Coments

To Mike Battaglia,

> 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 like12-equal. What do you want to hear?For achieving the top scale (note that this time I wrote "top") it is perhaps necessary to sacrifice a bit of perfection. To make that a set of tone frequencies provide the top quality triads is more than a hard job. The number 2 works in different ways starting from the needed octaves when tuning the musical instrument with the expectance of obtaining pure fifths (relation 3/2)and perfect fourths (relation 4/3) where 3 has nothing to do with octaves. If nature would have been an ally of music, the perfect fifths and fourths might be for instance 3,2/2 and 4/3,2.> 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 deserves me the> right of make a mistake.Indeed.My objective is to demonstrate, that despite both scales are almost perfect, one of them, the Piagui scale, is better than the equal tempered and I believe that this is what happens. The aesthetic Piagui triad responses are not for nothing.> 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 at work. That is why> you are considering non conventional fractions to satisfy your object.I have no idea what "non-conventional fractions" you are talkingabout, nor do I understand what you mean by "satisfy my object."Conventional Major thirds, as I see, are eq-temp. 1,2599,,, Pythagoras major third of 1,265625 = (81/64), Aristoxenus or Just Intonation Major third 1,25 = 5/4; similarly, conventional minor third is the eq-temp. minor third, Non conventional major thirds are the Piagui major third 1,261344,,,,,, and non conventional minor thirds are your (19/16) = 1,1875 as well as 1,2 (Eb = 1,2 that works in the extended scale of Aristoxenus - Zarlino with 21 tones per octave, XVI century).Please remove "object" in the pre-precedent paragraph and write in its place ªpurpose". > 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 listof decimals memorized. Why is that E?You told me that you prefer to work with rational numbers, exponents, fractions rather than with inexact figures like E = 1,26134462288,,,. In fact, you are considering 1,2 and (19/16)= 1,1875 rather than 1,189207115. So experimentally I could follow your way by determining the fraction whose decimal value is very close to the original one provided the difference is lower than the schisma. Since you have elected (19/16) = 1,1875 instead of the conventional eq-t minor third, I found the fraction (1009/800) = 1,26125 sufficiently close to the irrational figureof 1,26134462288,,, Now you disapprove what you recommended some days ago or I misunderstood you. I will check the copy of your message. In the previous paragraph you wrote that you don´t know where I got 1009/800, I will explain it:Piagui major third =1,26134462288,,,,, ( (1,26134462288) 10(7) / 10(7) ) = [ 12613446,2288 / 10(7) ] = [ 12613446,2288 /100 x 5 x 5 x 5) ] / [ (10(7) /(100 x 5 x 5 x 5) ] =[ 1009,0756983 ] / [ (10(7)/(100 x 5x5x5) ] = Approximately equals to 1009 / 800. Can you please translate these things into cents and fractions? (Not yet Master)Cents aren't that hard, they're just a hundredth of a 12-et semitone. Here'sa converter: http://www.sengpielaudio.com/calculator-centsratio.htmI would be happy to help you once I can understand where these numberscome from.OK, I have already done that and the explanation is clear. The numbers detail how I got 1009 / 800 and opened the three pages of the blue web and did a few operations. Can you give me some orientation on how I can start the chart? I am eager to demonstrate that the equal tempered only deserves to be remembered. > 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 doesthis mean? Isn't the whole point of this discussion an evaluation ofthe claim that your scale offers "perfect harmony?"I had started writing about the western music and suddenly forgot the subject.> 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"?PENDING ANSWERxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxWell for starters let's just get the error from each interval. Definewhat you think a "perfect" major third and perfect fifth is. Mostpeople want a major third to be as close to 5/4 as possible, and aperfect fifth to be as close to 3/2.So get those intervals in cents, the "closest match" in the Piaguiscale in cents, and then we'll compare. Honestly, since your scale isa well-temperament, I don't know a better approach, but there might beone. I don't know.XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX> 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.PENDING COMMENTS. INTERESTING TASK !!!!!!XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXOK, the main point is: the flat perfect fifth is 5 cents flat, whichwill be distinguishable, especially when the notes are playedtogether, and will cause all sorts of problems for quartal and quintalchord 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 notesthat are a certain 'distance' apart. The same applies with 5/4 and3/2. 3/2, for example, consists of two notes that are about 702 centsapart. The 12 tone equal tempered scale has a note in it that is a 700cent interval away from the root. The perfect fifth of 12-equal, inother words, is very slightly flat from the pure 3/2. Cents make iteasy to compare how accurate a tuning or temperament is.XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX> 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 thatyou quoted. In what way do you want me to add them? Adding the errorfor 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 the6/5 minor third rather than the 19/16 minor third. 19/16 is a validchoice for a minor third as well, and it does have a slightlydifferent tone quality and character to it than 6/5. 12-equal handles6/5 just as well as Piagui does.Now I understandIf I am the only capable man of hearing the difference between 6/5 and19/16, then who's going to hear the difference between Piagui and12-equal?CORRECT!!!!!!> 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 Piaguiscale uses 1.189207115 just like equal tempered does. I don'tunderstand what you mean here.I had misunderstood you. Thought that you disapprove long tone frequency numbers including square roots. Sorry.> 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 theoctave. You did, however, arbitrarily define that 4 equal divisions ofthe octave is the "correct" diminished chord. A diminished chord couldbe a stack of 3 6/5. It could be a lot of things.-------------------------IT WAS ORDERED BY THE PROGRESSION OF MUSSICAL CELLS. I NOTICED THAT YOU DO NOT APPROVE THIS PROGRESSION. SHOULD YOU ACCEPT THAT I SEND YOU ITS 6 PAGES I WOULD BE HAPPY. JUST TELL ME. THE COLUMN OF REMARKS CONTAINS INTERESTING DATA --------------------------I don't understand where this conversation is going anymore. I'vetried my best to help you. This is what I've made of this whole mess:1) You think your scale offers perfect harmony because2) the C major triad is almost perfectly periodicAnd I am responding to you that a wave can be perfectly periodic andstill not be suitable for western harmony. Furthermore, you seem toinsist that the schisma is the lowest interval that can be detected byear, but I estimate the threshold to be much higher. That is myopinion. I think Piagui will be useful in certain circumstances andless useful in others. That is all I can say about it.About the schisma: The evaluation of the schisma as you know was done in the XVI century by Zarlino, Delezenne and others who devoted their time to these evaluations as it is written in "Teoría dela Música" by Joaquin Zamaçois. I think that their statement that (32805/32768)= 1,00112915039 is the smallest consonance that can be distinguished by the ear of man is correct. At this point I ask you to be highly reflective and receptive: I am sure that you have read in some message about de Progression of Musical Cells (612 cells per octave). The fundamental stone of the progression is the schisma, it is the first cell of it. It took me two years to develop the cyclical frequenciesfrom Do = 1 to 2Do = 2. If the frequency value of the schisma or its relation given above would be wrong I never would have talk about Piagui scales because the inexact value of that first cell would have made impossible the deduction of the semitone factors K and P , the considerable number of perfect fifths and fourths while the equal tempered does not have one.Through the tuning yahoogroups I got a new friend; I ignore his age and nationality. What I realized is that this new friend is very capable.-Mike Regards

MARIO PIZARRO

MARIO PIZARRO
Lima, June 30, 2008