Marpurg-Barbour article

Mike,

In your message you wrote:

I'm not sure what you mean when you say "perfect" harmony, but there will definitely be noticeable beating for any major triad in your scale, as will there be in equal temperament. Some of your fifths are 6 cents flat of a "just" third, and your thirds are either 16 cents sharp or 12 cents sharp, which will cause pronounced beating in any case. What exactly do you mean by "perfect" or "flawless" harmony?
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> When I was writing my book I detected several interesting frequency relations, besides the Natural Progression >of Musical Cells containing 612 cells from C=1 to 2C=2.; one of its features is the following: If you take any cell >and multiply it by 1,33333...... to produce a perfect fourth the product is another cell (twelve digits). Any other >cell when multiplied by 1,5 to make a perfect fifth, the product is another cell. I can send you the six pages >progression containing the cells 2(1/4), 2(1/2), 2(3/4) , (the fractions within parentheses are exponents); also >contains all the tones of the heptatonic and extended scales of Pythagoras and Aristoxenus (21 tones per >octave each extended set), but Tom Dent is of the opinion that it is not advisable to work with hundred of lines >of data and probably most if not all of Yahoo tuning members are of the same opinion and I don´t want to >generate a compromise. However if one of the members suggest me to send the progression I will be pleased >to do it.

>The data contained in the progression allowed the detection of the K and P semitone factors, these being the >most suitable for yielding aesthetic complex waves when the chord tone frequencies are computer-plotted and >added to show perfect harmony. The data also allowed to develope the three variants of Piagui scales..

>After examining the chord wave responses of major and minor Piagui and Tempered triads, the aesthetic and >periodic views of Piagui graphs contrasted with the disordered and chaotic responses of all the Tempered >printed responses so I called imperfect harmony to the combined or blended tone sounds of Tempered triads >and "perfect" or "flawless harmony" to the ones produced by the Piagui scales (An expression that means that >Piagui chords do not contain any level of failed harmony as showed by the printed graphs). I believe that this is >happening. On the other hand, the tempered harmony is not a bad harmony but imperfect and sufficiently close >to the Piagui harmony. That is why the Piagui tone frequencies are not far from the corresponding tempered >ones.

Mike;

>Your phrase: "Some of your fifths are 6 cents flat of a "just" third, and your thirds are either 16 cents sharp or 12 >cents sharp," I don´t understand it sufficiently because I am Electronic Engineer that only got the basic knowledges >on music; an old man (74).
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Brad,

>I will respond to your message in a matter of days. Before I need to be recovered; I had an accident in my >apartment, meanwhile I tell you that I acknowledge that I misunderstood the terms in the article and really the >correct understanding was not so easy to get directly. However, I already advanced with a deep analysis of it >and found important errors that you will know promptly. The H row works with different parameters and wrong >conditions. All the rows are incorrect and there is no match with any of the Piagui variants. Without doubt.

>Regards

MARIO PIZARRO

ELECTRONIC ENGINEER

piagui@...

> Mike;
>
>>Your phrase: "Some of your fifths are 6 cents flat of a "just" third, and
>> your thirds are either 16 cents sharp or 12 >cents sharp," I don´t
>> understand it sufficiently because I am Electronic Engineer that only got
>> the basic knowledges >on music; an old man (74).

Well, a "just" third is made up of two frequencies in which one is 5/4
of the other (1.25 * the other one). Your thirds are larger/wider than
this, and so there will be beating. Beating is the name we have for
the "disordered" response that you spoke of in your pictures - so
named because the changes in volume "beat" in time.

A just fifth is made up of two frequencies in which one is 3/2 of the
other one (1.5 * the other one). Most of your fifths are exactly just
(which is an improvement over equal temperament). However, the rest of
them are actually narrower than that, so there will be beating.

The improvement in fifths may be what you're referring to when you say
that the harmony is more 'perfect' than in equal temperament. However,
major triads that have perfect fifths also are the same ones with the
wider thirds, so they will be more "out of tune" than even equal
temperament.

It's also possible that you've hit upon an even-beating "noble number"
interval, which we've been discussing lately, but I don't even want to
start on that right now.

-Mike

Mike,

It seems that I sent you some graphs of triad responses for comparing tempered and Piagui harmonies, I don�t remember if I did it or not. Those "harmony photos" are important results of the research I believe. Since the Piagui triad responses are periodic and aesthetic while the tempered ones are disordered and approaching to chaotic �Don`t you think that these features are enough to conclude that Piagui harmony is much better than tempered?. I am guessing that "pictures" in your phrase: "Beating is the name we have for the "disordered" response that you spoke of in your pictures - " is referred to the "harmony photos" I mentioned above. Please advise.
Regards

MARIO PIZARRO
June 19 -- 12:05 pm
piagui@...
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----- Original Message ----- From: "Mike Battaglia" <battaglia01@...>
To: <tuning@yahoogroups.com>
Sent: Wednesday, June 18, 2008 2:46 AM
Subject: Re: [tuning] Marpurg-Barbour article

> Mike;
>
>>Your phrase: "Some of your fifths are 6 cents flat of a "just" third, and
>> your thirds are either 16 cents sharp or 12 >cents sharp," I don�t
>> understand it sufficiently because I am Electronic Engineer that only got
>> the basic knowledges >on music; an old man (74).

Well, a "just" third is made up of two frequencies in which one is 5/4
of the other (1.25 * the other one). Your thirds are larger/wider than
this, and so there will be beating. Beating is the name we have for
the "disordered" response that you spoke of in your pictures - so
named because the changes in volume "beat" in time.

A just fifth is made up of two frequencies in which one is 3/2 of the
other one (1.5 * the other one). Most of your fifths are exactly just
(which is an improvement over equal temperament). However, the rest of
them are actually narrower than that, so there will be beating.

The improvement in fifths may be what you're referring to when you say
that the harmony is more 'perfect' than in equal temperament. However,
major triads that have perfect fifths also are the same ones with the
wider thirds, so they will be more "out of tune" than even equal
temperament.

It's also possible that you've hit upon an even-beating "noble number"
interval, which we've been discussing lately, but I don't even want to
start on that right now.

-Mike

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