Pentatonic - an idle Sunday question

[hans]

The division of an octave into twelve equal steps (12 tone equal temperament, or 12-TET) is to my understanding not arbitrary, but artificial.
...........

That's a kindofa oxymoronic point, mate.

If one chooses an artificial thing, one has made an arbitrary choice.

No. I meant arbitrary as in:
"Determined by chance, whim, or impulse, and not by necessity, reason, or principle"
(http://www.thefreedictionary.com/arbitrary)

There is reason and principle in twelve tone equal temperament, but it does not occur naturally and it is an artificial construct.

[david_h]

What puzzles me is how F (the fourth) is derived at. It is not present in the series of harmonic frequencies. C is obviously the fifth above F. But playing just a C does not produce a F. So F seems to be a construction, perhaps born out of desire to modulate. Has anyone got thoughts on this?

Good morning hans, I now have my spreadsheet. I think the answer is along the lines of talasiga's response to that. It is not that the G is 'in' the C and the F is not. It is that several harmonics of the two notes match.

Take a C and the C an octave above: the successive harmonics of the higher note are the alternate, even harmonics of the lower note.
Take a C and the G above on the same octave: Harmonics 2, 4, 6, 8... of the G are harmonics 3, 6, 9, 12 ... of the C
For a C and the F above: Harmonics 3, 6, 9, 12 ... of the F are harmonics 4, 8, 12, 16 .. of the C

As the integer ratios get larger the number of harmonics that match get less. By the time it gets to the second in the scale there are not many that match (in either of the two 'options'). The pentatonic scales (see, I am getting back on topic) are made up of notes that share a quite a few harmonics, hence their harmoniousness.

It is not so much that the notes of the scale are 'in' the harmonics, making F a problem, but that some of the harmonics of the notes in the scale are amongst some of the harmonics in the C, and rather a lot of them for F. Although this is more obvious to our ears when we hear the notes together it influences how we hear them in melody.

Helmholtz went into it in great detail, including scales other than the western one and I understand that most of his work still stands. The only vaguely relevant link I have to hand is this one http://www.archive.org/stream/onsensati ... 3/mode/1up (to a page that is not legible in the facsimile version that is still in print)

[hans]

David, I do not see talasiga's response to my questions, but I can see where you are driving at:

Do you mean that F should be admitted to the major C scale on strength of its strong harmonicity with C? I can understand that approach, since going a fifth down will result in exactly this very strong harmonicity between F and C. But then could we not go down a fifth further to Bb, to find strong harmonicity again, and lay claim to Bb as part of diatonic just C major likewise? Or perhaps the case only applies to F, one fifth down (or a fourth up), since within the harmonics of F there is the major triad of CEG, which is the strongest triad found in the note of C. Bb only contains D and C, but no corresponding triad with C.

Contrary to you I see that G is in C: it is the third harmonics, and all multiples of it are also part of the arithmetic series of the harmonic frequencies of C. It is clearly present in the tone (caveat: the clearness depending on how strong the harmonics are, which is instrument dependant). And this is true for all the other notes found as pure harmonics: E D B and A.

That is the reason I got excited to realise the presence of a pentatonic scale within each tone: CDEGBC for C, or raaga Hamsadhwani, which is considered to be a very "upbeat", happy raaga, from what I read googling around, and which I feel when improvising within this mode. I imagine this is because of the strong major triads, and the missing fourth and sixth, which cuts out going into D minor. (On D whistle: DEF#AC#D with DF#A and AC#E major, and no EGB minor triad, no going to E minor. Or alternative on D whistle, on G: GABDF#G)

[david_h]

Hans. Maybe I was just thinking that the general idea of talasiga's post after mine was more than way I was looking at it.

I didn't mean to say that the G was not in the C, more that if we look at the harmonics of a note in the scale and those in the root then something other than one being in the other becomes more prominant and 'explains' the fourth. Some harmonics match exactly (with just intonation) and most of the others don't clash.

Another way of looking at it is that not only is the fundemental of the G in the C, but many other harmonics are as well. For the F the fundamental is not there, but many of the harmonics are, more than for most other notes in the scale.

I think Helmholtz's discussion of consonance and dissonance, the way he used the terms, is enough for me to feel content that there is an explanation along those lines. Although psychoacoustics has come a long way since his time he was one of its founders. The physics and maths of it still works but the physiology and psychology have progressed. His explanation of dissonance as due to a percieved 'roughness' of the sound caused by the beating of harmonics is still mentioned in texts.

One aspect of beats that is not often mentioned explicitly is that if two notes differ by 1Hz the fundamental beats at 1Hz but the harmonics beat at progressively faster frequencies. As the frequency difference of the fundamental increases we go from something that may be OK in a tremelo harmonica (or two whistlers ) if you like that sort of thing to something that would be regarded as a clash in most situations. The same things are happening to some shared harmonics when other pairs of notes ina a scale are not in tune, and more so for pairs of notes one of which is not in the scale; the illustration I linked above is Helmholtz's 'visualisation' of that.

The part where Helmholtz is a bit dissapointing for me is that he seems to writing for a readership mainly interested in harmony, so the reasons why it is relevant in melody gets a brief mention as some sort of memory effect.

[talasiga]

Hans, thanks for clarifying you use of "arbitrary".

Again, at a tangent, but addressing what I imagine predicates a lot of the recent discussion:-
TET 12 notes is the limitation (from the microtonal/JI perspective) wholly attached to
keyboards only
and perhaps only partially attached to some set key mechanism instruments (eg, Boehm flute) and fretted instruments whose strings cannot be easily bent.

Does not really limit fiddle instruments (violin, viola, sarangi) and even diatonic tone hole instruments such as bansuri and Irish whistle which attract a niche tradition of SLIDING into notes and, of course, the supreme instrument, the human voice .....
(Include in this latter list, the Middle Eastern lute, the sarod and the sitar with its movable frets etc).

Even on my latest flute addition -Boehm, one can get shades by rolling the mouth (if you knoew what I mean).

[highland-piper]

Frequences are multiples of 32 giving a "scientific" middle C of 256Hz. I've just choosen this base frequency to have some convenient whole numbers. I used note names corresponding to abc convention to mark the octaves. I left out names for notes which are not part of the common major scale (harmonics 7, 11, 13, 14, 17, 19).

Interestingly, the 7th harmonic is used in modern highland piping to tune the note G, which would be Bb in the key of C (i.e., a flat 7th). It's 33 cents flat, as compared to ET. In the past they sometimes used a 9:5 G, which would be out of tune with the drones.

What puzzles me is how F (the fourth) is derived at. It is not present in the series of harmonic frequencies. C is obviously the fifth above F. But playing just a C does not produce a F. So F seems to be a construction, perhaps born out of desire to modulate. Has anyone got thoughts on this?

I don't think anyone ever got any of these through a direct application of the theory, but through the practice of playing one note against another. If you have a C drone, the F pops out. Funny thing though, until fairly recently there were two different fourths used in highland pipes. The one that's in tune, and also a sharper one that's out of tune with the drones, but a proper just interval -- 27:20

[hans]

The part where Helmholtz is a bit disappointing for me is that he seems to writing for a readership mainly interested in harmony, so the reasons why it is relevant in melody gets a brief mention as some sort of memory effect.

Well that's what it is, from a sensory perspective, no? A note is played, then another. They do not sound at the same time, so they do not combine to form consonances or dissonances. All we have is the short term memory, fresh in the mind, of the previous note, and the sensation of the current note, and we perceive harmony or disharmony from what we hear and what is fresh in the mind, as memory. And as the melody progresses the first note retreats further and further in this short term memory, and becomes more and more irrelevant to take part in a comparison of what one hears at present and what has just passed. I expect that pure intervals of high harmonicity like a fifth and a fourth in relation to the present note will be felt that little much longer, and so will chords, which amplify the harmonicity. But I am speculating.

Thanks for the link to Helmholtz, interesting diagrams and reading!

[david_h]

Yes, my dissapointment was that if he had discussed it more his thoughts would have been fascinating. The memories don't just fade though, they leave us expecting things and wondering what comes next etc.

Not always just memory though. On the radio this morning there was a piece about the Greek money crisis that included sound of singing in an orthodox church - the reverb time was so long it took some time for me to decide that it might just be one person singing/chanting

[hans]

Yes, my disappointment was that if he had discussed it more his thoughts would have been fascinating. The memories don't just fade though, they leave us expecting things and wondering what comes next etc.

Helmholtz does mention this a bit when writing about scales and the development of the sharp seventh "leading" note: how we come to expect to hear the octave note rather than cringing about some dissonances introduced with the seventh, in particular in relationship to the preceding notes (cringing is my expression here ).