Why is it stated that the 11th overtone is an F# (#4), when the 11th overtone actually sits squarely between the F (IV) and the F# (#IV)?
Also why is it stated that the 13th overtone is an A (majVI), when the 13th overtone is actually closer to a b6?
Thank you
#4 is the natural child of the overtone series
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An open letter from Alice Russell. June 21, 2011, Brookline, Massachusetts. 1. DO NOT make insulting, mean spirited remarks about anyone or their work; there are a plethora of sites where you can rant unfettered. If you attack someone personally, your comments will be removed. You can post it, but I'm not paying for it. Go elsewhere, and let those artists who are actually interested in discussion and learning have the floor. 2. There will be NO posting of or links to copyrighted material without permission of the copyright owner. That's the law. And if you respect the work of people who make meaningful contributions, you should have no problem following this policy. 3. I appreciate many of the postings from so many of you. Please don't feel you have to spend your time "defending" the LCC to those who come here with the express purpose of disproving it. George worked for decades to disprove it himself; if you know his music, there's no question that it has gravity. And a final word: George was famous for his refusal to lower his standards in all areas of his life, no matter the cost. He twice refused concerts of his music at Lincoln Center Jazz because of their early position on what was authentically jazz. So save any speculation about the level of him as an artist and a man. The quotes on our websites were not written by George; they were written by critics/writers/scholars/fans over many years. Sincerely, Alice
An open letter from Alice Russell. June 21, 2011, Brookline, Massachusetts. 1. DO NOT make insulting, mean spirited remarks about anyone or their work; there are a plethora of sites where you can rant unfettered. If you attack someone personally, your comments will be removed. You can post it, but I'm not paying for it. Go elsewhere, and let those artists who are actually interested in discussion and learning have the floor. 2. There will be NO posting of or links to copyrighted material without permission of the copyright owner. That's the law. And if you respect the work of people who make meaningful contributions, you should have no problem following this policy. 3. I appreciate many of the postings from so many of you. Please don't feel you have to spend your time "defending" the LCC to those who come here with the express purpose of disproving it. George worked for decades to disprove it himself; if you know his music, there's no question that it has gravity. And a final word: George was famous for his refusal to lower his standards in all areas of his life, no matter the cost. He twice refused concerts of his music at Lincoln Center Jazz because of their early position on what was authentically jazz. So save any speculation about the level of him as an artist and a man. The quotes on our websites were not written by George; they were written by critics/writers/scholars/fans over many years. Sincerely, Alice
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Jeff Brent
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Yes, I cannot answer your second question without having to find the page near the back of the concept. If it lists it, which I hope. However... There is quite a reasonable foundation for the +IV being closer to the harmonic series' "actual" tone. At the bottom note of page three it says that the tone F# has a frequency of 600 cents while F natural equals 500 cents. He also states that the harmonic series "true" overtone is equal to 551 cents, 1/100 of a semitone closer to F# than F natural.
It is not the "facts" that made me a concept junkie... It was hearing the stability of an C major 13 +11 compared to the C major 13th chord. Facts?
- http://www.lydianchromaticconcept.com/lccoto.html
--Just found this!
I hope I helped in some way... Please tell me if I was of any help. I also hope I didn't rant too terribly much!
It is not the "facts" that made me a concept junkie... It was hearing the stability of an C major 13 +11 compared to the C major 13th chord. Facts?
- http://www.lydianchromaticconcept.com/lccoto.html
--Just found this!
I hope I helped in some way... Please tell me if I was of any help. I also hope I didn't rant too terribly much!
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Jeff Brent
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- Location: L.A.
Using A440 (4A) as the point of reference, we have:
440Hz X 11 = 4840Hz (the 11th overtone)
Tempered 8D = 4698.64Hz (a distance of 141.36Hz away from 4840Hz)
Tempered 8D# = 4978.03Hz (a distance of 138.03Hz away from 4840Hz)
This means that tempered 8D# is only 3.333Hz closer than tempered 8D to the 11th overtone of A440.
The difference of 3.33Hz when compared to a frequency of 4840Hz is not only not discernible to the human ear, but represents only a 0.069% difference (not 1.00% [1%= 1 cent]).
In addition, the following proof shows that the Natural D is much closer to the Natural 11th overtone of A than either the natural D# OR the tempered D#:
One can find a natural D by either calculating one perfect 5th below A, or by calculating eleven perfect 5ths above A.
Eleven perfect 5ths above A = D4757.365 (a distance of 82.635Hz away from 4840Hz)
One perfect 5th below A = D4693.333 (a distance of 146.667Hz away from 4840Hz)
The average distance of these two natural D’s from the 11th overtone (4840Hz) is 114.651Hz
One can find a natural D#/Eb by either calculating six perfect 5ths below A, or by calculating six perfect 5ths above A.
Six perfect 5ths above A = D#5011.875 (a distance of 171.875Hz away from 4840Hz)
Six perfect 5ths below A = Eb4944.411 (a distance of 104.411Hz away from 4840Hz)
The average distance of these two natural D#’s from the 11th overtone (4840Hz) is 138.143Hz
Therefore, Natural 8D is an average of 23.492Hz closer to the Natural 11th overtone of A440 than natural 8D#
(over 20Hz closer than the difference between tempered 8D# and tempered 8D to the 11th overtone).
While this does not qualify the 11th overtone to be a perfect fourth, it does prove that the 11th overtone is NOT an augmented fourth.
Is there some flaw in the above figures?
440Hz X 11 = 4840Hz (the 11th overtone)
Tempered 8D = 4698.64Hz (a distance of 141.36Hz away from 4840Hz)
Tempered 8D# = 4978.03Hz (a distance of 138.03Hz away from 4840Hz)
This means that tempered 8D# is only 3.333Hz closer than tempered 8D to the 11th overtone of A440.
The difference of 3.33Hz when compared to a frequency of 4840Hz is not only not discernible to the human ear, but represents only a 0.069% difference (not 1.00% [1%= 1 cent]).
In addition, the following proof shows that the Natural D is much closer to the Natural 11th overtone of A than either the natural D# OR the tempered D#:
One can find a natural D by either calculating one perfect 5th below A, or by calculating eleven perfect 5ths above A.
Eleven perfect 5ths above A = D4757.365 (a distance of 82.635Hz away from 4840Hz)
One perfect 5th below A = D4693.333 (a distance of 146.667Hz away from 4840Hz)
The average distance of these two natural D’s from the 11th overtone (4840Hz) is 114.651Hz
One can find a natural D#/Eb by either calculating six perfect 5ths below A, or by calculating six perfect 5ths above A.
Six perfect 5ths above A = D#5011.875 (a distance of 171.875Hz away from 4840Hz)
Six perfect 5ths below A = Eb4944.411 (a distance of 104.411Hz away from 4840Hz)
The average distance of these two natural D#’s from the 11th overtone (4840Hz) is 138.143Hz
Therefore, Natural 8D is an average of 23.492Hz closer to the Natural 11th overtone of A440 than natural 8D#
(over 20Hz closer than the difference between tempered 8D# and tempered 8D to the 11th overtone).
While this does not qualify the 11th overtone to be a perfect fourth, it does prove that the 11th overtone is NOT an augmented fourth.
Is there some flaw in the above figures?
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sandywilliams
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bobappleton
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Well, it's quite an interesting illustration of how art and mathematics are two different languages. Your arithmetical analysis may be just too "literal" to see or hear the point.Jeff Brent wrote:Is there some flaw in the above figures?
To relate science and music you have to move to the larger vision of someone like James Gleick - for example. See Judy Lochhead's "Hearing Chaos" referred to under the "Hearing Absolute" thread.
b
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Jeff Brent
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While the above laws of physics are expressed mathematically, I am concerned about the statement on pg 224, that says "something so powerfully evident in nature" couldn't be kept a secret.
Doesn't the term "so powerfully evident in nature" directly mean "the laws of physics"?
The physical facts that the 11th overtone is neither a perfect 4th nor an augmented 4th, and that the 13th overtone is not a major 6th
invalidate not only the chart on page 130 [typo correction -> page 230], but one of Russell's primary premises.
Doesn't this strike anyone as a little disconcerting?
Especially due to the repeated references throughout the book to the overtone series as one source for the justification of the concept.
Doesn't the term "so powerfully evident in nature" directly mean "the laws of physics"?
The physical facts that the 11th overtone is neither a perfect 4th nor an augmented 4th, and that the 13th overtone is not a major 6th
invalidate not only the chart on page 130 [typo correction -> page 230], but one of Russell's primary premises.
Doesn't this strike anyone as a little disconcerting?
Especially due to the repeated references throughout the book to the overtone series as one source for the justification of the concept.
Last edited by Jeff Brent on Mon Oct 20, 2008 12:44 pm, edited 1 time in total.
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bobappleton
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[quote="Jeff Brent"]
Is there some flaw in the above figures?[/quote]
yes. you cannot use linear math to analyze geometric progressions as it appears you are doing here. the 11th overtone of the series is 5.513179424 half-steps above the 8th overtone according to:
R = 2^(F/12) [read as 2 to the exponent (power) of the quantity (F/12)] where R is the harmonic ratio between two frequencies and F is the number of frets or half-steps of equal-temperament of the given ratio. F solves as:
F = 12 log R/log 2
R = 11/8 (the ratio of the 11th overtone to the tone three perfect octaves above the fundamental)
although i derived the above formulas from basic principles, they may be found in the new harvard dictionary of music.
so, we may establish that an equal-tempered fourth is equivalent to 5.00 half-steps and an equal-tempered augmented fourth is equivalent to 6.00 half-steps, we may infer that the 11th overtone at 5.51 half-steps is ever-so-slightly closer to an augmented fourth by a mere 0.01 half-steps or 1 cent, but is clearly not a perfect fourth - more like a quarter-step higher. furthermore, the fact that the 13th overtone at 8.41 half-steps is closer to a b6th than a natural sixth is overlooked in many texts of harmonic theory where the "overtone scale" of the 8th through 16th harmonics are used to justify a "scale" of some kind, which actually produces this:
C D E F# G Ab Bb B C
when rounded off to the nearest equal-tempered half-steps shouldn't trouble us greatly, other than inferring that using pure harmonic theory to justify any equal tempered scale is problematic at best. the establishment of the "circle of fifths" several centuries ago seems the best solution to merge the conflicts between harmonic theory and equal temperament, the error of a harmonically pure fifth (R=3/2) being a mere .02 half-steps. i have said before and will state again that it is my belief that whoever discovered, created, or otherwise implemented twelve-tone equal temperament has saved us from a whole lot of trouble...
please allow me opportunity to address some of these other concerns in time, because the same logic as proposed, that harmonic theory and equal-temperament are at odds with each other, may be used to expose all harmonic-based theory as meaningless, which clearly seems not to be the case. in other words, russell's theory is no more at odds with the overtone series than traditional theory, and i am troubled that one would propose to dismiss russell's concept because of the same harmonic irregularities that have plagued music theorists since the time of pythagoras.
Is there some flaw in the above figures?[/quote]
yes. you cannot use linear math to analyze geometric progressions as it appears you are doing here. the 11th overtone of the series is 5.513179424 half-steps above the 8th overtone according to:
R = 2^(F/12) [read as 2 to the exponent (power) of the quantity (F/12)] where R is the harmonic ratio between two frequencies and F is the number of frets or half-steps of equal-temperament of the given ratio. F solves as:
F = 12 log R/log 2
R = 11/8 (the ratio of the 11th overtone to the tone three perfect octaves above the fundamental)
although i derived the above formulas from basic principles, they may be found in the new harvard dictionary of music.
so, we may establish that an equal-tempered fourth is equivalent to 5.00 half-steps and an equal-tempered augmented fourth is equivalent to 6.00 half-steps, we may infer that the 11th overtone at 5.51 half-steps is ever-so-slightly closer to an augmented fourth by a mere 0.01 half-steps or 1 cent, but is clearly not a perfect fourth - more like a quarter-step higher. furthermore, the fact that the 13th overtone at 8.41 half-steps is closer to a b6th than a natural sixth is overlooked in many texts of harmonic theory where the "overtone scale" of the 8th through 16th harmonics are used to justify a "scale" of some kind, which actually produces this:
C D E F# G Ab Bb B C
when rounded off to the nearest equal-tempered half-steps shouldn't trouble us greatly, other than inferring that using pure harmonic theory to justify any equal tempered scale is problematic at best. the establishment of the "circle of fifths" several centuries ago seems the best solution to merge the conflicts between harmonic theory and equal temperament, the error of a harmonically pure fifth (R=3/2) being a mere .02 half-steps. i have said before and will state again that it is my belief that whoever discovered, created, or otherwise implemented twelve-tone equal temperament has saved us from a whole lot of trouble...
please allow me opportunity to address some of these other concerns in time, because the same logic as proposed, that harmonic theory and equal-temperament are at odds with each other, may be used to expose all harmonic-based theory as meaningless, which clearly seems not to be the case. in other words, russell's theory is no more at odds with the overtone series than traditional theory, and i am troubled that one would propose to dismiss russell's concept because of the same harmonic irregularities that have plagued music theorists since the time of pythagoras.
the reason you cannot use linear math to analyze intervallic relationships in music is as follows:
let us state that A above middle C = 440 Hz
and that E is a harmonically perfect fifth above at 660 Hz, and
A an octave obove A (440) is 880 Hz
linear math indicates that E (660) is exactly halfway between 440 and 880, being 220 Hz up or down from the "A"s, the "linear" halfway point...
but we also know that E is (approximately) seven half-steps above and five half-steps below A and that Eb is six half-steps above or below, the "logarithmic" halfway point...
because of the octave (A=55,110,220,440,880,etc...) and its geometric progression, you must use a logarithmic scale to measure ratios (intervals)...
some commonly used intervals and their equal-tempered measurements include:
perfect fifth (ideal ratio 3/2) = 7.02 half-steps
perfect fourth (ideal ratio 4/3) = 4.98 half-steps
major third (ideal ratio 5/4) = 3.86 half-steps
minor third (ideal ratio 6/5) = 3.16 half-steps
remember, R (harmonic ratio) = 2^(F/12) [f=half-steps]
let us state that A above middle C = 440 Hz
and that E is a harmonically perfect fifth above at 660 Hz, and
A an octave obove A (440) is 880 Hz
linear math indicates that E (660) is exactly halfway between 440 and 880, being 220 Hz up or down from the "A"s, the "linear" halfway point...
but we also know that E is (approximately) seven half-steps above and five half-steps below A and that Eb is six half-steps above or below, the "logarithmic" halfway point...
because of the octave (A=55,110,220,440,880,etc...) and its geometric progression, you must use a logarithmic scale to measure ratios (intervals)...
some commonly used intervals and their equal-tempered measurements include:
perfect fifth (ideal ratio 3/2) = 7.02 half-steps
perfect fourth (ideal ratio 4/3) = 4.98 half-steps
major third (ideal ratio 5/4) = 3.86 half-steps
minor third (ideal ratio 6/5) = 3.16 half-steps
remember, R (harmonic ratio) = 2^(F/12) [f=half-steps]
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Jeff Brent
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I would amend the above to read "we may infer that the 11th overtone is ever-so-slightly closer to a tempered augmented fourth (but the natural perfect 4th is, in fact, closer) "dogbite wrote:we may infer that the 11th overtone at 5.51 half-steps is ever-so-slightly closer to an augmented fourth
We appear to be in agreement here.dogbite wrote:the fact that the 13th overtone at 8.41 half-steps is closer to a b6th than a natural sixth is overlooked in many texts of harmonic theory where the "overtone scale" of the 8th through 16th harmonics are used to justify a "scale" of some kind.
I also agree with this statement. Which is why I think that Russell shouldn't have used "harmonic theory to justify" an "equal tempered scale".dogbite wrote:using pure harmonic theory to justify any equal tempered scale is problematic at best.
I am using logarithmic relationships here. A quick look at any guitar neck is proof that music is logrithmical.dogbite wrote:the reason you cannot use linear math to analyze intervallic relationships in music is as follows: ...
Did I say that I was dismissing Russell's entire concept? I thought that I was saying that one of his justifications was unjustified.dogbite wrote:I am troubled that one would propose to dismiss Russell's concept because of the same harmonic irregularities that have plagued music theorists since the time of Pythagoras.
Is there anyone here that can truthfully say (now that they now know that the 11th and 13th overtones are not respectively a #4 and a maj6th) that they actually believe the chart on page 130 is correct?
[typo correction -> page 230]
Last edited by Jeff Brent on Mon Oct 20, 2008 12:43 pm, edited 1 time in total.
jeff,
first let me state that i don't believe that mathematics is the ultimate solution to the inherent discrepancies of harmonic-based theory, other than the establishment of fixed pitches (more or less) within our current twelve-tone equal-temperament system. the system is not perfect, but it is a good system, which allows the freedom to transpose to any of the twelve tonal centers without worry of whether or not harmonic intervals are better or worse in the new key, for they are all equal. i believe that it is what our ears tell us that will decide what is best; therefore, in the end we may simply agree to disagree. the following is mere opinion, completely my own, and is not meant to dismiss any of your astute observations. i, for one, am a champion of equal temperament for many reasons, some of which are defined here - and if we agree that equal-temperament is an acceptable solution to these inherent discrepancies, then perhaps at least some of any apparent disagreement we may have will be rendered moot...
respectfully, i honestly have no wish to quarrel with you, but some of your math is off. not all, but you simply cannot use differences in Hz and percentage of frequencies to define intervallic relationships in music, since they are linear measurements of a logarithmic curve. the measurement of intervals in cents (1/1200 of an octave) is well established in the harmonic theories of hindemith and rameau, as well as many others. furthermore, i have read your pdf and have found it interesting; however, i cannot help believing that you are looking at an elephant through a microscope, to borrow our "elephants in the forest" vision previously mentioned. follow me here:
perfect fourth = 4/3 = 4.98 half-steps above fundamental, rounded to the nearest cent, or 498 cents, 4.980449991 half-steps to ten digits:
4/3 = 2^(F/12)
F = (12 log (4/3))/log 2
F= 4.98...
alternate source: see new harvard dictionary of music.
eleventh harmonic = 11/8 = 5.51 half-steps above fundamental, rounded to the nearest cent, or 551 cents, 5.513179424 half-steps to ten digits:
11/8 = 2^(F/12)
F = (12 log (11/8))/log 2
F= 5.51...
and
5.513179424
minus
4.980449991
equals
0.532729432
half-steps from a harmonically pure perfect fourth (4/3) and the eleventh overtone (11/8), clearly more than a quarter-step.
you said:
"440Hz X 11 = 4840Hz (the 11th overtone)
Tempered 8D = 4698.64Hz (a distance of 141.36Hz away from 4840Hz)
Tempered 8D# = 4978.03Hz (a distance of 138.03Hz away from 4840Hz)
This means that tempered 8D# is only 3.333Hz closer than tempered 8D to the 11th overtone of A440.
The difference of 3.33Hz when compared to a frequency of 4840Hz is not only not discernible to the human ear, but represents only a 0.069% difference (not 1.00% [1%= 1 cent]).
In addition, the following proof shows that the Natural D is much closer to the Natural 11th overtone of A than either the natural D# OR the tempered D#:
One can find a natural D by either calculating one perfect 5th below A, or by calculating eleven perfect 5ths above A.
Eleven perfect 5ths above A = D4757.365 (a distance of 82.635Hz away from 4840Hz)
One perfect 5th below A = D4693.333 (a distance of 146.667Hz away from 4840Hz)
The average distance of these two natural D’s from the 11th overtone (4840Hz) is 114.651Hz"
difference in Hz is a linear measurement, jeff. when you state "difference of 3.33Hz", this is a linear (first degree equation) measurement, and "0.069% difference" of Hz, also a first degree equation.
furthermore, calculating eleven perfect fifths (3/2)^11 above the fundamental when compared to a perfect fifth below (4/3) introduces an error known as the "pythagorean comma", an error of over 23 cents (more precisely 0.234600104 half-steps) which is perhaps the most convincing evidence that "whoever discovered, created, or otherwise implemented twelve-tone equal temperament has saved us from a whole lot of trouble."
jeff, i am pleased that you took the time to look at all this and i do not want to give you the impression that i am dismissing your observations out of hand. to the contrary, i look forward to what is likely to be a spirited and illuminating discussion.
i believe the LCC to be based on equal temperament through the circle of fifths and that analysis of it to be most appropriate in this domain; however, like traditional harmonic theory, it has its basis in the natural overtone series - the real discrepancies being averaged out in the approximately 2 cent error in each fifth as they occur. i also believe that it is easy to stay long enough in the microtonal nature of the harmonic series for nothing at all to make sense - in other words, we must discover how best to apply the series for our purposes, which i will assume to be the composition, improvisation, and analysis of music.
here is an excerpt from posts at AAJ in which these things were brought up in which you may see more of my perspective. forgive that it is quite lengthy and was not directly written to address your concerns, but that of others at AAJ:
begin paste of repost from AAJ
"The half-step is described in classical harmonic theory as an ideal frequency ratio of 16/15, which is approximately equal to 1.12 half-steps. This means that an equal-tempered instrument will play a half-step which is approximately 0.12 half-steps flat, compared to the ideal frequency ratio of 16/15. 0.12 half-steps is not a large error compared to that of other intervals, most notably thirds and sixths; therefore, one may argue that the perfect harmonic ratio of the half-step is not grossly distorted when playing them on equal-tempered instruments.
Since the sixteenth harmonic is several perfect octaves above the fundamental tone (root), it may be deduced that the "interval tonic" of a half-step or "minor second" is in fact, the upper tone. I believe what Russell is trying to say is that the Lydian scale has a natural tendency to resolve to its fifth degree, because of the half-step between the fourth and fifth tones; and its eighth (or first) degree, because of the half-step between the seventh and eighth (or first) tones. Both tones resolved by half-step in this manner are chord tones of the Lydian scale's tonic triad; therefore, the Lydian mode can be said to "resolve to its tonic triad", in other words, to itself.
Classical music theory describes resolution of the Major scale (Ionian mode) in terms of the Dominant to Tonic (V7 I) chord progression. The tritone (third and seventh) of the V7 chord (B and F for G7 in the key of C major) resolve to the root and third (C and E) of the tonic triad:
F
.......E
.......C
B
or
.......C
B
F
.......E
I was attempting to show how the diminished fifth interval of the V7 chord in the key of C resolves inward in the first example and how the augmented fourth interval of the V7 chord in the key of C resolves outward in the second example; I hope the spacing was not unclear (the tritone may be described as either an augmented fourth or a diminished fifth, depending upon the voicing of the chord).
What I am trying to point out is that the Ionian mode resolves to its tonic triad through resolution from its V7 chord (chords in groups, in motion through time), as opposed to the Lydian mode, which resolves to itself (a major chord in isolation, at rest).
Nowhere in Russell's writings does he advocate the abandonment of the Ionian mode Major scale. You have to get several chapters into it before seeing this: He is describing the Major (Ionian) scale as being used for chords in motion, for chord progressions - he calls this "Horizontal Tonal Gravity"...
He is describing the Lydian scale as being used for a major chord in isolation, by itself - he calls this "Vertical Tonal Gravity"...
I must point out that I am a student of the Lydian Chromatic Concept; I am not an authorized or "official" instructor of it. I am simply calling it as I see it. There is more that I may post in the future, to hopefully provide answers to those who have questions and clear up misunderstandings. It is not; however, my intention to teach the LCC online. That would be inappropriate, since I sell books too, and I sure as heck wouldn't sell any if I posted the whole text here and now for all to see, would I...
It is also not my intention to sell books for Russell either, only to suggest that if you are intrigued by these thoughts, to investigate further. I know that some of the first questions will be, "what new is offered here in terms of actual musical practice, what of the other chord types, or of counterpoint, voice leading, etc..." - One thing at a time, folks. I've posted a lot here to chew on, now that the dialectic has calmed down some and please, everybody, no more "slappy fights"! It's boring...
But to end on a positive note, I look forward to hearing the thoughts, questions, commentary, backtalk, and myriad perspectives of the participants of this forum. There are some very smart people out there, in "both camps"; let's keep it upliftng.
Db"
begin part two of two - repost from AAJ
"Let me try to answer your question this way. Vertical Tonal Gravity (VTG for short) is further defined as: melody derived from the chord, at "that moment in time". Melodies derived from VTG are not "concerned" with the previous chord or the next. The closest analogy would be the familiar concept of a "chordscale", where the tonal environment is dependent entirely upon the chord of the moment. Of course, in the rapidly changing harmonic environment of Be-Bop, this would be a difficult proposition to say the least; to change the scale of choice for every single chord would be navigable for only the players with the most fine-tuned skills...
This is in opposition to the idea of Horizontal Tonal Gravity or HTG, where the "tonic stations" or chord(s) of resolution are what define the melodic material. Em7 A7 Dm7 G7 Cm7 F7 Bbmaj7 for example has a final resting point on Bb - HTG allows us to play scales such as Bb major or Bb Blues over the entire progression. But look closer: Em7 A7 would normally resolve to D major, so how about a melody consisting of elements from D major resolving at some point to Bb, or C major (because of the resolving tendency of Dm7 G7) to Bb, or D to C, and finally Bb. There are many choices.
If alterations of the chords of "tension" (non-tonic) are present, this changes nothing, as HTG is concerned with where chords are "going", rather than where they "are", as with VTG.
In a sense, nothing "new" has appeared here - we are either reacting to chords one at a time (VTG) or in logical groups of resolution (HTG). Russell provides a concise list of scales for use in VTG, and a different list for HTG, although they are really part of the same "larger" list. All of the scales are listed in a very orderly manner which allows the player to choose from the most "inside" to the most "outside" sounds. This is where the "new" melodic ideas come in:
Russell's list of "Principal" scales provide me with ideas that I just plain wouldn't have thought of, and the list is not really that large. The fourth edition of the LCC (published in 2001) describes eleven scales for use with both VTG and HTG. All of the commonly used jazz scales are described, albeit with different language, but what is very interesting to me is that some unexpected cases show up (such as the Ab whole-tone scale played against an A minor chord)
As well as the fact that the process for transposing this "list of principal scales" for use with VTG in particular is, in a sense, quite simple. I may someday confess to both you and Russell (but perhaps not today) that I use a list that is ever so slightly different to suit my needs, but I believe that the premise for his method is elegant and compelling. It is difficult language for the novice, as you say. Mick Goodrick described the LCC as an example of "derivative" thinking - for those who don't have Goodrick's book, I think he was talking about "parent scales" as the source of modes...
In closing, I will remind everyone that the views expressed here are that of my current understanding, and therefore may be totally wrong. And I most certainly do not speak for Russell. Thanks again, Bill.
Db
p.s. I think that the paraphrasing you cited was from the now out of print 1959 edition and is not included in the 2001 fourth edition. I do not know if its omission is significant.
You had other questions upon which I will contemplate, and I'm sure others will chime in."
end paste of repost from AAJ...
i truly hope that i have given us much to contemplate. do not hesitate to tell us what you think, as i'm sure you will; but please do not get too hung up on the charts in the book as they are not the substance of it. rather, the core of the LCC is "tonal gravity" - and every harmonic-based theory i've ever seen states that the tonic of an interval of a fifth is its lower tone, and this is the basis of the LCC. however, let me repeat that i am not in any way an official voice of the LCC, merely a student thereof, and like you jeff, a performer, author, and an instructor of music.
db
ps - "the chart on page 130"??? - my copy doesn't have a chart on this page, lemme know which one you're referring to...
first let me state that i don't believe that mathematics is the ultimate solution to the inherent discrepancies of harmonic-based theory, other than the establishment of fixed pitches (more or less) within our current twelve-tone equal-temperament system. the system is not perfect, but it is a good system, which allows the freedom to transpose to any of the twelve tonal centers without worry of whether or not harmonic intervals are better or worse in the new key, for they are all equal. i believe that it is what our ears tell us that will decide what is best; therefore, in the end we may simply agree to disagree. the following is mere opinion, completely my own, and is not meant to dismiss any of your astute observations. i, for one, am a champion of equal temperament for many reasons, some of which are defined here - and if we agree that equal-temperament is an acceptable solution to these inherent discrepancies, then perhaps at least some of any apparent disagreement we may have will be rendered moot...
respectfully, i honestly have no wish to quarrel with you, but some of your math is off. not all, but you simply cannot use differences in Hz and percentage of frequencies to define intervallic relationships in music, since they are linear measurements of a logarithmic curve. the measurement of intervals in cents (1/1200 of an octave) is well established in the harmonic theories of hindemith and rameau, as well as many others. furthermore, i have read your pdf and have found it interesting; however, i cannot help believing that you are looking at an elephant through a microscope, to borrow our "elephants in the forest" vision previously mentioned. follow me here:
perfect fourth = 4/3 = 4.98 half-steps above fundamental, rounded to the nearest cent, or 498 cents, 4.980449991 half-steps to ten digits:
4/3 = 2^(F/12)
F = (12 log (4/3))/log 2
F= 4.98...
alternate source: see new harvard dictionary of music.
eleventh harmonic = 11/8 = 5.51 half-steps above fundamental, rounded to the nearest cent, or 551 cents, 5.513179424 half-steps to ten digits:
11/8 = 2^(F/12)
F = (12 log (11/8))/log 2
F= 5.51...
and
5.513179424
minus
4.980449991
equals
0.532729432
half-steps from a harmonically pure perfect fourth (4/3) and the eleventh overtone (11/8), clearly more than a quarter-step.
you said:
"440Hz X 11 = 4840Hz (the 11th overtone)
Tempered 8D = 4698.64Hz (a distance of 141.36Hz away from 4840Hz)
Tempered 8D# = 4978.03Hz (a distance of 138.03Hz away from 4840Hz)
This means that tempered 8D# is only 3.333Hz closer than tempered 8D to the 11th overtone of A440.
The difference of 3.33Hz when compared to a frequency of 4840Hz is not only not discernible to the human ear, but represents only a 0.069% difference (not 1.00% [1%= 1 cent]).
In addition, the following proof shows that the Natural D is much closer to the Natural 11th overtone of A than either the natural D# OR the tempered D#:
One can find a natural D by either calculating one perfect 5th below A, or by calculating eleven perfect 5ths above A.
Eleven perfect 5ths above A = D4757.365 (a distance of 82.635Hz away from 4840Hz)
One perfect 5th below A = D4693.333 (a distance of 146.667Hz away from 4840Hz)
The average distance of these two natural D’s from the 11th overtone (4840Hz) is 114.651Hz"
difference in Hz is a linear measurement, jeff. when you state "difference of 3.33Hz", this is a linear (first degree equation) measurement, and "0.069% difference" of Hz, also a first degree equation.
furthermore, calculating eleven perfect fifths (3/2)^11 above the fundamental when compared to a perfect fifth below (4/3) introduces an error known as the "pythagorean comma", an error of over 23 cents (more precisely 0.234600104 half-steps) which is perhaps the most convincing evidence that "whoever discovered, created, or otherwise implemented twelve-tone equal temperament has saved us from a whole lot of trouble."
jeff, i am pleased that you took the time to look at all this and i do not want to give you the impression that i am dismissing your observations out of hand. to the contrary, i look forward to what is likely to be a spirited and illuminating discussion.
i believe the LCC to be based on equal temperament through the circle of fifths and that analysis of it to be most appropriate in this domain; however, like traditional harmonic theory, it has its basis in the natural overtone series - the real discrepancies being averaged out in the approximately 2 cent error in each fifth as they occur. i also believe that it is easy to stay long enough in the microtonal nature of the harmonic series for nothing at all to make sense - in other words, we must discover how best to apply the series for our purposes, which i will assume to be the composition, improvisation, and analysis of music.
here is an excerpt from posts at AAJ in which these things were brought up in which you may see more of my perspective. forgive that it is quite lengthy and was not directly written to address your concerns, but that of others at AAJ:
begin paste of repost from AAJ
"The half-step is described in classical harmonic theory as an ideal frequency ratio of 16/15, which is approximately equal to 1.12 half-steps. This means that an equal-tempered instrument will play a half-step which is approximately 0.12 half-steps flat, compared to the ideal frequency ratio of 16/15. 0.12 half-steps is not a large error compared to that of other intervals, most notably thirds and sixths; therefore, one may argue that the perfect harmonic ratio of the half-step is not grossly distorted when playing them on equal-tempered instruments.
Since the sixteenth harmonic is several perfect octaves above the fundamental tone (root), it may be deduced that the "interval tonic" of a half-step or "minor second" is in fact, the upper tone. I believe what Russell is trying to say is that the Lydian scale has a natural tendency to resolve to its fifth degree, because of the half-step between the fourth and fifth tones; and its eighth (or first) degree, because of the half-step between the seventh and eighth (or first) tones. Both tones resolved by half-step in this manner are chord tones of the Lydian scale's tonic triad; therefore, the Lydian mode can be said to "resolve to its tonic triad", in other words, to itself.
Classical music theory describes resolution of the Major scale (Ionian mode) in terms of the Dominant to Tonic (V7 I) chord progression. The tritone (third and seventh) of the V7 chord (B and F for G7 in the key of C major) resolve to the root and third (C and E) of the tonic triad:
F
.......E
.......C
B
or
.......C
B
F
.......E
I was attempting to show how the diminished fifth interval of the V7 chord in the key of C resolves inward in the first example and how the augmented fourth interval of the V7 chord in the key of C resolves outward in the second example; I hope the spacing was not unclear (the tritone may be described as either an augmented fourth or a diminished fifth, depending upon the voicing of the chord).
What I am trying to point out is that the Ionian mode resolves to its tonic triad through resolution from its V7 chord (chords in groups, in motion through time), as opposed to the Lydian mode, which resolves to itself (a major chord in isolation, at rest).
Nowhere in Russell's writings does he advocate the abandonment of the Ionian mode Major scale. You have to get several chapters into it before seeing this: He is describing the Major (Ionian) scale as being used for chords in motion, for chord progressions - he calls this "Horizontal Tonal Gravity"...
He is describing the Lydian scale as being used for a major chord in isolation, by itself - he calls this "Vertical Tonal Gravity"...
I must point out that I am a student of the Lydian Chromatic Concept; I am not an authorized or "official" instructor of it. I am simply calling it as I see it. There is more that I may post in the future, to hopefully provide answers to those who have questions and clear up misunderstandings. It is not; however, my intention to teach the LCC online. That would be inappropriate, since I sell books too, and I sure as heck wouldn't sell any if I posted the whole text here and now for all to see, would I...
It is also not my intention to sell books for Russell either, only to suggest that if you are intrigued by these thoughts, to investigate further. I know that some of the first questions will be, "what new is offered here in terms of actual musical practice, what of the other chord types, or of counterpoint, voice leading, etc..." - One thing at a time, folks. I've posted a lot here to chew on, now that the dialectic has calmed down some and please, everybody, no more "slappy fights"! It's boring...
But to end on a positive note, I look forward to hearing the thoughts, questions, commentary, backtalk, and myriad perspectives of the participants of this forum. There are some very smart people out there, in "both camps"; let's keep it upliftng.
Db"
begin part two of two - repost from AAJ
"Let me try to answer your question this way. Vertical Tonal Gravity (VTG for short) is further defined as: melody derived from the chord, at "that moment in time". Melodies derived from VTG are not "concerned" with the previous chord or the next. The closest analogy would be the familiar concept of a "chordscale", where the tonal environment is dependent entirely upon the chord of the moment. Of course, in the rapidly changing harmonic environment of Be-Bop, this would be a difficult proposition to say the least; to change the scale of choice for every single chord would be navigable for only the players with the most fine-tuned skills...
This is in opposition to the idea of Horizontal Tonal Gravity or HTG, where the "tonic stations" or chord(s) of resolution are what define the melodic material. Em7 A7 Dm7 G7 Cm7 F7 Bbmaj7 for example has a final resting point on Bb - HTG allows us to play scales such as Bb major or Bb Blues over the entire progression. But look closer: Em7 A7 would normally resolve to D major, so how about a melody consisting of elements from D major resolving at some point to Bb, or C major (because of the resolving tendency of Dm7 G7) to Bb, or D to C, and finally Bb. There are many choices.
If alterations of the chords of "tension" (non-tonic) are present, this changes nothing, as HTG is concerned with where chords are "going", rather than where they "are", as with VTG.
In a sense, nothing "new" has appeared here - we are either reacting to chords one at a time (VTG) or in logical groups of resolution (HTG). Russell provides a concise list of scales for use in VTG, and a different list for HTG, although they are really part of the same "larger" list. All of the scales are listed in a very orderly manner which allows the player to choose from the most "inside" to the most "outside" sounds. This is where the "new" melodic ideas come in:
Russell's list of "Principal" scales provide me with ideas that I just plain wouldn't have thought of, and the list is not really that large. The fourth edition of the LCC (published in 2001) describes eleven scales for use with both VTG and HTG. All of the commonly used jazz scales are described, albeit with different language, but what is very interesting to me is that some unexpected cases show up (such as the Ab whole-tone scale played against an A minor chord)
As well as the fact that the process for transposing this "list of principal scales" for use with VTG in particular is, in a sense, quite simple. I may someday confess to both you and Russell (but perhaps not today) that I use a list that is ever so slightly different to suit my needs, but I believe that the premise for his method is elegant and compelling. It is difficult language for the novice, as you say. Mick Goodrick described the LCC as an example of "derivative" thinking - for those who don't have Goodrick's book, I think he was talking about "parent scales" as the source of modes...
In closing, I will remind everyone that the views expressed here are that of my current understanding, and therefore may be totally wrong. And I most certainly do not speak for Russell. Thanks again, Bill.
Db
p.s. I think that the paraphrasing you cited was from the now out of print 1959 edition and is not included in the 2001 fourth edition. I do not know if its omission is significant.
You had other questions upon which I will contemplate, and I'm sure others will chime in."
end paste of repost from AAJ...
i truly hope that i have given us much to contemplate. do not hesitate to tell us what you think, as i'm sure you will; but please do not get too hung up on the charts in the book as they are not the substance of it. rather, the core of the LCC is "tonal gravity" - and every harmonic-based theory i've ever seen states that the tonic of an interval of a fifth is its lower tone, and this is the basis of the LCC. however, let me repeat that i am not in any way an official voice of the LCC, merely a student thereof, and like you jeff, a performer, author, and an instructor of music.
db
ps - "the chart on page 130"??? - my copy doesn't have a chart on this page, lemme know which one you're referring to...
-
Jeff Brent
- Posts: 10
- Joined: Sat Oct 18, 2008 4:13 pm
- Location: L.A.
I meant the chart Page 230 - (sorry for the typo).
~
If (as motherlode says) the glaring inaccuracies of the "Overtone Series" chart on page 2 example I:3 are a "time worn" issue, that strongly implies that GR is completely aware that that pg2 exI:3 chart contains significant errors, yet chooses to prominently include these errors in his work anyway.
Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda.
One would think that if GR were aware of these discrepancies, the honorable course of action would have been to delete all references to said chart.
~
If, however, GR is unaware of the inaccuracies in this chart, then this can hardly be a "time worn" issue.
And someone would need to bring this to his attention.
~
If (as motherlode says) the glaring inaccuracies of the "Overtone Series" chart on page 2 example I:3 are a "time worn" issue, that strongly implies that GR is completely aware that that pg2 exI:3 chart contains significant errors, yet chooses to prominently include these errors in his work anyway.
Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda.
One would think that if GR were aware of these discrepancies, the honorable course of action would have been to delete all references to said chart.
~
If, however, GR is unaware of the inaccuracies in this chart, then this can hardly be a "time worn" issue.
And someone would need to bring this to his attention.
"I meant the chart Page [b]2[/b]30 - (sorry for the typo).
~
If (as motherlode says) the glaring inaccuracies of the "Overtone Series" chart on page 2 example I:3 are a "time worn" issue, that strongly implies that GR is completely aware that that pg2 exI:3 chart contains significant errors, yet chooses to prominently include these errors in his work anyway.
Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda.
One would think that if GR were aware of these discrepancies, the honorable course of action would have been to delete all references to said chart.
~
If, however, GR is unaware of the inaccuracies in this chart, then this can hardly be a "time worn" issue.
And someone would need to bring this to his attention."
on the first page of your pdf you stated, "As instructed, I entered into this with an open mind, hoping to find the inner truth within this concept." and you stated above several phrases, such as "the glaring inaccuracies of the "Overtone Series" chart on page 2" and "Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda."
can you honestly state that you are conversing here with an open mind?
let's see:
1) "as instructed" - instructed by whom? can you honestly state that your posts do not appear to have an agenda?
2) when motherlode says "time worn", perhaps you are unaware that this argument has been going on at AAJ in several threads for well over a year now...
3) here is the math behind the chart you speak of, one overtone at a time, beginning with the fundamental, and then continuing with sequential overtones:
1 - 0.00 half-steps above fundamental
2 - 12.00 half-steps above fundamental
3 - 7.02 half-steps above first octave
4 - 12.00 half-steps above first octave
5 - 3.86 half-steps above second octave
6 - 7.02 half-steps above second octave
7 - 9.69 half-steps above second octave
8 - 12.00 half-steps above second octave
9 - 2.04 half-steps above third octave
10 - 3.86 half-steps above third octave
11 - 5.51 half-steps above third octave
12 - 7.02 half-steps above third octave
13 - 8.41 half-steps above third octave
14 - 9.69 half-steps above third octave
15 - 10.88 half-steps above third octave
16 - 12.00 half-steps above third octave
17 - 1.05 half-steps above fourth octave
18 - 2.04 half-steps above fourth octave
19 - 2.98 half-steps above fourth octave
20 - 3.86 half-steps above fourth octave
and i see only one overtone interval that we agree that is questionable: the thirteenth (13) which is closer to Ab than A natural when C is used as the fundamental, and you refer to this as "glaring inaccuracies"???
perhaps mr. russell is aware of this single oversight, and perhaps not, but is the LCC based on either the eleventh or thirteenth harmonic? no, it is based upon the fifth. the chart says so, and mr. russell has said so.
"Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda."
you should go into politics. i would greatly enjoy discussing the substance of the LCC with you; however, if this is what you call "with an open mind," then one can only speculate as to what your "agenda" is.
if you are willing to drop the political talking points, i would be willing to discuss this further with you. let me know.
db
~
If (as motherlode says) the glaring inaccuracies of the "Overtone Series" chart on page 2 example I:3 are a "time worn" issue, that strongly implies that GR is completely aware that that pg2 exI:3 chart contains significant errors, yet chooses to prominently include these errors in his work anyway.
Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda.
One would think that if GR were aware of these discrepancies, the honorable course of action would have been to delete all references to said chart.
~
If, however, GR is unaware of the inaccuracies in this chart, then this can hardly be a "time worn" issue.
And someone would need to bring this to his attention."
on the first page of your pdf you stated, "As instructed, I entered into this with an open mind, hoping to find the inner truth within this concept." and you stated above several phrases, such as "the glaring inaccuracies of the "Overtone Series" chart on page 2" and "Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda."
can you honestly state that you are conversing here with an open mind?
let's see:
1) "as instructed" - instructed by whom? can you honestly state that your posts do not appear to have an agenda?
2) when motherlode says "time worn", perhaps you are unaware that this argument has been going on at AAJ in several threads for well over a year now...
3) here is the math behind the chart you speak of, one overtone at a time, beginning with the fundamental, and then continuing with sequential overtones:
1 - 0.00 half-steps above fundamental
2 - 12.00 half-steps above fundamental
3 - 7.02 half-steps above first octave
4 - 12.00 half-steps above first octave
5 - 3.86 half-steps above second octave
6 - 7.02 half-steps above second octave
7 - 9.69 half-steps above second octave
8 - 12.00 half-steps above second octave
9 - 2.04 half-steps above third octave
10 - 3.86 half-steps above third octave
11 - 5.51 half-steps above third octave
12 - 7.02 half-steps above third octave
13 - 8.41 half-steps above third octave
14 - 9.69 half-steps above third octave
15 - 10.88 half-steps above third octave
16 - 12.00 half-steps above third octave
17 - 1.05 half-steps above fourth octave
18 - 2.04 half-steps above fourth octave
19 - 2.98 half-steps above fourth octave
20 - 3.86 half-steps above fourth octave
and i see only one overtone interval that we agree that is questionable: the thirteenth (13) which is closer to Ab than A natural when C is used as the fundamental, and you refer to this as "glaring inaccuracies"???
perhaps mr. russell is aware of this single oversight, and perhaps not, but is the LCC based on either the eleventh or thirteenth harmonic? no, it is based upon the fifth. the chart says so, and mr. russell has said so.
"Using these falsehoods as justification points, basically boils down to promoting lies and half-truths to further his agenda."
you should go into politics. i would greatly enjoy discussing the substance of the LCC with you; however, if this is what you call "with an open mind," then one can only speculate as to what your "agenda" is.
if you are willing to drop the political talking points, i would be willing to discuss this further with you. let me know.
db
-
Jeff Brent
- Posts: 10
- Joined: Sat Oct 18, 2008 4:13 pm
- Location: L.A.
In the first few lines of the Foreword, it is stated "... it is strongly suggested that you give these ideas your complete openness and attention and ... let go of your preconceptions of the theoretical foundations of Western music."dogbite wrote:"as instructed" - instructed by whom? can you honestly state that your posts do not appear to have an agenda?
I have no problem letting go of my preconceptions of "the theoretical foundations of Western music". During my lengthy travels around the globe I have been exposed first-hand to many types of non-Western music in Africa, the Middle East, the Indian subcontinent and the Far East.
However, I do have a problem with letting go of the laws of physics. If perhaps Mr Wasserman had stated in his foreword to the fourth edition of the LCC that I should completely suspend belief in the laws of physics, I might have had an easier time swallowing the basic premise that the #4 is the natural child of the overtone series.
To me these inaccuracies are glaring, but seemingly not to any one else here.dogbite wrote: I see only one overtone interval that we agree that is questionable: the thirteenth (13) which is closer to Ab than A natural when C is used as the fundamental, and you refer to this as "glaring inaccuracies"???
The lie here is that the 13th overtone is a major 6th. And even though it is closer to being a b6th, it is NEITHER A b6th NOR a major 6th.
The half-truth here is that the 11th overtone is an augmented fourth. One eensy-weensy 1/100 of a semi-tone is not enough evidence to convince any rational mind that the 11th overtone is in actuality an augmented 4th.
The 11th overtone is NEITHER a perfect fourth NOR an augmented fourth.
The Overtone Series chart (pg 2 ex I:3) is a chart that is found reproduced in many many music books. Mr Russell simply reprinted this chart in his book and obviously believed it to be accurate.dogbite wrote:perhaps Mr. Russell is aware of this single oversight, and perhaps not, but is the LCC based on either the eleventh or thirteenth harmonic? no, it is based upon the fifth. the chart says so, and Mr. Russell has said so.
In this case, his only initial guilt is naively using flawed data to support his premises without checking the facts first.
However, if he IS aware of the flaws in said chart, then he is guilty of propagating falsehoods.
And how could he not be aware of the flaws in that chart? The book has been around in one form or another for 55 years, and it is incredibly hard to believe that one of the hundreds of intelligent and inquisitive minds who have read the book wouldn't have spotted this and pointed it out to him.
~
Dogbite,
I thank you for taking the time to read through my pdf (whether or not you agree with any or all of it). It is nice to know that somebody has given it more than just a passing glance.
Since you have read that article, you already know that scales are formed via simple radial symmetry of consonants and not via uni-directional ascending fifths (GR's hypothesis).
Which could be seen as an "agenda".
For the benefit of those who have not seen the article on radial symmetrical scales and its consequence "the derivation of radial symmetrical altered scales", I intend to begin a thread entitled "How does the tonic imply the #4?"
But I want to see this thread end of its own volition first, as I consider the above to be a more or less separate topic.