Q about Law of Octaves

I'm hoping that someone, maybe Ark, could help me to understand the theoretics behind The Law of Octaves as it is presented in ISOTM. I've been reading the pages 123-125 over and over again, trying to understand it precisely, but I'm kind of stuck at a few points that may sound trivial but which I really would like to understand:

G says:

"The principle of dividing into eight unequal parts the period, in which the vibrations are doubled, is based upon the observation of the non-uniform increase of vibrations in the entire octave, and separate 'steps' of the octave show acceleration and retardation at different moments of its development. [...] In very remote times one of the schools found that it was possible to apply this formula to music. In this way was obtained the seven-tone musical scale which was known in the most distant antiquity, then forgotten, and then discovered or 'found' again" (ISOTM, 124).

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So, basically in the figures 7 and 8 on page 124 (the horizontal axis between frequencies f1 and (2xf1) there is a time component embedded? I mean, one can't (at least in 3D) measure acceleration without time dependency? I guess one could represent the process with to axes: (f) being vertical and (t) being horizontal.
So, if I understand this correctly, the 8 points were originally not chosen on the basis of consonance (music). Instead they are measurement points, like samples, of the acceleration of the frequency, am I wrong? And between every check point (do, re, mi...) the 'time gap' (delta t) is the same, am I completely off track here? What does the ratios 9/8, 10/9, 16/15 actually represent, in units?

So between (mi)-(fa), and (si)-(do) the frequency hasn't increased/decreased as much as at the other intervals - in the same amount of time (delta t)? So basically I've been wondering if G. did choose exactly eight check points because it was easier to present that way, or if there was another reason? I mean the 'sample rate' could have been e.g. much higher?

I tried also to find any research on this phenomenon, but so far without real success - which actually wasn't that surprising. Has there been any attempts to scientifically prove this discontinuity of vibration or is the phenomenon 'unmeasurable'?

Sorry for the many and maybe unclear questions!

Aragorn said:

I tried also to find any research on this phenomenon, but so far without real success - which actually wasn't that surprising. Has there been any attempts to scientifically prove this discontinuity of vibration or is the phenomenon 'unmeasurable'?

Sorry for the many and maybe unclear questions!

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according to my extremely hazy understanding of the whole thing, it is also inately related to the different, and distinct, levels of 'food' whereby certain substances can be used by the body, which then can enter into different states, that can then further allow the use of higher substances, or not, depending on which way it goes. Depending on the particular use of the different substances at different steps, where the process can go in one of several ways (such as the different use of the energy of the sex centre which has only a possibility of being transmuted into higher substances, by the appropriate conscious application) this then allows a kind of progression with distinct steps, and the 2 'intervals' within the octave kind of represent those steps where the process meets 'resistance' and has the likelyhood of looping back round to a lower state. So, because these substances are objectively distinct this is related to the steps of the octave. This manifests in all kinds of ways in the real world, such that actions initially move forward based on an initial 'energy' but then inescapably meet these 'intervals' of resistance, whereby 'something' has to be done to keep going forward, otherwise the mechanical path takes over, and the initial action is diverted back round, so that objective progress is halted. So maybe it's measurable, but not from our current state of being unable to utilise the higher hydrogens? Sorry, that's all rather fuzzy, and I'm probably waay out. Without coming to a better understanding of the whole 'hydrogens' thing, which confuses the hell out o' me, I'm afraid I'm in speculation land.

Aragorn said:

So, basically in the figures 7 and 8 on page 124 (the horizontal axis between frequencies f1 and (2xf1) there is a time component embedded? I mean, one can't (at least in 3D) measure acceleration without time dependency? I guess one could represent the process with to axes: (f) being vertical and (t) being horizontal.

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Hi Aragorn, I think the law of octaves can be viewed more in terms of a "primordial cosmic law." As I understand it the law of octaves or the law of seven is in everything, at all levels of reality, and is represented both in the world of time (that determines how vibrations grow or decline in the temporal world) and is also represented within the eternal world such as in the color spectrum which is not really time dependent in the sense you mention above but the law determines the essential underlying “structure” of reality, such as, for example, the seven levels of the visible light spectrum, the seven densities, etc. So I think the law applies both horizontally and vertically and is always there and in everything at all levels of reality. At least that's how I understand it so far. In the book 'In Search Of The Miraculous' Gurdjieff speaks of it in terms of laws such as “The Law of Octaves” or “The Law of Seven” and in his book Beelzebub's Tales he refers to it as the "Law of Heptaparaparshinokh."

Thanks Nomad and Kenlee. I think I'm beginning to get the 'big picture' of the law. Maybe it's a waste of time to be hung up on these small details that I mentioned, but the engineer in me would like to 'get the numbers right'. Maybe it would have been enough if G had just stated what the law of seven/octaves is and how it can be found in everything, without the numerical examples. But since he did represent it in such detail it probably has some value and I'd like to understand the calculus right. I think I'll have to retrieve my physics and acoustics books and start reading...

Okay, after reading my old copy of a book called 'The Science Of Sound' by Thomas D. Rossing (very good book BTW), I feel pretty stupid. The thing that G goes on to 'calculate' is only the usual way a Just Scale is constructed. In case someone else should wonder about 'the numbers' here's what Rossing has to say:

The scale of just intonation (or just diatonic scale) is based on the major triad, a group of three notes that sound particularly harmonious (for example, C, E, G). The notes of the major triad are spaced in two intervals: a major third (C:E) and a minor third (E:G). When these intervals are made as consonant as possible, the notes in the major triad are found to have frequencies in the ratios 4:5:6.
[...]

Frequency Ratios n the Just Scale
This may be illustrated in the key of C as follows. First we let the notes of the tonic chord (C,E,G) have the ratios 4:5:6 and set C=1; we have now determined C=1, E=5/4, G=6/4=3/2. Next we let the notes of the dominant chord (G, B, D) be in ratio 4:5:6; this determines B=5/4 x 3/2=15/8 and D=3/2 x 3/2=9/4.Dropping D down into the same octave as C makes D=9/8. Now we require that the subdominant chord (F, A, C) likewise be in the ratio 4:5:6; in this case we work from C an octave up (C=2), and obtain F=2 / (3/2)=4/3 and A=2 / (6/5)=5/3. Putting this all together, we obtain the frequency ratios for the just scale in C-major: C=1, D=9/8, E=5/4, F=4/3, G=3/2, A=5/3, B=15/8, C=2.

Note that E is a major third above C, F is a perfect fourth above C, and G is a perfect fifth above C. Next consider the intervals between successive notes: D to E is (5/4) / (9/8)=10/9; E to F is (4/3) / (5/4)=16/15; G to F is (3/2) / (4/3)=9/8, and so on. In fact, if we write all the ratios as in Fig. 9.1 we observe that there are only three different intervals and they have ratios 9/8, 10/9, and 16/15. The interval corresponding to 9/8 is called a major whole tone, and the 16/15 interval a semitone. In the just scale, there are three major whole tones, two minor whole tones, and two semitones (Rossing 1990, 172-173).

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So that explained 'the numbers'. Then I noticed that on page 125 of ISOTM G. says:

"A study of the structure of the seven-tone musical scale gives a very good foundation for understanding the cosmic law of octaves"(Ouspensky 1949, 125).

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And on the next page he says:

"In this way the structure of the musical seven-tone scale gives a scheme of the cosmic law of 'intervals', or absent semitones. In this respect when octaves are spoken of in a 'cosmic' or 'mechanical' sense, only those intervals between mi-fa and si-do are called 'intervals'"(Ouspensky 1949, 126).

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So all this finally revealed to me that Gurjieff was only using the musical scale (the just scale) as a 'visualizing aid' because it fits so nicely to the concept. Because I tried to understand units and stuff taking the musical scale representation too 'literally' I got confused. And since G says that the appliance of the law of octaves to music became AFTER this law was known to ancient science, the number of 'check points' being 7 must have come from other knowledge. We can only guess what that knowledge was, OSIT.

Thanks for using this thread to clear things up in my rusty brain! :D

Yes, the way instruments are tuned is different in Western music now than a few hundred years ago. Now the octave is equally divided (equal temperament) into 12 chromatic tones, ie, half steps (aka semitones). Historically, certain intervals were desired to sound "good", ie, in tune- those being so-called "perfect" intervals like the fourth, fifth, and octave, and to a lesser extent thirds. What you will hear if octaves, fifths, and even a major chord are "in tune" according to just temperament (and actually, even today many musicians aim for this) is that another note seems to sound above the two or three notes played. It often feels/or sounds like a buzzing, and indeed another note is heard- higher up in the overtone series a note or notes will be in harmony with lower partials. Interesting to note that if two or more lower notes are sounded together in tune, a third (or more) higher tones, not necessarily the same letter name as the ones sounded, will occur. Networking for a common goal that neither of the original members could produce alone? Not sure if that's what G intended here but it is interesting to think about.

I agree with your final summary Aragorn that the musical scale was used here more as a familiar analogy than anything else although who knows, maybe it is symbolic of something else?

FWIW: If ya'll want to check out some music in other temperaments, look for music played on period/historical instruments, generally of the Renaissance and before. The Baroque period is when things finally started to settle in on equal temperament although there was much discrepancy for some time.

Hi Aragon
Warning, the following is only my opinion as food for thought, it is not verified fact.

Have you read how the law of seven is expounded in Beelzebubs Tales? It is quite different. In BTs Beelzebub states man has only become aware of two particularities on the Heptaparaparshinokh. And in ISOTM only 2 are expounded. Perhaps Ouspensky only received partial teaching as G's method at that time was to throw ideas and let students try to complete the gaps. In BT's Beelzebub introduces harnel aoot which for me completely changed my understanding of the law.
You are right to notice that the musical octave is a 'representation'. Example If I strike the note G on a piano , then G as a 'new Do' created by an 'external force' does not automatically proceed to the note A then to B then to C. G will remain G untill the vibrations peter out or untill I grab a piano key and tighten or loosen the string, therefore the musical octave is a medium through which the law could be 'represented'
But if you have a wish to observe the law of seven in action then observation of it's action as regard your own interaction with the world might be a good place to start. Eg I write an insulting post about you on this forum. Where does the Do start, with My posting or with your reading of the post?
How might the octave proceed from your registration that
1. I have insulted you
2. your cognisance (or not) of choice (conscious v mechanical) .
3. Your reaction internally and externally - (will you mechanically 'feed the moon' or provide your 'sun' with some 'fuel for light and heat' etc etc. Consider the role of intentional suffering in this transformation etc)

Thanks D Rusak and Stevie! I'm looking forward to read Beelzebub's Tales, when I'm ready for it.

Another question on the law of octave, relating to the human organism as explained on page 183 of ISOTM:

Why are the carbons (present in the organism) in the food, air and impression octaves not counted as part of the octave? As in why don't they get a note, only the passive 'oxygens' and their resultant 'nitrogens' ring a note.

I read on Wikipedia a long time ago that if you hear two notes with the ratio 5:6, that your ears will create a tone so that you hear 4:5:6. The information has been scraped off of Wikipedia so I can't point to it. It may be 3:4:5 instead, I don't remember it very well. It could be that it was removed because it was wrong; It's been hard to find every time I looked for it.

I just wanted to include that here in case it's important.

Parallel said:

Another question on the law of octave, relating to the human organism as explained on page 183 of ISOTM:

Why are the carbons (present in the organism) in the food, air and impression octaves not counted as part of the octave? As in why don't they get a note, only the passive 'oxygens' and their resultant 'nitrogens' ring a note.

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I think it has to do with the concept of 'carbons' being present in the organism already. The octave in particular being talked about is food, and its transformation by the organism. The 'frequency' is given, but not a note name.

Kris

monotonic said:

I read on Wikipedia a long time ago that if you hear two notes with the ratio 5:6, that your ears will create a tone so that you hear 4:5:6. The information has been scraped off of Wikipedia so I can't point to it. It may be 3:4:5 instead, I don't remember it very well. It could be that it was removed because it was wrong; It's been hard to find every time I looked for it.

I just wanted to include that here in case it's important.

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This phenomena, hearing a third pitch while two are sounding, is called a resultant pitch. I think it is more than just a psychological trick. When two different frequency pitches blend, the difference between them sounds as well. I haven't finished my research into this yet, but I strongly suspect that this aspect of music has a lot to do with what G called objective music. The math on calculating resultants is rather simple...basic arithmetic. If we take an octave starting at 24 vibrations the scale will be : do24 re27 mi30 fa32 sol36 la40 si45 do48. If we take sol and mi, the fifth and the third respectively, 36-30=6. do6 sounds two octaves lower than the octave in question. This ties in with a clue G gave in one of the lectures recounted in "Views from the Real World". To paraphrase: A man should have a total number of 30. All of his postures should number 30. If he has 12 thought postures and 8 postures of body, he must of necessity have 10 emotional postures. :the ratio given 8:10:12, for clarity let us triple it, becomes 24:30:36... The very frequencies (ratio wise) of do-mi-sol, or a major triad. I think this lecture was a hint of the proper balance of vibrations of the moving center-feeling center-thinking center.

Kris

Not exactly the same thing but I noticed some number connections with scales too.

There's 8 notes on a scale that sounds complete, and the last note is the same as the first just one octave up - i.e. if we start at C on a piano and play the scale, we end with the next higher C. If we stay on key and play the scale 1 octave higher so it sounds complete we have to start on C again (the higher one up from previous). So you could posit that the last C is more a part of the next octave, as the first note of an octave is the first defining note of the rest of the scale and can't be taken out (otherwise musically you'd expect a different scale is being played if one starts at the 2 note of the scale).

So then there are 7 distinguishable tonal degrees that make up a complete sounding scale. 7 has also been the number of perfection, or God's number too. "And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work, which God created and made." There's also 7 heavens in the Talmud. The Cs also say 7th density is union with the one, DCM etc.

Just an observation I was pondering recently. I know you can find number connections to really anything if you really want though.

alkhemst said:

Not exactly the same thing but I noticed some number connections with scales too.

There's 8 notes on a scale that sounds complete, and the last note is the same as the first just one octave up - i.e. if we start at C on a piano and play the scale, we end with the next higher C. If we stay on key and play the scale 1 octave higher so it sounds complete we have to start on C again (the higher one up from previous). So you could posit that the last C is more a part of the next octave, as the first note of an octave is the first defining note of the rest of the scale and can't be taken out (otherwise musically you'd expect a different scale is being played if one starts at the 2 note of the scale).

So then there are 7 distinguishable tonal degrees that make up a complete sounding scale. 7 has also been the number of perfection, or God's number too. "And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work, which God created and made." There's also 7 heavens in the Talmud. The Cs also say 7th density is union with the one, DCM etc.

Just an observation I was pondering recently. I know you can find number connections to really anything if you really want though.

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I just want to add to what you said that there are ascending scales, and descending scales - evolutionary and creative respectively. In either case, the resulting Do is contained within the initial Do and completes the cycle, while beginning a new one potentially.

An interesting observation made by Russel A. Smith in "Cosmic Secrets" is the overall structure of the octave is in miniature within the octave at the missing semitone intervals. Mi-Fa/Fa-Sol and Si-Do/Do-Re. Using the previous frequencies I cited, that would be 30 - 32/32 - 36. (2. 4. ). And 45 - 48/48 - 54. (3. 6. ). The increase doubles, just as succeeding octaves double, although the difference is pronounced.

Kris

RflctnOfU said:

I just want to add to what you said that there are ascending scales, and descending scales - evolutionary and creative respectively. In either case, the resulting Do is contained within the initial Do and completes the cycle, while beginning a new one potentially.

An interesting observation made by Russel A. Smith in "Cosmic Secrets" is the overall structure of the octave is in miniature within the octave at the missing semitone intervals. Mi-Fa/Fa-Sol and Si-Do/Do-Re. Using the previous frequencies I cited, that would be 30 - 32/32 - 36. (2. 4. ). And 45 - 48/48 - 54. (3. 6. ). The increase doubles, just as succeeding octaves double, although the difference is pronounced.

Kris

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It's sounds Asymptote ... If a frog jumps half the distance to a wall with every hop, is it possible for him to ever reach the wall? OR if an instrument continues to play C an octave higher with each subsequent note, is is possible for it to reach silence?