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Eastern Tone System (Was: Microtones)

This is the first section (without particular order) of my promised article. Don't hesitate to express comments.

Regards.
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NOTES ON EASTERN TONE SYSTEM
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I - FORMATION OF THE TONE SCALE by SUPERCEEDING FIFTHS:

This is a well-known and widely used practice; Pythagorean Method. It was also known and supported in Eastern theory. Now, we’re going to form a scale on D (Yegâh) going by perfect fifths. We will not "invent" the pitches; that’s we’ll presume the existance of 17 intervals per octave and try to find their "place" within this system.

Our way is rather simple. We will progress upward and downward (upward with perfect fourths; inversion of perfect fifths) and try to intersect both directions so that we will have a closed system, a circle.

By fifths, we get the following pitches:

D, A, E, B, F#, C#, G#, D#, A#

And the following ones by fourths:

D, G, C, F, Bb, Eb, Ab, Db, Gb

Now, we all know that the cycle of fifths is never closed (the initial pitch can never be obtained) unless any kind of 'tempering' is applied. At this point, we will interfere in the system and do a couple of little tricks;

First, we will progress the A# to Gb, that’s we will define the Gb as the fifth of A#. (Remember we’re not working in 12tet so that for instance C# is NOT Db. In this circumstance, A# is rather higher than Bb and Gb is lower than F#.) Thus, we have reached the same note, Gb, in both ways. Now, we’ll have a look at their intervallic rations and frequencies (relative to D = 220 Hz.) and explore if they are or can be treated as the same pitch.

We have progressed 9 steps by fifths and 8 steps by fourths to reach the Gb. Let’s formulize both of them:

Fifths   -> (3/2)^9 = ~34.44; divided by octaves => ~1.20
Fourths   -> (4/3)^8 = ~10.00; divided by octaves => ~1.25

220 * ~1.20 = ~264.3 Hz; F + 721 pitch bend
220 * ~1.25 = ~274.7 Hz; F# - 640 pitch bend

You can get the result of these intervallic rations in sequencers as Noteworthy Composer. There is a slight audible difference in the frequencies of our two Gb notes (2735 pb), not much wider than the equally tempered quartertone (2048 pb). This gives us another feature of Eastern tone system: Pitches are not fixed to a single frequency; instead, they are sounded flexibly within a range of frequencies not wider than a quartertone. The two frequencies we have obtained give us the minimum and maximum limits of the Gb (Evc); neutral third of D (Yegâh), in Pythagorean method.

Thus, we have resulted our progression in 17 pitches, moreover, closed our system to a circle:

D, Eb, D#, E, F, Gb, F#, G, Ab, G#, A, Bb, A#, B, C, Db, C#

We will continue our survey further on Eastern tone system.

Ertugrul Inanç,
http://ertugrulinanc.8m.com
!link(ertugrulinanc@yahoo.com=mailto:ertugrulinanc@yahoo.com)

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Credits:

Yalçin Tura, who has clarified and explained the system in his book, “Türk Mûsikísinin Mes’eleleri”

Fred Nachbaur, author of “Harmonic Analyser”, a freeware yet powerful software that calculates the pitch bending and cent deviation values for a given interval (harmonic ration).

Re: Eastern Tone System (Was: Microtones)

Reply #1 –
Before we go any further with this interesting subject, could you please tell me what 12tet means?

Thanks,

Jeremy

Re: Eastern Tone System (Was: Microtones)

Reply #2 –
12tet = 12 tone, equal temperament. The twelve tones are evenly spaced, this is the "temperament" system that midi uses. It's also what fretted instruments such as guitars are pretty much limited to. Its main "feature" is that all keys sound relatively the same, and it's more-or-less what we've become accustomed to especially in pop music. As compared to the harmonic series, the fifth in 12tet is off by just a smidge, but is pretty close. However, the major third and perfect fourth deviate more from the pythagorean "ideal".

12tet is not to be confused with "well" temperaments, which in fact give each key a unique tonality while minimising intervals that sound "bad."

Re: Eastern Tone System (Was: Microtones)

Reply #3 –
Fred

I don't normally query your wise words but I think the error for perfect 4ths and 5ths is identical at .113%. Did you mean minor thirds or sixths?

And also I thought that Bach's 'Well Tempered Clavier' was written to show equal temperament, but I'm pepared to be educated!

Peter

Re: Eastern Tone System (Was: Microtones)

Reply #4 –
Peter, yes you're right. P4 and P5 are both "off" by the same amount, about 2 cents, albeit in opposite directions. Also true that m6 shows the greatest deviation, about 16 cents, with tritone not far behind at about 12 cents, and m2, M7 around 10 cents (all values relative to 12tet).

The idea that Bach's "Well-tempered" refers to 12tet is, as far as I've gleaned, a common but erroneous interpretation. While it's true that 12tet will sound the same in any key, it also loses any key-based nuances. The "Well" tempers (Kirnberger and Werckmeister to name but two) are euphonious in any key (as Bach set out to demonstrate in WTC) while still retaining a unique tonality for every key.

(Shameless plug department): I've just published a suite of three pieces (an electric guitar concerto) that uses a well temperament, should anyone be inclined to have a listen. It's at my ampcast site for free download. And no, I don't have a well-tempered guitar. But my computer does. :-)

Re: Eastern Tone System (Was: Microtones)

Reply #5 –
Since a 5th plus a 4th add up to an 8ve, it would seem that an adjustment in the size of one would require a corresponding (but opposing) adjustment in the size of the other. Of course, this adjustment, whatever its size, would be a larger proportion of a 4th than it would be of a 5th, since a 4th is a smaller interval to start with. Would this be noticeable? I doubt it.

In any event, Fred is correct about Bach's 48. The clavier in question wasn't equal-tempered.

Re: Eastern Tone System (Was: Microtones)

Reply #6 –
Actually the mechanism for 'adding' intervals is multiplicative, not additive in terms of frequency. So a fifth plus a fourth is claculated as (3/2)*(4/3) = 2 as one would expect.

Looking at frequencies, the error for the fourth is smaller than the error for the fifth exactly in the proportion of their actual frequencies, and therefore the errors are identical (error/frequency is the same in both cases)

Peter

Re: Eastern Tone System (Was: Microtones)

Reply #7 –
True if you're speaking of frequencies, but this isn't the only way to look at it. You can also speak in terms of cents, and in these terms any given interval is in fact larger in proportion to a 4th (500 cents) than to a 5th (700 cents).

Re: Eastern Tone System (Was: Microtones)

Reply #8 –
My only "instrument" is voice. Without a fixed-pitch instrumental accompaniment (such as piano or organ, voices in harmony can tune to each other. Thus, an /a capella/ choir can attain a "more perfect" fifth than a tempered instrument.

Does this apply in non-Western song as well?

Re: Eastern Tone System (Was: Microtones)

Reply #9 –
We could easily get into long, drawn-out discussions on this subject (it's certainly happened on rec.music.theory and other groups) while missing out on the practical aspect.
There are, as I see it, two basic reasons why you'd want to experiment with temperaments: the first is for reasons of authenticity for rendering musics from specific times and ethnicity. I believe this forms the basis for Ertugrul's original post on this thread.

The second is to give a whole new bag of tools for new music. Try out some temperaments, and decide for yourself whether they're useful to you. For instance, you might find as I do that the "well" tempers can give a nice silky feel to your music. OTOH, some of the just intonation temperaments sound decidedly "wonky". So, I keep that in mind in case I ever write a piece where I /want/ that particular wonkiness! :)

Re: Eastern Tone System (Was: Microtones)

Reply #10 –
Peter, I think your reply #9 is addressed to me, but if it is I'm confused. You wrote (in reply 3) "the error for perfect 4ths and 5ths is identical at .113%". You also wrote (in reply 6) "the error for the fourth is smaller than the error for the fifth exactly in the proportion of their actual frequencies".

Now you say "we're not discussing intervals". In what sense can this be true?

Re: Eastern Tone System (Was: Microtones)

Reply #11 –
Grant

I was using the expression perfect fourth to indicate a particular pitch a perfect fourth above a tonic (or root) note. You are of course correct in asserting (Msg #5) that the error compared to this interval is greater because the interval is smaller but I don't believe that this comparison means anything since the tonic may or may not be present for the comparison to be made.

Peter

Re: Eastern Tone System (Was: Microtones)

Reply #12 –
As far as I'm aware, *all* discussions of tuning systems are based on the quality of the intervals produced, not the position of any particular note relative to some abstract standard. If you do any reading in this area you'll see what I mean.

If I take an equal-tempered piano and lower every pitch by 50 cents, I still have an equal-tempered piano. This is because it's not the individual pitches that are important, but the relationships (read: intervals) between them.

Re: Eastern Tone System (Was: Microtones)

Reply #13 –
Typo in the article:

Fifths -> (3/2)^9 = ~38.44; divided by octaves => ~1.20; NOT ~34.44.

(Thanks to Kemal KARAOSMANOGLU)

Ertugrul

Re: Eastern Tone System (Was: Microtones)

Reply #14 –
But we're not talking about intervals, we are discussing how much a particular note differs in equal temperamnent from it's theoretical pitch. The root has nothing to do with it (apart from defining the pitch series).

Re: Eastern Tone System (Was: Microtones)

Reply #15 –
Grant,

I'm not going to get involved in an interminable discussion on this. And there is no need for you to be dismissive and insulting - I have done considerable reading on the matter including Helmholtz's massive tome.

I respect your knowledge and thought in many fields but we all have weaknesses and mathematical matters are obviously not included in your strong points. You frequently point out errors in other submissions - please be man enough to accept the return compliment when you raise red herrings as in msg #5, and deliberately mistate what I have said, as in msg #13. Life is too short.

Peter

Re: Eastern Tone System (Was: Microtones)

Reply #16 –
Peter,

Of course we don't have to continue this forever (nor do I have any such intention), but I do think a couple of things should be cleared up.

First, your accusation that I've deliberately misstated what you've said is in some sense a compliment, since if I've done any misstatement it was entirely unintentional.

Second, I don't think there's any call for you to be questinging my mathematical competence. I fully understand that musical intervals are determined by vibration ratios, and that adding intervals physically involves multiplying these ratios. However, although tuning is physically accomplished by adjusting vibration rates in conformance with these rules, I would assert that the human *experience* of intervals is better described by the cents system. This is based on the observation that all intervals of a given size (i.e., ratio) are subjectively identical, and moreover that the process of combining two intervals is subjectively an additive process. The human brain experiences a rise of, say, a semitone as the same phenomenon regardless of the actual change in cycles per second - and it experiences a whole tone as twice the size of a semitone.

Given that Fred's original comment about 12tet vs. the harmonic series explicitly referred to the "sound" of the various intervals, I thought I was on fairly firm ground in couching my comments in terms of the additive cents system, which, as I said, I believe conforms best to how sounds and intervals are actually experienced.

As to the "red herring" in reply #5, I don't see it. My first statement exactly echoes an observation by Fred in the previous reply (which hadn't been posted when I wrote mine). The second asserts that the discrepancy (in cents) is a larger proportion of a 4th than of a 5th, which you yourself in reply #12 agreed is true. The problem must be that you object to its relevance, which may be the crux of the matter. You seem to be asserting that you can discuss a tuning system without regard to the sound of the intervals it produces. I don't see how else to interpret your statements that (1) "I was using the expression perfect fourth to indicate a particular pitch a perfect fourth above a tonic (or root) note" and (2) "I don't believe that this comparison [between different tunings of the same interval] means anything since the tonic may or may not be present." Are you saying that it is meaningful to compare, say, the G of equal-tempered C major with the G of Pythagorean C major abstractly, without regard to the intervals these notes make with the other notes of their respective scales? If not, then I apologize for misstating your view (though not intentionally) and would appreciate a clarification. If so, then I must disagree. In my opinion the only meaningful musical comparison between these notes must involve the listener's subjective reaction to them, which in turn must be based on their relations with the other notes of the scale, above all the tonic.

Finally, I apologize for the comment about "do[ing] any reading in this area". This was written out of frustration at our inability to agree on terms. It was uncalled-for and I withdraw it.

- Grant

Re: Eastern Tone System (Was: Microtones)

Reply #17 –
Well here's my final 2 cents worth.

1. Apology accepted, and I was wrong to question your mathematical ability.

2. I don't disagree about measuring errors and intervals in cents if that is more meaningful. After all cents are just frequency ratios expressed in terms of 1200*Log2.

3. The thread (started by Ertugrul) is about approximating one theoretical note to another. This assumes a tonic frequency from which notes at other intervals are calculated. The error comparison between the target frequency and the trial frequency is meaningfully expressed as a ratio (or an interval) of those two frequencies, and that error calculation is independent of the tonic frequency. Furthermore it remains constant for a note at that interval from any tonic, regardless of the original tonic frequency chosen.

4. Your IMHO 'Red Herring'. You feel that the error calculation can be extended to include a comparison with the tonic. I don't believe this is meaningful since it gives different error values for the 4th and the 11th. However if you compare the subdominant error a fifth below the tonic with the dominant error a fifth above then the errors are equal, which is what I said all along.

Peter

Re: Eastern Tone System (Was: Microtones)

Reply #18 –
Peter,

About 4ths and 11ths: your right, this is exactly the point at which my earlier comments are vulnerable. I only wish you'd made your objection to this clearer at the time; much typing, offense and apology might have been saved.

The question I was raising, only as an aside, was how the ear & brain make their decisions about whether, and by how much, one tuning of a note or interval differs from another. On reflection I think a number of things go into this, but probably not any sense of the discrepancy as a proportion of the total interval. (And even when I was entertaining this notion, I at least had the foresight to speculate that it was negligible.)

There are a couple of places in your latest replay, by the way, where you state things with which I agree as though you thought I wouldn't agree with them. I would go through these in detail but I really think it's time to put this one to bed.

Grant

Re: Eastern Tone System (Was: Microtones)

Reply #19 –
Guys,

Following your valuable discussion is a pleasure but can't we stay a little on topic? I mean don't [any of] you really have anything to discuss on the main subject, that is Eastern Tone Sytem?

Best Regards.

Re: Eastern Tone System (Was: Microtones)

Reply #20 –
Point taken, Ertugrul, but you're too late. I think we were already done.

Re: Eastern Tone System (Was: Microtones)

Reply #21 –
Yes, I do have a point of confusion... you said,

>First, we will progress the A# to Gb, that’s we will define the Gb as the fifth of A#.

I'm sorry, I don't see how that works. "Normally" the fifth of A# would be E#. Did I miss something?

Re: Eastern Tone System (Was: Microtones)

Reply #22 –
No, you didn't. It's still E#. Just that I installed Gb as E#. :)

If you continue downward the article, you'll get the solution. There is a slight audible difference (in frequencies) with E# and Gb (which is far lower than F#) so that we can assume them as lower and higher limits of a certain 'pitch area'.

I have a drawn sheme of the situation and I'll try to get it into Cyberia someway.

Regards.

Re: Eastern Tone System (Was: Microtones)

Reply #23 –
Ah, yes! Now it makes sense. I agree, a diagram would really help.

Re: Eastern Tone System (Was: Microtones)

Reply #24 –
I am currently reading a history of the Balkans. It says that at the Congress of Berlin (ca. 1877) entertainment was provided by an orchestra that played music appropriate to the various nations present (France, Britain, Russia, Austria-Hungary, Germany, and I'm not sure who else, maybe Italy). The Ottoman Empire was there, but the musicians did not want to deal with the Turkish tone system, so instead played "Rondo a la Turk" and another piece of Western music with a supposedly Turkish theme. The Ottomans were severely insulted by this.

Re: Eastern Tone System (Was: Microtones)

Reply #25 –
Has there been a staff of equal ratios throughout?

Re: Eastern Tone System (Was: Microtones)

Reply #26 –
Has there ever been a staff of totally equal ratios?