# intonation and math

April 7, 2007 at 02:41 AM · I wonder how well it is known that the actually pitch for the E from a G major scale isn't the same pitch as the E string would be if the violin is tuned in perfect fifths.

When perfect fifths (C,G,D,A and E) are used the distance between the C and the E is nine-eigths squared, or rather two major seconds (the E is the higher part of the ratio the C the lower).

This is eighty-one sixty-fourths.

When, as is the case with a major scale based on the overtone series, the ratio is not based on fifths but on being a major third the ratio is five fourths or rather eighty sixty-fourths

81/64 perfect fifths, 80/64 major third

Thus the one with perfect fifths is higher

The same thing happens with the A string for a C major scale.

I was just playing on my violin improvising and noticed that I was above the open string E and wondered whether I had overcompensated and thus would be on the scale (well. Twice the "distance" between the two ratios above G)

Regardless, I think that's very important information. In the time of Plato or Josquin Desprez this was part of how they taught math I think (using actual occurences of natural harmony rather than numbers which are meant to add up to some quantity that is an illusion because it's just about possession).

Note the Ratio denotes frequency of the vibrations

In a major third, five fourths means the upper note has vibrations during the same time that the lower note has four.

## Replies (32)

April 7, 2007 at 08:17 AM · By the way, what I'm saying is that the secondary dominant's La and the subdominant's La are different pitches.

The open E string would be the pitch for the secondary dominant for G (A, C-sharp, E).

Also, with intervals, when you go up an interval you are multiplying the note you started from by the ratio that goes with the interval (NOT adding). Thus two major seconds (which is the ratio 9/8) ends up being 9/8 times 9/8 or 9/8 squared.

So when you are thinking intervals your brain is actually doing multiplication to synthesize the sound you here, when you are adding an interval. In a way, this is a whole different way to look at rhythm (the frequency is a rhythm and when you multiply up an interval you are increasing the rhythm).

When you look at the kind of tree of harmonics that is built up to create a simple scale in the middle range of the piano (or the bottom end of the violin's range) the base notes (or the roots of the tree) – their actual frequency is low enough for you to hear the individual waves (if you could hear them).

The mind does amazing multidimensional things....

April 7, 2007 at 03:39 PM · wow. if we could add 2 inertial frames of reference, we just might have the kernel of Roelof's Special Theory of Musical Relativity.

April 8, 2007 at 01:46 AM · For the seriously interested..the first 15 chapters of W.A. Mathieu's book "Harmonic Experience" contain wonderful and practical exercises for learning to hear and use a more platonic system of intonation...

April 8, 2007 at 01:57 AM · Platonic, or Pythagorean?

April 8, 2007 at 02:24 AM · pythagorean. sorry.

but platonic because sometimes it's an ideal rather than an actual relationship between notes. :)

April 8, 2007 at 02:36 AM · Nice! :)

April 8, 2007 at 01:28 PM · The intervals go with the overtone series:

2/1 ration is an octave,

3/2 is a fifth

4/3 is a fourth

5/4 is a major third

(notice that 2/1 is the ratio of the first two vibrations of the overtone series, 3/2 the second two: The second vibrates twice as fast as the first, the third three times as fast etc.)

6/5 and 7/6 are both considered minor thirds and 8/7 and 9/8 are both considered major seconds...

If I were to explain how a tree of resonance is formed which makes a major scale:

Notice that the third and 6th vibrations of the overtone series form an octave, this octave in turn is the first two vibrations of a new overtone series: this sequence happens twice to form the notes of a whole scale and you end up with, in a C major scale, FAC from the first series, CEG from the second and then GBD from the third which add up to the notes of what would seem to be the C major scale but is more the notes of an F lydian scale (for a "C major" scale you would need another branch of the tree to get the secondary dominant which is a necessary ingredient, you would need D, F-sharp A as well, and then you would have two different pitches for A).

This is the tree whose roots go beyond human hearing as pitches, but you could hear the roots as rhythms.

Note that the first partial or harmonic of an overtone series is the second vibration.

April 8, 2007 at 01:24 PM · With all this said, the mind is sensitive enough to adjust pitches on its own and if you would really listen to it you would hear that it does this all by itself (because it is a product of evolution not of compromise). A sensitive person trying to really sink into the way an expressive phrase moves would hear all these things by themselves because it is natural architecture in space and wave patterns.

April 15, 2007 at 06:21 PM · Thanks for making my head explode.

April 15, 2007 at 06:29 PM · So basically what you are saying is that pitch is relative to the preceding note and also to the degree of the scale thus a g in the scale of c major which is the fifth will be different to a g in d major where it is the fourth degree?

April 16, 2007 at 04:15 PM · I once started a thread somewhere about intonation and maths. Usually it causes consternation among the readers.

In the end, it has to sound right, and the practitioner need not understand the maths in order to understand the sound.

Yet in India, there is a longstanding tradition of mathematical rigor as a part of poetry and music, so that is not to say that one shouldn't consider maths as a neccessary aspect of musical thought. Rather I observe that for European music, maths seems to be considered nonessential.

And as Buri once suggested, the thirds are always quite fluid relative to the 4ths and fifths--they are very contextual.

July 19, 2007 at 07:10 AM · "So basically what you are saying is that pitch is relative to the preceding note and also to the degree of the scale thus a g in the scale of c major which is the fifth will be different to a g in d major where it is the fourth degree? "

???

No I didn't say that. A G a fourth above a major second above C is the same as that a fifth above C.

9/8 times 4/3 equals 3/2.

They come out the same

However, The A that is a major third above a fourth above C (4/3 times 6/5) is a different pitch from the A that is a perfect fifth above a major second above C (9/8 times 3/2)......

July 19, 2007 at 06:04 PM · Well, I am indeed much consterned :-p even though as a "computer scientist" -- well, that's what I supposedly studied anyhow ;-) -- I probably should be quite enthused to learn more about it, platonic or otherwise... ;-p

_Man_

July 19, 2007 at 06:20 PM · >With all this said, the mind is sensitive enough to adjust pitches on its own and if you would really listen to it you would hear that it does this all by itself (because it is a product of evolution not of compromise). A sensitive person trying to really sink into the way an expressive phrase moves would hear all these things by themselves because it is natural architecture in space and wave patterns.

Yup. When I asked my teacher just how I was supposed to tell the difference when playing a specific note in any given key signature, she said "sing it first." And that it would happen on its own. (I have decades of experience in singing, BTW, and only two years on the violin.)

And yup on the second half of your above comment. Some notes, when I'm listening to some music, just feel so achinging right, and I think it's that tiny bit of change something other than my brain is processing. Or that my mind agrees with the violinist's interpretation.

Very interesting stuff you posted here. I'm going to print it all out and read it from time to time and maybe some year (or decade) down the road, it will all sink in.

July 19, 2007 at 10:14 PM · All of the above math is well and good (I guess), but it doesn't mean anything when you have to play with a piano or harpsichord.

July 19, 2007 at 11:16 PM · It's crucial to know about this when playing in a string quartet, as playing perfectly in tune together often takes some planning.

We sometimes have to discuss which open string to base the key note to, and which pitch to end up after modulating.

Once, when we played the 2nd movement of the Mozart quartet 421 (which is in F major), we decided to try tuning the F to the C string. The 1st chord is in F major, with F played in the cello. The 2nd chord is G minor, with the cello dropping down to Bb. That means that the G played by the 1st violin in that chord has to be played lower than the G string, which feels slightly strange to a violinist. It did work, but we later decided to base the key note to the A string instead, even though you then get problems with the lower cello and viola strings.

It's worth knowing that it's possible to construct a major and a minor chord using natural intervals. However, it's not possible to do so with diminished and augmented chords; the 3rds have to be narrowed in the former and widened in the latter.

July 20, 2007 at 05:14 AM · Ok, but I'll bet you guys played out of tune anyway.

July 20, 2007 at 11:33 AM · "Ok, but I'll bet you guys played out of tune anyway"

Yes, but what is your point Scott?

July 20, 2007 at 05:27 PM · I forgot.

July 22, 2007 at 05:49 PM · BTW--all this talk of math may be relevant to diatonic music,

but I'm not sure if it can help with octatonic or 12-tone music.

July 22, 2007 at 05:59 PM · Math? hmm, I smell Calculus. When ratios and variance are reduced to string names, I instinct limits and first derivatives.

Obviously, I'm still trying to wake up.

July 22, 2007 at 11:57 PM · I think what you're smelling is your own colophany dust. And it's not pretty.

July 23, 2007 at 10:36 AM · "BTW--all this talk of math may be relevant to diatonic music,

but I'm not sure if it can help with octatonic or 12-tone music."

Yes, I agree that natural intonation is only really relevant to diatonic music, and with instruments that are not restricted by temperament.

The octatonic scale is closely related to the diminished chord isn't it? In that case, you wouldn't be able to play this scale using natural intervals.

July 23, 2007 at 02:26 PM · Interesting--much better than today's alternatives. I'm certain of this.

July 23, 2007 at 02:36 PM · Man I just haven't had enough coffee yet to be reading this.

You people are cruel.

August 29, 2007 at 05:49 AM · Neil, you wrote "Once, when we played the 2nd movement of the Mozart quartet 421 (which is in F major), we decided to try tuning the F to the C string. The 1st chord is in F major, with F played in the cello. The 2nd chord is G minor, with the cello dropping down to Bb. That means that the G played by the 1st violin in that chord has to be played lower than the G string, which feels slightly strange to a violinist. "

According to my math: the G would only be lower than the G string if one were trying to play a perfect fifth below the D which is a major third above the b flat. That chord wouldn't really have a perfect fifth in it though, the G should be were it was, it's the D being a major third above the B flat (which is a fifth below the F) which is lower than a D which is a fifth above the G. Those are all notes in the chord then, although the D isn't a perfect fifth above the G. In other words, the D would be lower than the open D string but you could keep the G the same.

That's how the math works...

August 29, 2007 at 06:13 AM · Scott Cole wrote "BTW--all this talk of math may be relevant to diatonic music,

but I'm not sure if it can help with octatonic or 12-tone music."

Well, it's not that difficult to realize that 12-tone music, where there are only 12 notes is based on the well tempered scale (that also is mathematical by the way). In just (pure) intonation there is no limit to the amount of notes you can have (this is why we have the well tempered system otherwise you would need an unlimited amount of keys on a keyboard to accomodate the pitch differences).

12 tone music is simply a way of playing with a matrix of 12 notes. If you study the music you will see that for it to resonate with the human ear and have meaning it does involve notes that have resonance with each other. That again involves math and balance.

August 29, 2007 at 11:52 AM · Hi Roelof. Maybe I didn’t explain very well, but here’s the math’s to illustrate the point I was trying to make.

Let’s assume that the quartet tunes their A string to 440Hz, and that all the strings are tuned to natural 5ths, and that they are going to tune using natural intervals. Firstly, here are the frequencies of the strings in a quartet.

Violin; E=660, A=440, D=293.333, G=195.556

Viola; A=440, D=293.333, G=195.556, C=130.370

Cello; A=220, D=146.667, G=97.778, C=65.185

The first chord is the piece is F major (F in the cello, A in the 1st violin). Let’s assume that everyone tunes their 1st note to the cello (or viola) C string. If you tune the A in the 1st violin to the C string, it will be at 434.567Hz (which is obviously going to be lower than the A string), and the cello’s F will be 173.82Hz.

In the 2nd chord, the cello drops down a natural 5th to Bb (the chord here is G minor, 1st inversion), and the violin plays the note G. The frequency of the cello Bb is therefore 115.884HZ. If the 1st violinist tunes their note G to the cello’s Bb, their frequency will be 386.282.Hz. If this note were an octave lower (i.e. at the pitch of the G string), it would be 193.141Hz. As their G string is tuned at 195.556Hz, you can see that the note they play is harmonically lower than their G sting.

I hope that makes sense!

August 29, 2007 at 12:24 PM · Roelof wrote "According to my math: the G would only be lower than the G string if one were trying to play a perfect fifth below the D which is a major third above the b flat. That chord wouldn't really have a perfect fifth in it though."

If I understood you’re point, then I’m afraid you’re math is incorrect.

Basically, a natural 5th interval can contain a natural major and minor 3rd. I’ll demonstrate this mathematically.

On a violin, if you tune the A string to 440Hz, the E string will be 660Hz if tuned as a natural 5th to the A (440 * 1.5).

In the case of a major chord, the 1st interval is a major 3rd, which in this case is C#. The frequency is therefore 550Hz (440 * 5/4). The next interval is a minor 3rd. A minor 3rd up from the C# is 660Hz (550 * 6/5), which exactly the same pitch as the E string. As you can see, both these natural 3rds fit neatly into a perfect 5th. You can apply the same math to the minor chord.

Therefore, in your above augment, if the D is in tune to the Bb, then so is the G; both will also either be in tune to the G string, or both out or tune to it…it can’t be one or the other.

August 29, 2007 at 04:34 PM · I find playing string quartets a lot of fun precisely because of the problems with intonation.

August 29, 2007 at 11:56 PM · I didn't say that it was a major third and a minor third (5/4 times 6/5 which is 30/20 or 3/2 which is a fifth).

I said you started from the B-flat being a fifth below the F, that the D a major third above that note (the B-flat) doesn't have a perfect fifth relationship with the G which is two fifths above the F (9/4 or 9/8 when an octave lower).

The G would be 9/8 above the F then, the B flat a fourth above the F would be 4/3. A major third above that is 4/3 times 5/4 with is 5/3 (that is the D).

9/8 (the G) and 5/3 (the D) don't have a perfect fifth relationship. A perfect fifth has the ration 3/2 and 9/8 times 3/2 is not 5/3, it is 27/16 which is slightly larger.

Thus, if you left the pitch of the open string violin to be less than a perfect fifth below the D (which is a major third above the B-flat) then it would be just intonation. And it would be in tune.

The secondary Dominant (which the g minor chord isn't) would be a different D. That would indeed be a perfect fifth above the G (the open G). I handled all of this already.

It seems you retained the pitch of the D but tried to lower the G to make a perfect fifth below it.

Again, the g minor chord in F major does not consist of the Major third and the minor third you were refering to.

In just intonation C major, the D and the A aren't a perfect fifth apart.

Again, all of this changes when you use the secondary dominant (in C major that is the D major which functions as dominante to the dominant G). Then you do use perfect fifth.

So you actually have to A's on is 5/3 and the other is 9/8 times 3/2 which is 27/16.

Since, F sharp (or in the case of F major b natural) is not considered part of the scale then it is the 5/3 that takes precedent.

This all becomes really confusing when you consider that you cannot have the key of C major without an F-sharp in it ( go look at any piece in C major and see that you need the secondary dominant as part of the harmony to set up the tonic of C, in fact you need that note more than you need the f natural).

Thus you could argue that C major in reality is C lydian (this has an f-sharp) and you could argue that in reality the G minor chord of F or the D minor chord of C are arguably borrowed chords....

Confusing enough!?

Now, the chord that would be e minor in C or a minor in F. These chords (in just intonation) would be made out of a major third and a minor third (this the D minor chord in C and the G minor chord in F aren't the same type of chord and the other).

If you go to a store which sells synthesizers with prefabricated scales in just intonation: they will have the tuning I am refering to.

August 30, 2007 at 12:08 AM · Neil you said this "Therefore, in your above augment, if the D is in tune to the Bb, then so is the G; both will also either be in tune to the G string, or both out or tune to it…it can’t be one or the other. "

That isn't what I said.

The G is in tune with the F (two fifths below it) that is a different interval than a major third. This is also what I started my whole thread about I believe. You cannot assume that all minor chords are the same, because they aren't. The G minor chord in F is made up of a whole different matrix of overtone connections than the A minor chord would be. The D minor chord in F would be the same type of chord mathematically as the A minor chord but the G chord is different in just intonation.

You'll truly have to look closer.

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