oud notes frequencies

Frequencies of Musical notes

Quarter tones (half flats etc)

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tuning-warning! contains math

nb: the phrase "in tune" has two primary meanings. one is simply that two notes have a relationship, determined by cultural context, which everyone agrees is pleasing. the other is based on the acoustical principles of harmonic resonance, in which "in-tune" intervals are those that resonate in relationship determined by integer ratios. A ratio of 279:117 might be acoustically "in-tune" but wouldn't be musically in-tune to anybody, because the ratios are far too complex. Conversely, a singer who sings 882Hz while the piano player plays 440Hz is acoustically out of tune, but it's close enough to the octave (880Hz) that most people wouldn't mind too much and would say it's "in-tune." In the acoustical sense equal temperament has all the intervals out of tune except the octaves, yet musically we can call this "in tune" with respect to its cultural context. What's important to realize is that no matter what the cultural context, the acoustical intonation plays a role as a sort of reference point, consciously or not.

That said:
It's a misconception to think of western musicians as playing in equal temperament, unless they're playing piano or guitar. Temperaments are a compromise which allow instruments with fixed pitches (like pianos, harps, guitars, mandolins, pipie organs) to modulate freely from one key to another.
vocal groups are almost always performing in 5-limit just intonation (meaning the first 5 overtones are the tuning reference, octave, fifth, and 3rd). American music sometimes uses 7-limit harmony (meaning the first 7 overtones are used as tuning notes), which comes from the African influence (blues harmony). blugrass and gospel groups are primarily 5-limit just intonation with ocassion 7-limit harmony for "bluesy" parts.
String quartets play in an approximation of just intonation (5-limit), but are limited by the open strings of their instuments to some extent. They tune their strings to pure 5ths, though, not tempered 5ths. Wind instruments do the same, although they have some limits based on the placement of the finger holes/valves. When performing with pianos or guitars, the more flexible instruments make adjustments to match equal temperament when necessary, but it's really a mixture of different tuning systems (a good example of this is the beach boys--pure 5 limit "barbershop" type harmonies which are slightly adjusted here and there to accomodate the guitars & keyboards). Certain modulations (those based on diminished or augmented harmony) require something very close to equal temperament, which performers have to accommodate. Also, chromatic scale passages of more than a few notes will only sound right in equal temperament.

the oud, in my experience, is tuned to pure 4ths, which are slightly smaller than tempered 4ths (likewise, pure 5ths are slightly larger than tempered 5ths). There is a slight issue that arises in the tunings of the F and A strings, though.

here's an explanation of what's happening when you tune:
an octave is a ratio of 2:1, meaning for every 2 times the octave vibrates, the fundamental vibrates one time. in hertz, this simple: A440 is an octave below A880.
a 5th is a ratio of 3:2 E660, A440
a 3rd is a ratio of 5:4 C#550, A440
4ths are inverted 5ths, and so have a ratio of 4:3
to get a descending interval, you just flip the ratio (octave below is 1:2, 5th below is 2:3, 3rd below is 4:5, 4th below is 3:4) Ratios are multiplied to get other intervals.

now look at tuning an oud CFADGC:

start with the low C (we'll pretend it's 100Hz for simplicity)
tune the high C to the low C (2 octaves, 2:1x2:1=4:1) so the high C is 400 Hz
tune the G to the low C (an octave and a 5th, 3:2x2:1=3:1), so the G is 300Hz (you get the same result if you do up a 5th, down an octave (3:2x1:2) from the 400 Hz high C)
tune the D to the G (down a 4th, 3:4), so the D is 225Hz.
Tune the A to the D (down a 4th, 3:4) so the A is 168.75Hz
Tune the F to the low C (up a 4th, 4:3) so the F is 133.33Hz
Now here's the weird part: tune the A to the F (up a 3rd, 5:4) and you get: 166.667Hz, which is 2+Hz lower than the A we got before. this is a difference of about 22 cents-not small! ( a half-step is 100 cents). This is almost 1/9 of a whole step (whole step=200 cents, 200/9=22.22_), which surprise, surprise, is the Koma (comma) of turkish theoriticians. An actual in-tune whole step is 204 cents (remember that the "cents" system is designed around equal temperament, so 200 cents is an equal-tempered whole step), which would make 1/9 22.66_ (and .66_ cents difference is not discernable by the ear).

pythagoras proposed a system of tuning in which thirds were ignored and only octaves and fifths are used, this is called 2-limit or pythagorean tuning. It seems that this is what is used for TUNING the oud (in order to avoid the two different tunings for the "A" string), but the basis for playing the music is what I think of as "just intonation + some pythagorean tuning because of open strings + quarter tones and microtonal inflections".
The just intervals are the reference points, but players will deviate from them in order to create a sort of melodic tension; you feel the "pull" of the microtonal inflections. My personal theory about Rast is that it's bittersweet character is, on some level, the result of the E half-flat being pulled almost equally in both directions at once (although I have no justification for this, it's just how I feel).

if you're wondering how the in-tune intervals compare to equal temperament, here's a quick rundown:

m2-depends on context, but usually higher (12 cents)
M2-higher (4 cents)
m3-higher (16 cents)
M3-lower (14 cents)
P4-lower (2 cents)
A4-lower (10 cents)
d5-higher (10 cents)
P5-higher (2 cents)
m6-higher (14 cents)
M6-depends on context, either lower (6 cents, usually in the key of F, sometimes in C) or higher (6 cents, keys of D and G, sometimes in C)
m7-depends again on context, sometimes higher (18 cents, usu. keys of F, C) most often lower (4 cents, keys of D, G, A)
M7-lower (12 cents)

my contextual suggestions about the keys are based on the arabic tuning, and have to do with the fact that the open strings are fixed.

For the half flat and half sharp intervals, all bets are off

of course, this stuff is just a description of acoustical laws pertaining to tuning and resonance. What's amazing is how our ears/brains are finely attuned to this phenomenon; our experience understands it before we grasp the mathematical relationships. I do find it helpful to be aware of the pure tuning of the different notes, though (especially if I play with a pianist and have to adjust to him (or her))

I realize this probably makes no sense to a lot of people . . . but maybe a couple will find it interesting or even helpful