Yahoo Tuning Groups Ultimate Backup tuning Re decimal vs cents

>"Base two logarithms are more of a mathematical fact of life than a system."
Comes to think of it, base 2 logarithms seem to not only lend themselves to
the octave, but all TET-type scales.

The problem with that, as I see it, is that any interval different than the
fixed "semi-tone" step size in any logarithmic system looks "off" mathematically
IE 213 looks more "off key" than 200 no matter how it actually sounds.
Coincidentally that would very likely cause people working with such systems to
try and use either a single fixed interval size (200) or things half way
in-between two interval sizes (250)...because it looks tidier numerically in
that system. I can only imagine that partly explains why when many people
think "micro-tonal" they think 24TET and not, say, 22TET.

Hence (as I see it):
1) Base-2 is biased to octaves (not a huge bias as many scales include
octaves)
2) Base-x is biased to TET-type scales and encourages the scale designer to
not use uneven interval sizes to make the scale look numerically tidy. A fairly
significant bias IMVHO, but still fine for many types of scales/tunings using
tones not-so-far from TET tunings.
3) The cent has the two biases above plus the IMVHO much greater bias of
marking everything relative to compliance with the 12TET semi-tone, a special
case of "base-X" compliance which may well encourages scale creators to make and
use scales (as much like 12TET IE "what they know first") as possible.
4) Decimals have essentially no bias, but not optimization for anything
either. It's like building with custom bricks instead of pre-made/pre-shaped
walls.
5) Fractions have a bias toward ratios that summarize to low-limit ratios,
which may create a bias against use of ratios not summarize-able into
low-numbered fractions. I actually think fractions could be an ideal format as
it can be tricked into being both optimized and non-biased. That is, f we
consider things like tone classes (including non-common-practice ones) and
tempering in the equation. One solution, I figure, may be to have a format which
aims to summarize most fractions in x/11 or less format in something like the
following fashion:

(fraction) 121/80 (class) 3/2
(fraction) 18/11 (class) 18/11
(fraction) 22/15 (class) 3/2
(fraction) 32/25 (class) 9/7

........the tough part becomes figuring out and, in general, agreeing on what
fractions count as boundaries to all potential tonal classes.....

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> The problem with that, as I see it, is that any interval different than the
> fixed "semi-tone" step size in any logarithmic system looks "off" mathematically
> IE 213 looks more "off key" than 200 no matter how it actually sounds.

You seem to think your strange psychological quirks are shared by everyone. Have you ever met anyone else who expressed such a view?

> 4) Decimals have essentially no bias, but not optimization for anything
> either.

Decimals have a clear bias, and it's amazing hear you claim this. Outr hearing is much more nearly logarithmic than it is linear, and hence decimal makes intervals look bigger than they sound as you go higher in pitch, and smaller if you go lower. Decimal says an octave down and a fifth up are the same size of interval, which is clearly false, and the bias gets worse as you push that farther: it says infinitely far down in pitch is the same as an octave up.
I actually think fractions could be an ideal format as
> it can be tricked into being both optimized and non-biased.

This is just silly. Fractions are fine for just intonation but require a whole new area of effort for tempering, and make the final result much harder to read (decimal is worse than cents in this regard also, btw.) If you are going to do this, you should at least be able to claim that you are improving the sound somehow, as Jaques Dudon claims for his recurrence relation versions of temperaments.

That is, f we
> consider things like tone classes (including non-common-practice ones) and
> tempering in the equation. One solution, I figure, may be to have a format which
> aims to summarize most fractions in x/11 or less format in something like the
> following fashion:
>
> (fraction) 121/80 (class) 3/2
> (fraction) 18/11 (class) 18/11
> (fraction) 22/15 (class) 3/2
> (fraction) 32/25 (class) 9/7
>
> ........the tough part becomes figuring out and, in general, agreeing on what
> fractions count as boundaries to all potential tonal classes.....

There IS no one size fits all answer like the above to this sort of thing. If you learn nothing else here, at least learn that. You want to say 22/15 should always be interpreted as a seriously out of tune (flat by 45/44!) fifth, but what if I think it is a minor third above a neutral third instead, for instance?

🔗Michael <djtrancendance@...>

>> The problem with that, as I see it, is that any interval different than the

>> fixed "semi-tone" step size in any logarithmic system looks "off"
>>mathematically
>>
>> IE 213 looks more "off key" than 200 no matter how it actually sounds.
>"You seem to think your strange psychological quirks are shared by everyone.
>Have you ever met anyone else who expressed such a view?"
Seriously, do I have to have some huge survey (perhaps with hundreds of
participants) to express an opinion on this list?

Of course many (and likely most) people find it easier to
analyze/compare/picture smaller numbers/sets/etc. Try to graph y = 200,250,300
in your head. Now try y = 200,214,263. Which one seems more graceful and easy
to understand? As I see it...it's like asking someone to choose between
multiplying 12 * 12 in their head and 12.3 * 11.7. Is a survey really necessary
as proof?

>"Decimals have a clear bias, and it's amazing hear you claim this. Outr hearing
>is much more nearly logarithmic than it is linear, and hence decimal makes
>intervals look bigger than they sound as you go higher in pitch, and smaller if
>you go lower."
Of course, meanwhile, beating rate is based in linear not logarithmic factors
and critical band is on a skewed logarithmic curve, not a straight one (gets
narrower at high frequencies). So logarithmic has some psycho-acoustic analysis
advantages while linear-decimal has others. Not to mention that someone you
easily use a combination of both IE 1.0245^x. Unfortunately though, as I
explained before, that would assume Equal Temperament, and much of the world of
tuning is NOT Equal Temperament.

________________________________
From: genewardsmith <genewardsmith@...>
To: tuning@yahoogroups.com
Sent: Tue, July 6, 2010 6:44:13 PM
Subject: [tuning] Re: Re decimal vs cents

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> The problem with that, as I see it, is that any interval different than the
> fixed "semi-tone" step size in any logarithmic system looks "off"
>mathematically
>
> IE 213 looks more "off key" than 200 no matter how it actually sounds.

You seem to think your strange psychological quirks are shared by everyone. Have
you ever met anyone else who expressed such a view?

> 4) Decimals have essentially no bias, but not optimization for anything
> either.

Decimals have a clear bias, and it's amazing hear you claim this. Outr hearing
is much more nearly logarithmic than it is linear, and hence decimal makes
intervals look bigger than they sound as you go higher in pitch, and smaller if
you go lower. Decimal says an octave down and a fifth up are the same size of
interval, which is clearly false, and the bias gets worse as you push that
farther: it says infinitely far down in pitch is the same as an octave up.
I actually think fractions could be an ideal format as
> it can be tricked into being both optimized and non-biased.

This is just silly. Fractions are fine for just intonation but require a whole
new area of effort for tempering, and make the final result much harder to read
(decimal is worse than cents in this regard also, btw.) If you are going to do
this, you should at least be able to claim that you are improving the sound
somehow, as Jaques Dudon claims for his recurrence relation versions of
temperaments.

That is, f we
> consider things like tone classes (including non-common-practice ones) and
> tempering in the equation. One solution, I figure, may be to have a format
>which
>
> aims to summarize most fractions in x/11 or less format in something like the
> following fashion:
>
> (fraction) 121/80 (class) 3/2
> (fraction) 18/11 (class) 18/11
> (fraction) 22/15 (class) 3/2
> (fraction) 32/25 (class) 9/7
>
> ........the tough part becomes figuring out and, in general, agreeing on what
> fractions count as boundaries to all potential tonal classes.....

There IS no one size fits all answer like the above to this sort of thing. If
you learn nothing else here, at least learn that. You want to say 22/15 should
always be interpreted as a seriously out of tune (flat by 45/44!) fifth, but
what if I think it is a minor third above a neutral third instead, for instance?

🔗Michael <djtrancendance@...>

Sorry that last message auto-sent for some odd reason...

Me>>I actually think fractions could be an ideal format as
>> it can be tricked into being both optimized and non-biased.

Gene>"This is just silly. Fractions are fine for just intonation but require a
whole new area of effort for tempering, and make the final result much harder to
read (decimal is worse than cents in this regard also, btw.)"
How so is it much harder to read?

>"If you are going to do this, you should at least be able to claim that you are
>improving the sound somehow, as Jaques Dudon claims for his recurrence relation
>versions of temperaments."
I figure the system encourages you to improve the sound plenty. For one, you
can easily categorize classes of intervals which don't fit together evenly. IE,
to me it seems clear 8/5 and 16/11, though very near on a cent scale, are two
totally different interval classes with two very different
feelings/aesthetic-values useful in music in very different ways. If you
visualize it in cents or even decimal you may well think "oh, they look about
the same in value and probably sound about the same"...but that seems utterly
incorrect when you actually hear the mood of those dyads. Meanwhile take 22/15
(which looks much more complex due to larger numbers) vs. 3/2. Numerically very
different, but in many ways they have a similar feel...certainly a lot more so
than the rather close (in cents) 16/11 does to 3/2.

Erm...what can I say...fractions have class! :-)

Me>> ........the tough part becomes figuring out and, in general, agreeing on
what

>> fractions count as boundaries to all potential tonal classes.....

>"There IS no one size fits all answer like the above to this sort of thing."
That's why I say "in general". If we could, say, agree on a system favored
by 60% of people over all other alternatives I'd say we have a good system.

>"If you learn nothing else here, at least learn that. You want to say 22/15
>should always be interpreted as a seriously out of tune (flat by 45/44!) fifth,
>but what if I think it is a minor third above a neutral third instead, for
>instance?"

It can be both. What you've just said sounds akin to saying if 3 + 3 = 6
then 6 can't also be the square root of 36. You could even go the level of
saying each "class" has multiple derivations...but the point is to still
say/agree it belongs to a single class so it can be both agreed upon and favored
among a large percent of the musical community.

🔗Tony <leopold_plumtree@...>

🔗Michael <djtrancendance@...>

My bad...meant 22/15 vs. 16/11 AKA 1.466666 vs. 1.454545

________________________________
From: Tony <leopold_plumtree@...>
To: tuning@yahoogroups.com
Sent: Wed, July 7, 2010 12:59:22 AM
Subject: [tuning] Re: Re decimal vs cents

>IE,
> to me it seems clear 8/5 and 16/11, though very near on a cent scale, are two
> totally different interval classes with two very different
> feelings/aesthetic-values useful in music in very different ways.

Even 8:5 and 18:11 differ by 38.9 cents.

🔗Tony <leopold_plumtree@...>

🔗Michael <djtrancendance@...>

Tony>"No one into micro-tonalism is going to disregard 14.4 cents."
In this case, it's true: that 14 cents makes a world of a difference.

But consider 22/15 (1.4666) and 3/2 (1.5). They sound surprisingly alike
considering how far apart they are. Meanwhile 18/11 (1.63636) and 13/8 (1.625)
are very close but sound very different.

Thus I'm fairly convince size between intervals, even if given
logarithmically, is NOT a magic bullet so far as analyzing how alike two ratios
are. There can be many interval "classes" in a very large frequency space and
few within a large space (between 22/15 and 50/33, I believe, is such a space).

The problem I find with logarithmic systems rating tones by distance apart is
it seems to encourage people who use the system to rate tonal like-ness by
distance. My point is it's not distance that's important so far as mood, it's
tonal classes. And one extremely easy to handle/memorize way of denoting tonal
classes is in nearest fairly-simple fractions. It's a lot easier to memorize
50/33 than 719.181 (cents) and even easier to treat 50/33 as "a
brighter-sounding version in the 3/2 class" and memorize 3/2. Or even memorize
22/15 (1.4666) as "a darker version in the 3/2 class".

🔗Tony <leopold_plumtree@...>

> But consider 22/15 (1.4666) and 3/2 (1.5). They sound surprisingly alike
> considering how far apart they are. Meanwhile 18/11 (1.63636) and 13/8 (1.625)
> are very close but sound very different.
>
> Thus I'm fairly convince size between intervals, even if given
> logarithmically, is NOT a magic bullet so far as analyzing how alike two ratios
> are.
> The problem I find with logarithmic systems rating tones by distance apart is
> it seems to encourage people who use the system to rate tonal like-ness by
> distance.

To a point (often a few cents), tonal likeness sometimes does relate to distance, making logarithmic units useful for tempering and whatnot. Beyond that, I can't see how such units encourage anyone to go one way or another in characterizing and using intervals. You just as easily say people would be tempted to round integer ratios to ones of similar value with simpler numbers. In either case, I don't see it as a major issue. Both logarithmic and linear comparisons can be instructive.

>It's a lot easier to memorize
> 50/33 than 719.181 (cents)

That's pretty questionable. I get 719.354 cents for 50:33, BTW.

🔗Michael <djtrancendance@...>

>Me> The problem I find with logarithmic systems rating tones by distance
>apart is
>
>> it seems to encourage people who use the system to rate tonal like-ness by
>> distance.

Tony>"To a point (often a few cents), tonal likeness sometimes does relate to
distance"

True, but only in the extreme case of only a few cents does that appear to
hold "across the board". Often the difference you deal with are more like 10-20
cents where how many cents difference there are between two "similar" dyads can
change the mood dramatically.
Even something like 12 cents can make a huge difference in feel for certain
dyads IE (again) the 1.636363 DYAD vs. the 1.625 one yet sound like just a minor
de-tuning for others (IE 50/33 vs. 16/11).

>"Beyond that, I can't see how such units encourage anyone to go one way or
>another in characterizing and using intervals."
I figure it's easier to define memorize-able tone classes in fractional form
(11/9 being a lot easier to memorize than "about 347 cents". And when you can't
remember something, it makes sense to remember it as "between two points you can
memorize". And how do you usually guess how much it will sound like one of
those two points vs. the other? Distance...

>"You just as easily say people would be tempted to round integer ratios to ones
>of similar value with simpler numbers."
True...neither system is perfect.
But cents seem to imply a straight line rising scale that invites you to
compare things in "distance". Unless you sit down and calculate them, it's hard
to know 50/33 is larger than 3/2, for example, while the fact say 104 cents is
greater than 90 cents is blatantly obvious. And if that user "knows" what 110
cents and 100 cents sound like, that person is likely to just guess "well it
sounds a bit more like 100 cents because it's a bit closer". And, back to my
point, that kind of guess quite often is simply not true and gives a false sense
of security about what something "should sound like".

>"In either case, I don't see it as a major issue. Both logarithmic and linear
>comparisons can be instructive."
True, but I'm not talking about logarithmic vs. linear here. I'm talking
about fractions lending themselves more to thinking about tunings and scales as
tonal classes than logarithmic systems do.

-------------------------------------------

>>It's a lot easier to memorize
>> 50/33 than 719.181 (cents)
>That's pretty questionable. I get 719.354 cents for 50:33, BTW.
That was just a quick estimate and it doesn't change to point: I'm not trying
to prove how accurately I can convert from fractions to cents. If you still
want to be anal about it however...your more accurate answer 719.354 is no
easier to memorize than 719.181...and both are much trickier to memorize than
50/33.

Re decimal vs cents

🔗John H. Chalmers <JHCHALMERS@...>

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

🔗Michael <djtrancendance@...>

🔗Tony <leopold_plumtree@...>

🔗Tony <leopold_plumtree@...>

🔗Michael <djtrancendance@...>

🔗Tony <leopold_plumtree@...>

🔗Michael <djtrancendance@...>

🔗Tony <leopold_plumtree@...>

🔗Michael <djtrancendance@...>