn't even try them, you have no basis for making this claim. In
fact, both the Tenney and van Prooijen metrics say that 64:63 is the
smallest comma for its distance. The chart I showed you was saying as
much.

> Or, perhaps you can illustrate a formula that would take into
account the
> difference in tunability between two 10/7's and two 3/2's plus a
7/4...

I'd be happy to, except I don't think these are quite intervals
you're concerned with. Two 10/7s gives a 100/49, while two 3/2s plus
a 7/4 gives a 63/16. But you were talking about tuning commas --
small intervals. Once we're on the same page as to how one would tune
these commas in reality (not just on paper), then I'll be more than
happy to try to discuss this more quantitatively -- preferably on the
tuning-math list.

Best,
Paul

--- In tuning@yahoogroups.com, "Jacques Dudon (AEH)" <fotosonix@w...>
wrote:
> le 24/07/05 23:55, Jacques Dudon (AEH) à fotosonix@w... a écrit :
>
> Knowing my interest for harmonics, somebody told me recently of a book
> called "The Harmonic Experience", by W. A. Mathieu -
>
> The editor (Inner Traditions) writes on internet :
>
> "His theory of music reconciles the ancient harmonic system of just
> intonation with the modern system of twelve-tone temperament"...
>
> ... which leaves me highly dubious, but does anyone knows this book
and is
> it worthy of the 50$, or should be avoided ?

This book is full of gems, and if you spend a lot of time composing in
12-equal at the piano, its examples and insights may well be worth
hundreds or thousands of dollars to you. I've recommended this book to
many people. But apart from various theoretical disagreements I have
with it, I have to say that as far as the stated goal you quote above,
the book falls short. The "reconciliation" given by this book is a
shallow and forced one, largely because it completely glosses over the
crucial "middle path" of meantone tuning that Western music went
through for about three centuries, but also for various other reasons
I've discussed before (for example, his valid 'commas' include only
those which are *both* under 50 cents *and* vanish in 12-equal -- could
the goal of 12-equal be more preconceived?) If you can afford the book,
get it, and then let's discuss where it could be better.

Best,
Paul

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I think *you're* the one living in a pure paper, theoretical
> world :). How do you tune 64/63 using two 5ths and a single 7th?
> Maybe you mean two 4ths? Meanwhile, I can tune 50/49 easily using two
> 7:5s and a 2:1.

My understanding of Aaron's problem is that he is proposing to tune
things up to octave equivalence, and regards octaves as freebies.
Moreover I think tuning 63/64 is supposed to be the same as tuning
64/63; we get the same interval, at any rate.

> I'd be happy to, except I don't think these are quite intervals
> you're concerned with. Two 10/7s gives a 100/49, while two 3/2s plus
> a 7/4 gives a 63/16. But you were talking about tuning commas --
> small intervals. Once we're on the same page as to how one would tune
> these commas in reality (not just on paper), then I'll be more than
> happy to try to discuss this more quantitatively -- preferably on the
> tuning-math list.

It sounds like a plan, but how do you quantify tuning ease?

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I've discussed before (for example, his valid 'commas' include only
> those which are *both* under 50 cents *and* vanish in 12-equal -- could
> the goal of 12-equal be more preconceived?) If you can afford the book,
> get it, and then let's discuss where it could be better.

Even then I think he misses come clearly important septimal commas.

Dave Keenan wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > If we look at cents values for the square roots of semiconvergents, we
> > get this:
> >
> > |3 -2> 101.955
> > |-11 7> 56.843
> > |8 -5> 45.112
> > |27 -17> 33.382
> > |46 -29> 21.652
> > |-19 12> 11.730
> > |65 -41> 9.922
> > |149 -94> 8.115
> > |233 -147> 6.370
> > |401 -253> 2.692
> > |-84 53> 1.808
> > |-569 359> 0.9227
> > |485 -386> 0.8849
> >
> > It's still somewhat irregular, but less so.
> >
> > Anyway, the method you describe strikes me as ad hoc,
>
> Your method above, is indeed far more elegant. So if we take the first
> four semi-convergents of log2(3) and their apotome complements we have
> the same result (excluding the "historicals").

... (Snip!)

> > even if you
> > assume to start out with that you should base your boundries on the
> > 3-limit, and should take square roots when you do. If only there was a
> > consistent way to refine an equal division to a higher one, it would
> > be grand, but even lacking that it seems to me that categorizing
> > commas with respect to selected equal divisions might make more sense
> > than this.
>
> It all depends on your purpose, or the use you have for these size
> categories. It turns out that these
> square-roots-of-semiconvergents-of-log2(3) boundaries satisfy two
> purposes simultaneously.
>
> 1. To give precise boundaries to existing historical categories.
>
> 2. For most ratios used in tuning, to put each of the commas used to
> notate a given ratio, into a different size category.
>
> This lets us unambiguously refer to the septimal comma versus the
> septimal diesis or the septimal large diesis or the septimal kleisma
> etc. Although, as I mentioned in a previous post, the distinction
> between schisma and kleisma is not necessary for this purpose.

Very neat, guys!

This seems to me to be a very useful result;
one worthy of inclusion in any encyclopaedia of
microtonality. Monz?

Regards,
Yahya

--
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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> Very neat, guys!
>
> This seems to me to be a very useful result;
> one worthy of inclusion in any encyclopaedia of
> microtonality. Monz?

It's already in there -- almost.
http://www.tonalsoft.com/enc/c/comma.aspx

Monz,

Please correct the comma/small-diesis and limma/small-semitone
boundaries as requested earlier. Hit the Up Thread button a few times
to see them (near 33 and 80 cents).

-- Dave Keenan

On Monday 25 July 2005 2:04 pm, wallyesterpaulrus wrote:
> --- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
>
> wrote:
> > Gene and Paul,
> >
> > All of your theoretical considerations appear to be just that:
> >theoretical.
>
> I believe that's totally unfair and untrue, at least in my case. But
> I'm happy to turn this claim around and use it against you. I hope
> you'll see this in the spirit of fair play and good humor.
>
> > I've implemented your suggestions and sorted the lists of the
>
> output, using:
> > a) Hahn distance
> > b) Tenney distance
> > c) log Tenney distance
> >
> > Each time, something on the list was unsatisfactory in terms of:
>
> how easy is
>
> > this comma tuned?
> >
> > That is to say, what I'm talking about is far removed from the pure
>
> paper
>
> > world of '10/7' is as easy to tune as '5/3'.
>
> I never made that claim or anything like it, so I have no idea what
> this "pure paper" world you're talking about is.
>
> > Let me clearly rephrase my problem, so that you don't go
>
> resuggesting that my
>
> > problem has anything whatsoever to do with harmony on the lattice:
>
> what is
>
> > the maximully efficient comma, i.e. given the tradeoff between
>
> small size and
>
> > number of steps.
> >
> > I already think that any formula that would, for instance, give the
>
> septimal
>
> > comma after 50/49 on the list is flawed: given the ease of tuning 2
>
> fifths,
>
> > and then a harmonic 7th, and how relatively small this comma is, I
>
> think it
>
> > ought top 50/49 on any list, since two 10/7's are more difficult
>
> (at least
>
> > they would take me a significantly greater time to tune) than two
>
> 5ths and a
>
> > single 7th, and 50/49 is not as small as 64/63.
>
> I think *you're* the one living in a pure paper, theoretical
> world :). How do you tune 64/63 using two 5ths and a single 7th?
> Maybe you mean two 4ths?

Like Gene said elsewhere, because he understands what I'm saying, octaves are
freebies, and inversions can be used for ease.

> Meanwhile, I can tune 50/49 easily using two
> 7:5s and a 2:1. The 7:5s are harder than 4ths (4:3), but the 2:1 is
> much easier than the 7th (7:4), so the two intervals are pretty close
> in terms of 'tunability'. I can think of other ways of tuning 50/49
> that may appear slightly easier still, for example a 7:5, a 7:2, and
> a 1:5.
>
> > I have yet to find a technique that you suggested that jives with
>
> how easily
>
> > these comma are tuned: save the simple square lattice taxicab
>
> distance.
>
> > d) square lattice taxicab distance
> >
> > ....remains the most accurate
>
> It doesn't look like you've even thought about the alternatives, let
> alone really evaluated them. Meanwhile, you're simply ignoring the
> clear problems with your 'square lattice taxicab distance' (I'd
> add "with stripped 2s" or something for a mor accurate description)
> that have been brought up in this thread, and many others that
> haven't been brought up yet here.

You're accusing me of lying now. I wrote the python script and altered the
code per your suggestion. The results were not satisfying to say the least.
I may have made errors.

> I brought up the examples of 11/7
> vs. 77/64 or 128/77. You seem to be just ignoring those points.

These are not commas, so I'm not considering them. Again, I'm not interested
in the harmonic aspect of the lattice, just the comma tuning aspect.

> > Why this works you have every theoretical reason to dislike,
>
> however, your
>
> > metrics fail when considering 'ease'.
> > If you disagree, I challenge you to come up with a 7-limit comma
>
> smaller than
>
> > the septimal comma in three or less steps in *any* metric of your
>
> choosing.
>
> Which of the metrics claim there should be one? And how do you
> define "steps" in the case of Tenney or van Prooijen distance?
>
> > When you fail, you will see why it should be the top of the list.
>
> And you
>
> > will see why your metrics fail when they *don't* put it there.
>
> They don't fail. Perhaps you expected them to fail, but since you
> didn't even try them, you have no basis for making this claim.

Again, you're accusing me of lying. I *did* try them, but maybe not using your
script/formula. Perhaps you can send me your code, and we can compare notes.

> In
> fact, both the Tenney and van Prooijen metrics say that 64:63 is the
> smallest comma for its distance. The chart I showed you was saying as
> much.

Well, the only way this works is distance from 8/7 squared over log Tenney.
But how would you justify that arbitrary measure?

Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Dave Keenan wrote:
> >
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > > even if you assume to start out with that you
> > > should base your boundries on the 3-limit, and
> > > should take square roots when you do. If only there
> > > was a consistent way to refine an equal division
> > > to a higher one, it would be grand, but even lacking
> > > that it seems to me that categorizing commas with
> > > respect to selected equal divisions might make more
> > > sense than this.
> >
> > It all depends on your purpose, or the use you have for
> > these size categories. It turns out that these
> > square-roots-of-semiconvergents-of-log2(3) boundaries
> > satisfy two purposes simultaneously.
> >
> > 1. To give precise boundaries to existing historical
> > categories.
> >
> > 2. For most ratios used in tuning, to put each of the
> > commas used to notate a given ratio, into a different
> > size category.
> >
> > This lets us unambiguously refer to the septimal comma
> > versus the septimal diesis or the septimal large diesis
> > or the septimal kleisma etc. Although, as I mentioned
> > in a previous post, the distinction between schisma and
> > kleisma is not necessary for this purpose.
>
> Very neat, guys!
>
> This seems to me to be a very useful result;
> one worthy of inclusion in any encyclopaedia of
> microtonality. Monz?

In the "comma" page? If not there, then where do
you suggest?

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> > Very neat, guys!
> >
> > This seems to me to be a very useful result;
> > one worthy of inclusion in any encyclopaedia of
> > microtonality. Monz?
>
> It's already in there -- almost.
> http://www.tonalsoft.com/enc/c/comma.aspx
>
> Monz,
>
> Please correct the comma/small-diesis and
> limma/small-semitone boundaries as requested earlier.
> Hit the Up Thread button a few times
> to see them (near 33 and 80 cents).

I missed it, and don't trust this Yahoo interface
any farther than my nearby trashcan ... so please
post the link instead, or just quote the relevant
data.

-monz
http://tonalsoft.com
Tonescape microtonal music software

monz,

You wrote:

> Hi Yahya,
> -- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> > Dave Keenan wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > > > even if you assume to start out with that you
> > > > should base your boundries on the 3-limit, and
> > > > should take square roots when you do. If only there
> > > > was a consistent way to refine an equal division
> > > > to a higher one, it would be grand, but even lacking
> > > > that it seems to me that categorizing commas with
> > > > respect to selected equal divisions might make more
> > > > sense than this.
> > >
> > > It all depends on your purpose, or the use you have for
> > > these size categories. It turns out that these
> > > square-roots-of-semiconvergents-of-log2(3) boundaries
> > > satisfy two purposes simultaneously.
> > >
> > > 1. To give precise boundaries to existing historical
> > > categories.
> > >
> > > 2. For most ratios used in tuning, to put each of the
> > > commas used to notate a given ratio, into a different
> > > size category.
> > >
> > > This lets us unambiguously refer to the septimal comma
> > > versus the septimal diesis or the septimal large diesis
> > > or the septimal kleisma etc. Although, as I mentioned
> > > in a previous post, the distinction between schisma and
> > > kleisma is not necessary for this purpose.
> >
> > Very neat, guys!
> >
> > This seems to me to be a very useful result;
> > one worthy of inclusion in any encyclopaedia of
> > microtonality. Monz?
>
> In the "comma" page? If not there, then where do
> you suggest?
>
> -monz

You best know the organisation of the encyclopaedia,
I think ... Still, seems to me to perhaps warrant a
separate page on 'Interval Size Categories' or, more
generally, this could appear as part of a page on
'Naming Intervals' or 'Interval Names'. Then each
page on specific categories eg 'comma', 'diesis',
'kleisma' etc, could link to it.

Regards,
Yahya

--
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Checked by AVG Anti-Virus.
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on 7/22/05 10:53 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

>> Do you have a division of the octave for 351/350 ?
>
> 130 works for both 351/350 and 4096/4095. Other nice 351/350 divisions
> are 58, 72, 111, 183 and 241.

Gene,

I've met you. You're not enough of an idiot to be such a savant. Did you
really know that off the top of your head, from remembered experience?

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 7/22/05 10:53 AM, Gene Ward Smith <gwsmith@s...> wrote:
>
> >> Do you have a division of the octave for 351/350 ?
> >
> > 130 works for both 351/350 and 4096/4095. Other nice 351/350 divisions
> > are 58, 72, 111, 183 and 241.
>
> Gene,
>
> I've met you. You're not enough of an idiot to be such a savant.
Did you
> really know that off the top of your head, from remembered experience?

No, but it's the sort of thing I can find in a few seconds using my
Maple programs. I'd guess Graham could do something similar with
Python scripts.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...>
wrote:

> > I brought up the examples of 11/7
> > vs. 77/64 or 128/77. You seem to be just ignoring those points.
>
> These are not commas, so I'm not considering them. Again, I'm not
interested
> in the harmonic aspect of the lattice, just the comma tuning aspect.

OK, let's look at some commas then. How about 648:625 vs. 2048:2025.
Which one do you consider more 'tunable', and why?

> Again, you're accusing me of lying.

Ouch -- I'm terribly sorry about that.

>I *did* try them, but maybe not using your
> script/formula. Perhaps you can send me your code, and we can
>compare notes.

Hmm . . . I thought I gave you the formulas both here and in my
paper. I don't have any code to send you because the formulas are
very simple and I can always calculate them on the fly. Send me your
code and I'll have a go at correcting it.

>> > In
>> > fact, both the Tenney and van Prooijen metrics say that 64:63 is
the
>> > smallest comma for its distance. The chart I showed you was
saying as
>> > much.

>> Well, the only way this works is distance from 8/7 squared over
log Tenney. But how would you justify that arbitrary measure?

Huh? I don't get it. What do you mean?? Let me repeat: both the
Tenney and van Prooijen metrics say that 64:63 is the smallest comma
for its distance (or closer). What 8/7? What squared . . . what
arbitrary measure?

Sorry, I'm really trying to understand you, but the train seems to
have temporarily left the tracks . . .

Best,
Paul

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> Like Gene said elsewhere, because he understands what I'm saying,
octaves are
> freebies, and inversions can be used for ease.

Then I only need two steps to tune a 50/49: 7:5, and then another 7:5.

BTW, since you asked about other interval tunable in three steps -- in
three pretty easy steps, I can reach 126/125: 7:5, 3:5, and another
3:5. And 126/125 is a smaller interval than 64/63 ;)