Commas

To Mike Battaglia,

De:
"Mike Battaglia" <battaglia01@...>

Para:
"Mario Pizarro" <piagui@...>

Fecha:
27 Jun 2008, 10:28:56 PM

Asunto:
Fwd: [tuning] Distinguishable

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

< Your entire basis for calling it perfect is that the wave is slightly< closer to periodic. The wave of C major is NOT fully periodic, as it< would have to be in rational multiple relationships to be periodic.<< The wave of C Major is perfectly periodic. I sent you its response and<< measured the periodicity and stability of all triads during thirty minutes. When for the first time I looked at the Piagui graph responses, its aesthetic and apparent periodic display induced me to call them "perfect triads"; it would have been better to call them "almost perfect triads". In the other hand, the considerable effort I did along eight years to define the three variants of Piagui scales whose complicate deduction required peculiar and unusual mathematical methods deserve me the right of make a mistake.< The C and the G together will be periodic. The C and the E together< will NOT be periodic, as the two frequencies are in IRRATIONAL NUMBER< relationships to one another. The C and the G are in a rational 3/2< relationship to one another, meaning that for every 3 cycles of the G there will< have been 2 cycles of the C. This property does not apply to the C and< the E, the waves of which will NEVER sync up enough to be actually< periodic. If you have measured the C major triad's< waveform and determined it to be periodic, then you have made an error< somewhere, because it is PHYSICALLY IMPOSSIBLE.The musical instruments are mainly in hands of musicians andengineering. You Know that the exact values of some numeric relations in music are not possible to put to work. That is why you are considering non conventional fractions to satisfy your object. I understand and agree with you on the definitions given in thetwo previous paragraphs. At the moment and since the schisma (32805/32768) = 1,00112915039,,,is the smallest consonance that can be detected by the ear of man, I am just considering for tone E the frequency 1,26125 =(1009 / 800) rather than 1,26134482288. The deviation produced by 1,26125 with respectto the original frequency of E is (1,26125 / 1,26134482288) = 0,99992482398. An error of (1-- 0,99992482398) = 0,000075176 that is absolutely lower than the schisma´s deviation (1,00112915039,,, - 1) = 0,001129,,,, > The wave might be close enough to periodic to be indistinguishable,> but that major third will certainly be a far cry from 5/4.You are quite acquainted that the equal tempered major third is almost equal to (63/50) = 1,26 At one millimeter far from Piaguimajor third.The author W. T. Bartholomew in "Acoustics of Music" wrote:"Many of these scales persist in various parts of the world today. It is only in "Western music", where a harmonic development evolved, that the diatonic scale as we know it became predominant."> < My biggest problem with the scale is that the fifths that are better> < are only better by 2 cents, which is indistinguishable. The fifths> < that are worse, though, are flatter by 6 cents, which is very> < distinguishable.<> Until you give the total amount of indistinguishable and <> distinguishable cents in both scales Piagui and equal <> tempered, your incomplete report has no sense. It is time <> to know the truth in both sides. I know that truth, however <> prefer that another person that is not involved in the subject do it.< That is a lot of math. If you would like to provide us with a chart of< the errors from each interval, the RMS tuning error and TOP error and< all of that, as well as a corresponding chart from 12-tet, then I'd be< happy to compare and take a look and help you interpret what the< charts mean.I will see what I can do to make the needed approach at my way. It would be a limited discussion and explanation and hope that despite its incomplete final data it could give clear results. Would you explain me what do you mean by "the errors from each interval, the RMS tuning error and TOP error"?< I will say that you cannot claim that the flat perfect fifth, which is< 4 cents flat from 12-equal, will be indistinguishable, but that the< major third, which is 2 cents sharp from 12-equal, will be< distinguishable.As you have probably noted, I lived too far from cents. Last week I started to learn a bit about cents. At the moment my background is the acknowledge that each equal temperedsemitone is divided in one hundred cents, that is all, so this evening I will use my brain to understand the above paragraph.I will be better next month. In many occasions I read:,,,etc with ratio 9 : 10, or 182 cents.. ¿ WHAT ON THE EARTH...? I know that the 9 : 10 relation works in the Just Intonation scale of Aristoxenus but for a man whose background on cents is only the 1200 divisions of any octave ¿what is the connection between the ratio (9 : 10 ) and the number of cents of it?< It's a lose-lose: whenever the fifth is better than 12-equal, the< major third will be slightly worse, and whenever the major third is< better than 12-equal, the perfect fifth will significantly worse.I need a doctor.¿Did you ever added the deviation cents of all the fifths, fourths and thirds of the equal tempered scale?< The equal tempered scale has an approximation of 3/2 that is roughly 2< cents sharp, an approximation of 4/3 that is roughly 2 cents flat, an< approximation of the 5/4 major third that is roughly 14 cents sharp,< and an approximation of the 6/5 minor third that is roughly 16 cents< flat. If you treat the equal tempered minor third as an approximation< to 19/16, then it is only 2.5 cents sharp.This paragraph contains basic data for the projected chart of errors Piagui/equal tempered.>> <That C-E is not periodic by any measure of the scale - in order for>> <that dyad to be periodic, the two frequencies would have to be in some>> <kind of rational relationship to one another. That E-G is pretty close>> <to a 19/16 ratio and the C-G is exactly 3/2. That means that the C-E>> <is going to be pretty close to a 24/19 ratio. BUT, even though it's>> <going to be close to 24/19, having a scale that is close to periodic>> <still won't mean that it's "perfect" for what western music>> <traditionally wants, which is a major third that is close to 5/4. > About E-G = (1,5 / 1,26134462288 ) = Eb = (2)(1/4)) = 1,189207115> This quotient is the needed relation between E and G and at the same> time another support of E = 1,26134462288.The minor third that most people want to use is 6/5. The quarter of anoctave that is used in 12-tet and your scale is not 6/5 by any means,and is closer to 19/16. It is a workable approximation to 6/5 thatmost people won't notice: certainly does not make the scale offer"perfect" harmony. ¿How do you know that most people want to use 6/5 ? (19 /16) = 1,1875 --- You probably prefer (6/5) = 1,2Perhaps that scale is offering "perfect" harmony and you are the only capable man to recognize a "perfect" harmony. Piagui minor third and equal tempered minor third equals to 1,189207115. Possible new minor third in Piagui scales = (1189 / 1000) = 1,189. The inverse of the Deviation from 1,189207115 = (1,189 / 1,189207115) = 0,99982583773,The deviation is (1/0,99982583773) = 1,0001741926. Since 1,0001741926 is lower than the schisma (1,00112915039), it cannot be detected by the human ear.I don't know offhand what decimal numbers relate to what justrelationships. Please write using fractions and exponents, notdecimal.I will do it.> The tones (2)(1/4), (2)(1/2), (2)(3/4) and 2 are fundamental frequencies of> (Eb, F#, A and Do) of Piagui and equal tempered scale. Thanks to this> feature the only correct chord of equal tempered is DIMIN. DO, DIM. Eb, DIM F#, DIM> A.If you arbitrarily define the "correct" diminished chord as 4 equal divisions of the octave.Curiously when I was deducing the progression of 612 musicalcells using the schisma and two commas, the 4 equal divisions of the octave appeared spontaneously as well as many other basicparameters. Since I did not programmed these divisions I thought that they emerged by decision of nature and therefore the four cells are probably correct; as a matter of fact they are also workingin the equal tempered scale and will probably work in the scale you are "perfecting". So I did not arbitrarily defined how to call to the 4equal divisions of the octave.> Note Eb = (2)(1/4)) = 1,189207115> Regards> MARIO PIZARRO Mario, I will say this: You cannot "define" perfect harmony to besomething that is not in accordance with what the music community atlarge needs, and then claim that your scale offers perfect harmony.For the vast majority of western music out there, especially baroqueand classical pieces that were composed in some kind of meantonetuning, then your scale will be just as imperfect, if not slightlymore in certain situations, than 12 tone equal temperament. Mike,I have already explained in the second paragraph in this message why I called "perfect harmony" to the visual expression of the triad responses. Let us do better things like the chart of errors now that you have promised me that you will teach me how to interpret this chart.The fact that the wave appears to be periodic doesn't matter, as theHammond B3 organ's waves are all perfectly periodic, but the noteshave relationships to one another like 2808 / 2975. After a certainpoint, the ear can no longer hear the periodicity of the wave exceptfor a slight difference in beating. I don't know how you define "perfect," but the fact of the matter isthat periodicity does not mean "perfection" of a chord or a triad. Theother thing is that not only is the major third in the "perfect" chordeven FARTHER from 5/4 just, the perfect fifths in the "imperfectchords", of which there are four in piagui, are even further from justthan the "perfect" piagui thirds are. I think it is an interesting scale with interesting properties that incertain situations might lead to it being more advantageous than12-equal. I do not think it is a perfect scale that will handle themusical requirements of most music any better than 12-equal, but incertain situations, it may. -Mike

> When for the first time I looked at the Piagui graph responses,
> its aesthetic and apparent periodic display induced me to call them
> "perfect triads"; it would have been better to call them
> "almost perfect triads".

Then the Piagui scale offers "almost perfect harmony," just like
12-equal. What do you want to hear?

> In the other hand, the considerable effort I did along eight years
> to define the three variants of Piagui scales whose complicate deduction
> required peculiar and unusual mathematical methods deserve me the
> right of make a mistake.

Indeed.

> The musical instruments are mainly in hands of musicians and
> engineering. You Know that the exact values of some numeric
> relations in music are not possible to put to work. That is why
> you are considering non conventional fractions to satisfy your object.

I have no idea what "non-conventional fractions" you are talking
about, nor do I understand what you mean by "satisfy my object."

> I understand and agree with you on the definitions given in the
> two previous paragraphs. At the moment and since the schisma
> (32805/32768) = 1,00112915039,,,is the smallest consonance
> that can be detected by the ear of man, I am just considering for
> tone E the frequency 1,26125 =(1009 / 800) rather than
> 1,26134482288. The deviation produced by 1,26125 with respect
> to the original frequency of E is (1,26125 / 1,26134482288)
> = 0,99992482398. An error of (1-- 0,99992482398) = 0,000075176
> that is absolutely lower than the schisma´s deviation
> (1,00112915039,,, - 1) = 0,001129,,,,

I don't know where you got 1009/800. Once again, I don't have a list
of decimals memorized. Why is that E?

Can you please translate these things into cents and fractions? Cents
aren't that hard, they're just a hundredth of a 12-et semitone. Here's
a converter: http://www.sengpielaudio.com/calculator-centsratio.htm

I would be happy to help you once I can understand where these numbers
come from.

> You are quite acquainted that the equal tempered major third is
> almost equal to (63/50) = 1,26 At one millimeter far from Piagui
> major third.
>
> The author W. T. Bartholomew in "Acoustics of Music" wrote:
> "Many of these scales persist in various parts of the world today.
> It is only in "Western music", where a harmonic development
> evolved, that the diatonic scale as we know it became predominant."

What scales persist in various parts of the world today? What does
this mean? Isn't the whole point of this discussion an evaluation of
the claim that your scale offers "perfect harmony?"

> I will see what I can do to make the needed approach at my way.
> It would be a limited discussion and explanation and hope that
> despite its incomplete final data it could give clear results.
> Would you explain me what do you mean by
> "the errors from each interval, the RMS tuning
> error and TOP error"?

Well for starters let's just get the error from each interval. Define
what you think a "perfect" major third and perfect fifth is. Most
people want a major third to be as close to 5/4 as possible, and a
perfect fifth to be as close to 3/2.

So get those intervals in cents, the "closest match" in the Piagui
scale in cents, and then we'll compare. Honestly, since your scale is
a well-temperament, I don't know a better approach, but there might be
one. I don't know.

> As you have probably noted, I lived too far from cents.
> Last week I started to learn a bit about cents. At the moment
> my background is the acknowledge that each equal tempered
> semitone is divided in one hundred cents, that is all, so this
> evening I will use my brain to understand the above paragraph.
> I will be better next month.

OK, the main point is: the flat perfect fifth is 5 cents flat, which
will be distinguishable, especially when the notes are played
together, and will cause all sorts of problems for quartal and quintal
chord voicings. (stacked fourths and fifths)

> In many occasions I read:
> ,,,etc with ratio 9 : 10, or 182 cents…… ¿ WHAT ON THE EARTH…..?
> I know that the 9 : 10 relation works in the Just Intonation scale
> of Aristoxenus but for a man whose background on cents is only
> the 1200 divisions of any octave ¿what is the connection between
> the ratio (9 : 10 ) and the number of cents of it?

If you put the ratio 10/9, or 9:10 together, you will get two notes
that are a certain 'distance' apart. The same applies with 5/4 and
3/2. 3/2, for example, consists of two notes that are about 702 cents
apart. The 12 tone equal tempered scale has a note in it that is a 700
cent interval away from the root. The perfect fifth of 12-equal, in
other words, is very slightly flat from the pure 3/2. Cents make it
easy to compare how accurate a tuning or temperament is.

> I need a doctor.
> ¿Did you ever added the deviation cents of all the fifths,
> fourths and thirds of the equal tempered scale?

What do you mean? I posted the deviations right below the section that
you quoted. In what way do you want me to add them? Adding the error
for a fourth and a fifth will cancel out, for instance.

> This paragraph contains basic data for the projected chart
> of errors Piagui/equal tempered.

Right...

> ¿How do you know that most people want to use 6/5 ?
> (19 /16) = 1,1875 --- You probably prefer (6/5) = 1,2
> Perhaps that scale is offering "perfect" harmony and you are
> the only capable man to recognize a "perfect" harmony.

If you listen to barbershop quartets sing, they'll usually sing the
6/5 minor third rather than the 19/16 minor third. 19/16 is a valid
choice for a minor third as well, and it does have a slightly
different tone quality and character to it than 6/5. 12-equal handles
6/5 just as well as Piagui does.

If I am the only capable man of hearing the difference between 6/5 and
19/16, then who's going to hear the difference between Piagui and
12-equal?

> Piagui minor third and equal tempered minor third equals
> to 1,189207115. Possible new minor third in Piagui
> scales = (1189 / 1000) = 1,189. The inverse of the Deviation
> from 1,189207115 = (1,189 / 1,189207115) = 0,99982583773,
> The deviation is (1/0,99982583773) = 1,0001741926.
> Since 1,0001741926 is lower than the schisma (1,00112915039),
> it cannot be detected by the human ear.

Why is 1.189 the possible minor third in the Piagui scale? The Piagui
scale uses 1.189207115 just like equal tempered does. I don't
understand what you mean here.

> Curiously when I was deducing the progression of 612 musical
> cells using the schisma and two commas, the 4 equal divisions
> of the octave appeared spontaneously as well as many other basic
> parameters. Since I did not programmed these divisions I thought
> that they emerged by decision of nature and therefore the four
> cells are probably correct; as a matter of fact they are also working
> in the equal tempered scale and will probably work in the scale you
> are "perfecting". So I did not arbitrarily defined how to call to the 4
> equal divisions of the octave.

You did not arbitrarily define there being 4 equal divisions of the
octave. You did, however, arbitrarily define that 4 equal divisions of
the octave is the "correct" diminished chord. A diminished chord could
be a stack of 3 6/5. It could be a lot of things.

I don't understand where this conversation is going anymore. I've
tried my best to help you. This is what I've made of this whole mess:

1) You think your scale offers perfect harmony because
2) the C major triad is almost perfectly periodic

And I am responding to you that a wave can be perfectly periodic and
still not be suitable for western harmony. Furthermore, you seem to
insist that the schisma is the lowest interval that can be detected by
ear, but I estimate the threshold to be much higher. That is my
opinion. I think Piagui will be useful in certain circumstances and
less useful in others. That is all I can say about it.

-Mike

-Mike

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:

"At the moment and since the schisma (32805/32768) =
1,00112915039,,,is the smallest consonance that can be detected by the
ear of man, I am just considering for tone E the frequency 1,26125
=(1009 / 800) rather than 1,26134482288. The deviation produced by
1,26125 with respectto the original frequency of E is (1,26125 /
1,26134482288) = 0,99992482398. An error of (1-- 0,99992482398) =
0,000075176 that is absolutely lower than the schisma´s deviation
(1,00112915039,,, - 1) = 0,001129,,,,"

I don't know where you got this stuff, but you are operating under s
serious delusion. The smallest amount of deviation from a pure
interval which the ear can detect is by no means a constant. It varies
greatly with harmonic content and position of the interval within the
overall spectrum. Generally speaking, though, the lower the interval
and the richer the timbre, the finer is our ability to detect the
slightest deviation from pure.

That said, we detect detuning by the presence of either beats or
timbre shift (flanging) when the difference is too small to produce
beats. Neither of these have any fixed relationship to either ratios
or cents. I have repeatedly demonstrated to acoustics students that
they can hear the detuning between 200 Hz and 200.005 Hz with a
reasonably bright tone, which represents a deviation of only 0.043
cents, which is many magnitudes of order finer than your huge
"minimum" of the schisma, which is almost 2 cents! So I fear that your
basis for judging detectable errors is severely flawed.

"As you have probably noted, I lived too far from cents. Last week I
started to learn a bit about cents."

Better late than nevr.

"At the moment my background is the acknowledge that each equal
temperedsemitone is divided in one hundred cents, that is all, so this
evening I will use my brain to understand the above paragraph. I will
be better next month."

I hope it doesn't take an entire month to wrap your mind around the
concept, it is really raher simple (see below).

"In many occasions I read:,,,etc with ratio 9 : 10, or 182 cents.. ¿
WHAT ON THE EARTH...? I know that the 9 : 10 relation works in the
Just Intonation scale of Aristoxenus but for a man whose background on
cents is only the 1200 divisions of any octave ¿what is the connection
between the ratio (9 : 10 ) and the number of cents of it?"

The connection is expressed by this formula:

cents = log(ratio of the interval)/log(2) * 1200

Often the log(2) * 1200 is simplified into 3986, but since you need
some sort of calculating device with a log function to get the log of
the ratio anyway, there is no excuse for using this simplification
which only introduces errors.

Ciao,

P