samzayed
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Equal Tempered vs. Other tunings
(Thanks to Brian Prunka for mentioning this in a seperate thread. I think this deserves a new thread:
I've been tuning my oud with a tuner for years, mainly because of its convenience. I've been learning more about how equal tempered tuning was a
comprimise to make instruments able to play in many keys without tuning. That made me understand why many master oud players always look at me in
disgust when I pull out my tuner.
(1) So what is the traditional way of tuning? There is just, Pythagrious, and many others tuning sytems.
(2) How would I tune my oud using these other methods?
Interesting links
http://sonic-arts.org/monzo/arablute/arablute.htm
http://www.chrysalis-foundation.org/Al-Farabi's_'Uds.htm
Many thanks.
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Marina
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I think, tuner is fine.
"oud players always look at me in disgust when I pull out my tuner."
It's not you, it's them with a problem, or maybe your tuner is broken, so they don't like it!
 )
Or, if your try to find the half flat on the fingerboard with tuner - it's not right intonation...
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al-Halabi
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The Pythagorean intervals differ, to a larger or smaller degree, from the equal tempered ones. But for the purposes of oud tuning, using an electronic
tuner to get the tuning in fourths correctly is perfectly fine. The Pythagorean fourth is 498 cents while the equal tempered fourth is 500, a
difference of 2 cents that is not noticeable to our ears. It is the Pythagorean half-tone of 90 cents that is noticeably different from the
equal-tempered half-tone (100 cents), and the thirds (major and minor) are also noticeably differerent. To get these intervals to sound right on the
oud we have to depend on our ears and fingers. This is even more true of the microtonal intervals, which we know are fluid and mobile depending on the
maqam, the melodic progress, the tradition of the locality, etc. Theorists from medieval times until the present have actually differed in the way
they calculated these microtonal intervals, as can be seen on the good web sites that samzayed suggested. In the end we are left to produce correct
intonation based on knowledge of the music rather than the use of any measuring tool.
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samzayed
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ya Ustaz al-Halabi, many thanks for your articulate explanation. Your deep understanding in this subject and your willingness to share it is very
much appreciated. So now, I think the inaccuracy of equal temperement Brian refered to was specifically for identifying the notes on the fingerboard.
That would make the oud sound like a piano. . . It's making more sense now.
Marina, I just got a new tuner/metronome combo. Maybe they'll like this one 
Seriously though, I think the oud masters wanted me to wean off my reliance for the tuner and trust my ears. The tuner is my security blanket
sometimes . . .
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al-Halabi
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Salamtak ya samzayed. One way to reduce the dependence on a tuner is to use a song you know well to guide your tuning by ear. For example, the first
two notes of Lamma bada yatathana have an interval of a fourth. If you replicate that familiar interval from the song in the interval between each
pair of adjoining strings you will get a tuning in fourths that, with some tweaking, would be as exact as what you get with a tuner. Of course you
need to have the correct pitch for one course to start with.
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samzayed
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I did exactly that with this problem - I could never hear the difference between the E half flat when its played in Rast vs. Bayati. I always heard
that the E half flat in Bayati is lower than the Rast. I always played it somewhere in the middle and never correct. One day I played "Wayn a
Ramallah" which as this sequence F - E-half flat - C - D - D, and looked for the right E position. Then I played "Ghaneeli Shway Shway" C - D - E
half flat - E half flat and looked for the E's postion. Bingo! I probably jumped up for joy like the first caveman that invented fire. 
Ahh, microtonality, fire, water, and sun - the essentials of life 
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Jameel
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Sam, post some audio of that, will you?
I just looked at those links you posted. Whoa! That's some serious reading. Your brains are out of my league! I think I'd rather make ouds......
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samzayed
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I don't have my Mini Disc at the moment, and I don't have a mic for my comp. I'll show you when I see you next month
Those docs are a little too detailed for me too. I don't really understand the fractions aspect of it of intervals. Ustaz al-Halabi or anyone
else?
http://leb.net/rma/Articles/Samaie_Farhafza.pdf This is a nice paper.
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al-Halabi
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The ratios and fractions used in the calculation of intervals look more complicated than they actually are. Two of the basic calculations for interval
sizes and pitches go back to ancient times: the proportion of the string that is vibrating in relation to the total length of the string (the
frequency ratio); and the length of the vibrating segment of the string (the distance from the finger stopping the string to the bridge). These two
measurements are mathematically a function of each other. Today, the old ratios are commonly converted into cents, a modern logarithmic measure that
makes it easier to understand and compare intervals than some of the strange-looking ratios (an equal tempered semitone is equal to 100 cents, a whole
tone to 200 cents, and so on). All three types of measures of intervals now appear in texts and tables analyzing Middle Eastern intervals and tone
systems. I am not sure how much interest there is in this technical stuff, but here are some brief examples that might help to explain the basis of
these numbers and ratios.
On the course C, when you play the open string and then the F up the fingerboard you get an interval of a fourth. If you measure the position of your
finger on the F you will find that it is exactly ¼ of the string up the neck, so that ¾ of the string is actually vibrating. This interval of a
Pythagorean fourth is expressed as a ratio of 4/3 (or 4:3), showing the relation of the full string to the vibrating segment. This ratio can
alternatively be expressed in vibrating string length: in a string of 60 cm, the vibrating length that produces a fourth would be 45 cm. If you move
up on the same string by one whole tone from the F to G, you will be a perfect fifth from the open C. Your finger will be exactly 1/3 of the string up
the neck, with 2/3 of the string vibrating. A fifth is expressed as a ratio of 3/2.
Theorists added and subtracted intervals to arrive at the size of larger or smaller intervals. To add two intervals, the ratios of the two intervals
are multiplied by each other. For example, to add a fourth and a fifth (which we know combine to make a full octave) we do 4/3x3/2=2/1. 2/1 is the
ratio for the octave (the finger stopping the string is in the exact middle of the string, with only ½ the string vibrating). To subtract one
interval from another in order to get the difference between them you divide their two ratios by each other. A fifth minus a fourth, for example, is
calculated as 3/2 divided by 4/3, which produces 9/8. 9/8 is the ratio for a Pythagorean whole tone, which is the difference between a fifth and a
fourth. From this 9/8 ratio you can infer that 8/9 of the string is vibrating when you play the note D on the C course, and that your finger is 1/9th
up the string (or 60x1/9=6.66 cm from the nut on a string of 60 cm).
Using these kinds of mathematical methods along with experimentation with various vibrating string segments and the sounds they produce, theorists
going back to the ancient Greeks came up with calculations for the size of intervals and for the precise positioning of frets. Any of the interval
ratios can be converted into vibrating string lengths by inverting the ratio and multiplying it by the total length of the string. In this way you can
locate the exact spot on the fingerboard in which the string is to be stopped in order to produce a particular pitch. The interval ratios are also
convertible into Ellis cents. The 9/8 interval (the Pythagorean whole tone) is 204 cents (compared to the equal tempered tone of 200 cents), the 4/3
fourth is 498 cents (compared to 500), and the 3/2 fifth is 702 (compared to 700). Al-Farabi, for example, gave as 162/149 and 54/49 the ratios for
two microtonal intervals used at his time (10th century), both of which lie between the half tone and whole tone (which we often loosely call
"three-quarter tone" intervals). When converted to cents we find that they are equivalent to 145 cents and 168 cents. This is in fact the general
range within which we play the "three-quarter tone" intervals in various Arab maqams today. An exact three-quarter tone interval would be 150 cents,
but in practice it is only one possible spot in a larger zone which defines fluid microtones such as segah/sikah.
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palestine48
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While I am bad at tuning myself. What I am trying to do and my teacher is ok with it is to tune one of the courses to the tuner and match the other
in the pair to the tuned one just so i can get total matching in each pair. that is where, he said the tuner will not help.
P.S. i encourage posts like these, this is the main reason I want to belong to the forums so I can learn how to be a better player. Thanks and keep
it coming.
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samzayed
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wow, many thanks ya Ustaz!! That explains a lot and it clears up a lot for me
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eliot
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| Quote: | Originally posted by palestine48
While I am bad at tuning myself. What I am trying to do and my teacher is ok with it is to tune one of the courses to the tuner and match the other
in the pair to the tuned one just so i can get total matching in each pair. that is where, he said the tuner will not help. |
This is what I do - I tune the 2nd string A to 440 (currently) using a pitch fork, and tune all the rest to true perfect unbeating fourths. If I'm
doing a session and the piece focuses on E, I'll tune the 3rd string E to the tuner, and the rest by ear.
I can tell the difference from tuning it this way compared to using a guitar tuner for each note.
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Brian Prunka
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Al Halabi, your explanation of the ratios meaning is not how I understand it. It seems correct as far as it goes, but fails to get at the underlying
vibrational phenomena that apply to all music (not just strings)
The ratios are relative to other notes (generally the key, but can also be other notes in the key)
octave= 2:1, means the higher note vibrates 2 cycles for each cycle the tonic vibrates
fifth= 3:2, means the 5th vibrates 3 times for every 2 times the tonic vibrates
third= 5:4 (just intonation) means the 3rd vibrates 5 times for every 4 times the tonic vibrates.
These three intervals come from the first five notes of the overtone series (fundamental, octave, 12th (i.e., 5th + 8va), 2 octaves, and 17th (i.e.,
3rd + 15va)
in pythagorean intonation, all intervals are derived by multiplying octaves and 5ths (and their reciprocals, octave down (1:2) and fifth down (2:3))
Therefore a 9th, which is up a 5th, and then up another 5th is 3/2 x 3/2 = 9/4 (or 9:4). Taken down an octave (9/4 x 1/2) you get 9/8, a major 2nd.
A perfect 4th is up an octave and down a fifth (2/1 x 2/3 = 4/3). Etc.
A 6th would be P5+P5+P5 and down an octave (3/2 x 3/2 x 3/2 x 1/2 = 27/16)
It's worthwhile to try to figure out all 12 intervals this way to understand and hear relationships.
In letter names these would be:
D = C to G to D
F = C to high C, down to F
in the simplest, most common form of just intonation, intervals are derived by multiplying octaves, fifths and thirds. This is called 5-limit just
intonation (because only the first 5 notes of the overtone series are used).
A major 7th would be a fifth + a third (3/2 x 5/4 =15/8).
Take that down an octave (15/8 x 1/2 = 15/16) and you have a minor 2nd down. To get a m2 ascending, simply use the reciprocal (16/15).
A major 6 is up an octave, down a 5th, and up a 3rd (2/1 x 2/3 x 5/4 = 5/3)
Notice that this Major 6th is different from the one we got using the pythagorean system. (5/3 vs. 27/16)
There are other kinds of just intonation, the next most common being 7-limit, utilizing the first 7 intervals (the 7th interval being 7:4, a very low
m7 that is heard in some african music and american blues and jazz singers. Sing the goofy "and many more" tag to "happy birthday and odds are you
will sing a 7-limit minor 7th).
Sam, you're right that I was just referring to using a tuner to get the notes on the fingerboard. Al HAlabi is right that the difference between a
pythagorean 5th and an ET 5th is negligible. However, consider that every successive string compunds the error, so if you tune with a tuner, your A
string will be 6 cents flat relative to your C string, a minor but noticeable discrepancy in certain keys (D, G and A will make it most obvious).
Going down from C, G is 2 cents flat, D is 2 more cents flat, and A is 2 more cents flat. (or starting from A, the C ends up 6 cents sharp).
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Brian Prunka
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I just want to clarify that of course Al-Halabi's explanation is correct, I just wanted to add a layer of explanation (or confusion ) that brings it more into the realm of how it relates to music (i.e., sound waves
in relation to one another and in keys, not just abstract vibrating string lengths, which is indeed how the experiments were confirmed).
In addition, the method he describes is the only way to explain Arabic and Turkish microtones, which have no acoustical derivation in the way I
described it (i.e., they are not available in pythagorean or any usable form of just intonation).
I personally found the study of intonation to be immensely helpful in playing music and ear training.
The use of drones to solidify your sense of natural ratios (singing and playing) is highly recommended.
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Microber
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An interresting site about the subject of 'just intonation',
and an introduction to historical tuning.
As i'm not fluent in english, i don't understand everything.
Well, it's a good excuse, because i'm not sure i even should understand everything in french.
The guy (Kyle Gann) composes for 'Dysklavier' , a sort electronical piano mécanique that he tunes in just intonation.
http://www.kylegann.com/microtonality.html
Robert
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al-Halabi
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Brian, your points are all well taken. Understanding the underlying aspects of sound waves and how they relate to relative pitches and intonation is
definitely important. My comments were in response to a question by samzayed, who was unclear about the meaning of the ratios and other interval
measurements found in analyses of the writings of al-Farabi and other medieval theorists. I tried to explain in simple terms how these premodern
theorists did their calculations and what the ratios and other interval measurements that appear in various technical tables and texts about their
work stand for. Their methodology was very much rooted in the ratios derived from vibrating string lengths and the mathematical calculations based on
them. They commonly used fret and finger positions on the fingerboard of the oud as their descriptive framework for the series of intervals that made
up the tone system. In that sense, strings and their divisions were at the foundation of their empirical and theoretical work. In focusing on
vibrating strings my comments were aimed at describing the approach that guided this particular tradition of musical scholarship.
That said, it needs to be added that the work of the theorists in the premodern Middle East extended well beyond measuring strings and devising
ratios. It dealt with broader musical issues, including the vexing intonation of the microtonal pitches and the analysis of tetrachords and melodic
modes. It is interesting that the theorists struggled with the challenge of fitting the neutral or microtonal intervals into the Pythagorean
framework, coming up with various ways to rationalize the hybrid system that their region had developed as the basis of its music. One influential
theoretical solution to this quandary was the adoption of the comma/limma system, which became systematized by the thirteenth century. They also
struggled with the lack of agreement about the size of the microtonal intervals, which were played with varying intonations in premodern times as
today. Equal temperament, with its universal diffusion today, has only added another twist to debates on intonation in Middle Eastern music that go
back hundreds of years.
Intonation and temperament can be rather technical, but I agree with you that understanding their workings can deepen our understanding of the music
we play.
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