joey,
i was able to derive most of the values in the table by the following method. it appears to be derived from a half step being defined as a series of five perfect fourths:
C = 1/1
F = 4/3
Bb = 16/9
Eb = 32/27
Ab = 128/81
Db = 256/243
and using the interval 256/243 to define a chromatic series based upon this "pythagorean" half step, where the numerator is on top, followed by the denominator, and finally the quotient of the two. each successive set adds 256 to the numerator to generate the next half-step:
256
243
1.0535
512
243
2.1070
768
243
3.1605
1024
243
4.2140
1280
243
5.2675
1536
243
6.3210
1792
243
7.3745
2048
243
8.4280
2304
243
9.4815
2560
243
10.5350
2816
243
11.5885
3072
243
12.6420
as to the explanation of any conclusions made by mr. gratz in this regard, i cannot say :)
Reed Gratz's Pythagorean frequency ratios?
Moderators: bobappleton, sandywilliams
-
chespernevins
- Posts: 368
- Joined: Sat Sep 16, 2006 7:34 am
joey,
i didn't follow through the numbers until you asked the question. i paid them little attention because they looked so familiar, since the study of harmonic ratios generates lots of examples of exponents of 2s and 3s such as seen here. here's a way to simplify the chart in question:
let x = 256/243, a pythagorean half-step...
the "gratz series" is therefore:
x, 2x, 3x, 4x, etc...
this would be the overtone series of a pythogorean half-step would it not?
it most certainly is not a chromatic scale. what of it? i'm just not sure, but it certainly isn't what it seems to be...
the first number in the column on the left is 1.0595, which corresponds to the 12th root of 2, the equal-tempered half-step; however, i am currently unable to derive any of the other values in the left column - i'll keep at it in order to reproduce the algorithm...
again, what of it? i confess that i do not know. i just thought you should know that you weren't the only one who has left this chart scratching their head. i'm sure that you know full well that by definition, no harmonic ratio (by definition a rational number) could do more than to approximate a tritone (or any equal-tempered non-octave interval), an irrational number.
i didn't follow through the numbers until you asked the question. i paid them little attention because they looked so familiar, since the study of harmonic ratios generates lots of examples of exponents of 2s and 3s such as seen here. here's a way to simplify the chart in question:
let x = 256/243, a pythagorean half-step...
the "gratz series" is therefore:
x, 2x, 3x, 4x, etc...
this would be the overtone series of a pythogorean half-step would it not?
it most certainly is not a chromatic scale. what of it? i'm just not sure, but it certainly isn't what it seems to be...
the first number in the column on the left is 1.0595, which corresponds to the 12th root of 2, the equal-tempered half-step; however, i am currently unable to derive any of the other values in the left column - i'll keep at it in order to reproduce the algorithm...
again, what of it? i confess that i do not know. i just thought you should know that you weren't the only one who has left this chart scratching their head. i'm sure that you know full well that by definition, no harmonic ratio (by definition a rational number) could do more than to approximate a tritone (or any equal-tempered non-octave interval), an irrational number.
I think he's just trying to prove that the tritone is special somehow by virtue of being 'the exact centre' of the octave, and therefore the lydian scale is better than the major scale.
In my view, all he does is prove that 6 time an operation is half of 12 times an operation. No tritone produced by any kind of natural ratio is half of an octave.
In my view, all he does is prove that 6 time an operation is half of 12 times an operation. No tritone produced by any kind of natural ratio is half of an octave.
Re: Reed Gratz's Pythagorean frequency ratios?
The numbers are not "mine." See the Harvard Dictionary of Music (as mentioned on the previous page of the article) for those numbers. I had to take a look at that again myself as it's been 36 years since I wrote the article for George. He was particularly interested in including them in his book.
Best wishes!
Best wishes!
Re: Reed Gratz's Pythagorean frequency ratios?
I must say it's awesome that you have joined the forum, Mr. Gratz! Hopefully your presence will serve to fill in some gaps that some of us have in our understanding of LCC's underlying concepts.
Not sure if you've read any of my contributions to the forum, but if you have, you'll know that while I am in awe of the power LCC (as an organizing system) can give a person, I am at odds with some of the reasons and explanations for things (on a theoretical level).
Your article has been part of that, so I'd lke to bring up two things in this article that have been part of my friction, if you will, with the LCC: (and maybe you'll be able to twist the lens into focus for me.)
1. The imputing of almost sinister motives to the promotion of the major scale in Western music (at a minimum, faulty methods).
2. The logic behind the #4 "exact centre of the octave" idea. (as well as its implications were this true)
As to point 1:
I don't think that anyone (including John Backus or the Harvard Dictionary of Music) is suggesting that the the root or tonic of a major scale is the ACOUSTICAL FUNDAMENTAL of extended vertical chords derived from the major scale.
I think that any time Western music theory referred to the Pythagorean aspect of a scale, it was more of an organizational fact or tool than a suggestion that vertical chords sounded harmonious because of derivation from fifths.
When triads became "common practice", instruments (especially keyboards) were accordingly being tuned to so-called "just intonation", rather than pythagorean tuning.
This means that three separate but individually pure verticalities (the Tonic, Subdominant and Dominant triads - each expressing the acoustically "blending" pattern of found between partials 4, 5 and 6) could be sounded, which were likewise related to one another by an acoustically pure interval - the perfect fifth. There was little experimentation into larger vertical structures.
So, the major scale, as I see it, arose as a way to produce a small number of acoustically pure, yet simple, vertical structures, that could be used as a foundation for musical expression (mostly song-like, dance-like, social stuff, not impressionistic, vertically adventurous stuff). It suited it's purpose very well. It never claimed to also be the cornerstone for infinite expansion into vertical harmonic complexity.
The Lydian Scale is, of course, more suited to that side of things. But, in my view, not for the reasons that you or GR propose.
As to point 2:
It seems that you were asserting that the #4 blends with the fundamental ON ACCOUNT OF being "symmetrical". I don't see the logic for that.
One reason is that no naturally-occuring ratio for #4 is the "exact centre" of the octave. Unless you're broadening the definition of an octave to include the 13th tone in a ladder of fifths as a variant, there is only one "version" of the octave, and that is the 2:1 ratio. The #4 can only be made the "exact centre" of that octave by 'manipulated means', to borrow a term you used to account for the major scale's promotion.
To me, the reason the #4 blends well with the tonic is because it occupies the position of the 45th partial, and so relates to the tonic by a small number of simple relationships (two perfect fifths and a major third - or two 3:2 ratios and one 5:4 ratio). This, to me, is the most natural way to expand upon the harmonic unity/purity of the triad. This approach and the pythagorean one both lead to the Lydian scale as the ultimate vertical scale. However, it regards the Major scale, not as a manipulated, less-than-objective use of a pythagorean concept (as LCC views it), but as a very sensible combination of the vertical (acoustically perfect triads) and the horizontal (separate "places" to occupy with said triads).
It just seems like your article endeavors to fault Western music theory for being "guided toward the desired end" (the major scale) "by manipulated means", but then proceeds to promote the lydian scale (the desired end) by means no less manipulated.
The "exact centre - therefore symmetrical - therefore more objective" argument is just not something I can agree with. However, now that your part of the forum, maybe more light can be shed on what your reasoning really was. (and why you and GR were so hard on the poor major scale)
Anyway, welcome to the forum. You're definitely among the original LCC alumni.
Not sure if you've read any of my contributions to the forum, but if you have, you'll know that while I am in awe of the power LCC (as an organizing system) can give a person, I am at odds with some of the reasons and explanations for things (on a theoretical level).
Your article has been part of that, so I'd lke to bring up two things in this article that have been part of my friction, if you will, with the LCC: (and maybe you'll be able to twist the lens into focus for me.)
1. The imputing of almost sinister motives to the promotion of the major scale in Western music (at a minimum, faulty methods).
2. The logic behind the #4 "exact centre of the octave" idea. (as well as its implications were this true)
As to point 1:
I don't think that anyone (including John Backus or the Harvard Dictionary of Music) is suggesting that the the root or tonic of a major scale is the ACOUSTICAL FUNDAMENTAL of extended vertical chords derived from the major scale.
I think that any time Western music theory referred to the Pythagorean aspect of a scale, it was more of an organizational fact or tool than a suggestion that vertical chords sounded harmonious because of derivation from fifths.
When triads became "common practice", instruments (especially keyboards) were accordingly being tuned to so-called "just intonation", rather than pythagorean tuning.
This means that three separate but individually pure verticalities (the Tonic, Subdominant and Dominant triads - each expressing the acoustically "blending" pattern of found between partials 4, 5 and 6) could be sounded, which were likewise related to one another by an acoustically pure interval - the perfect fifth. There was little experimentation into larger vertical structures.
So, the major scale, as I see it, arose as a way to produce a small number of acoustically pure, yet simple, vertical structures, that could be used as a foundation for musical expression (mostly song-like, dance-like, social stuff, not impressionistic, vertically adventurous stuff). It suited it's purpose very well. It never claimed to also be the cornerstone for infinite expansion into vertical harmonic complexity.
The Lydian Scale is, of course, more suited to that side of things. But, in my view, not for the reasons that you or GR propose.
As to point 2:
It seems that you were asserting that the #4 blends with the fundamental ON ACCOUNT OF being "symmetrical". I don't see the logic for that.
One reason is that no naturally-occuring ratio for #4 is the "exact centre" of the octave. Unless you're broadening the definition of an octave to include the 13th tone in a ladder of fifths as a variant, there is only one "version" of the octave, and that is the 2:1 ratio. The #4 can only be made the "exact centre" of that octave by 'manipulated means', to borrow a term you used to account for the major scale's promotion.
To me, the reason the #4 blends well with the tonic is because it occupies the position of the 45th partial, and so relates to the tonic by a small number of simple relationships (two perfect fifths and a major third - or two 3:2 ratios and one 5:4 ratio). This, to me, is the most natural way to expand upon the harmonic unity/purity of the triad. This approach and the pythagorean one both lead to the Lydian scale as the ultimate vertical scale. However, it regards the Major scale, not as a manipulated, less-than-objective use of a pythagorean concept (as LCC views it), but as a very sensible combination of the vertical (acoustically perfect triads) and the horizontal (separate "places" to occupy with said triads).
It just seems like your article endeavors to fault Western music theory for being "guided toward the desired end" (the major scale) "by manipulated means", but then proceeds to promote the lydian scale (the desired end) by means no less manipulated.
The "exact centre - therefore symmetrical - therefore more objective" argument is just not something I can agree with. However, now that your part of the forum, maybe more light can be shed on what your reasoning really was. (and why you and GR were so hard on the poor major scale)
Anyway, welcome to the forum. You're definitely among the original LCC alumni.
Re: Reed Gratz's Pythagorean frequency ratios?
I appreciate your thoughtful comments. I would, at this point, just say that the Lydian Chromatic Concept is an INCLUSIVE approach to music, not exclusive. That is, it includes the teleological major-minor scale system (tonality) and the 'blending' music of, particularly, the lydian scale (and other modes). I recall George's love of the term Pan-Modal in describing that inclusiveness. You can hear and see his open-mindedness regarding this in his music. That, in my view, is the most important aspect of the Concept.
Best wishes,
Reed
Best wishes,
Reed