In Pythagorean tuning, what's the frequency of the tonic of each key?

Question

When playing a fretless instrument (like violin), what frequency should be used to tune beginning of each key?

I believe that for A major and A minor, 440Hz is the tonic. What will be the frequency of the C in C Major?

How about Bb major? What frequency should be the Bb tonic? Is it like the m2 of the A major scale? Maybe like the 7th of the C Major scale?

Is it tuned according by Just Intonation? Equal Temperament?

I'm rather looking for the formula that enables calculating the tonic of each key, rather than a list which could also be handy.

UPDATE
I'm afraid I wasn't so clear, so let me try to explain again.
In general I'm talking about fretless instruments in standard tuning (A=440Hz). The standard playing on these instruments would be Pythagorean. However, what if you want to play in Bb major. Do you use the Bb of Equal Temperament or what?

UPDATE on Caleb Hines' and leftaroundabout's answers.
This is almost obvious that the low Bb (465.12Hz) of the Pythagorean C Major (C=261.63Hz), and even the high A# (471.47Hz), both make no sense to use as the tonic of Bb major.
So you're right that the open-string A is anyway going to be a bit low for a Pythagorean-expecting ear. Anyway the question is what compromise do we take, is it the Bb of ET (466.16Hz) or the JI's (466.16Hz).
I'm curious what's the standard among orchestral string players (fiddlers should obviously tune to ET).

P.S. I picked Bb as an example, but the question applies to other keys like A# (question solved if answer is ET).

Answer 1

First of all, Pythagorean (PT), Just Intonation (JT) and Equal Temperament (ET) are different (families of) tunings. Therefore, note frequencies will be different in each case. You can find frequency charts for them on Wikipedia.

For any tuning, you need a reference frequency. Currently, 440 Hz for A above middle C is the most widely used standard. But historically it wasn't the case and some orchestras still tune differently.

Pythagorean tuning

PT is a family of tunings based on just perfect fifths, so it's a subset of the just intonation family. Perfect fifth is the transposition of the third harmonic of musical tones down to the same octave as the fundamental. Dividing a frequency by two transposes it down by one octave. Third harmonic's frequency is three times the fundamental. Therefore a perfect fifth is 3/2 the frequency of the fundamental.

PT works by starting on a selected frequency and moving in perfect fifths (and transposing them down to the same octave). So the frequency ratios are 1, 3/2, 9/8, 27/16... The general formula is 3^n/2^n (and then you transpose it back to your octave by dividing it by two as many times as necessary).

One problem is that a stack of perfect fifths can never add up to an octave. Circle of fifths cannot be closed. After 12 consecutive steps, starting from C, (C G D A E B F# C# G# D# A# E# B#) you end up with a B# that is close, but not quite equal to C; the ratio is 3^12/2^19 and it's about a quarter of a semitone sharper, which is very noticeable. In other words, C# ≠ Db in PT. As a result of this, if you want to live with only 12 notes, some of your fifths will be out of tune. It's called a wolf interval.

There is also another problem: The Pythagorean major third (81/64) is too sharp compared to the JT major third (5/4, see below). This renders this tuning mostly useless for triadic harmony.

Just Intonation

JT is based on integer ratios. It strives to make all intervals just (if we only make the fifths just, it's usually labeled as PT). For example, a major scale can be created with the major triads (ratios = 4:5:6) built on the fundamental, perfect fourth (4:3) and perfect fifth. It will give you C=1 D=9/8 E=5/4 F=4/3 G=3/2 A=5/3 B=15/8.

This is harmonically very pleasing as long as you stick to the I, IV, V, iii and vi triads. But the ii triad is out of tune. These are the wolf intervals of this particular JT major scale. You can fix it by lowering the D for example, but this will break the G chord, and trying to fix it will break something else. It's impossible to get all the chords right without adding new notes to the scale.

What are the frequency ratios of the notes with accidentals then? The answer is, it depends. The minor seventh (Bb) can have the 16/9, 9/5 or 7/4 ratio depending on what effect you want to achieve.

Again, you will need more than 12 notes to use JT, even in a single key.

Equal temperament

ET just divides the octave into 12 equal intervals. None of the intervals are "just" (save for the octave), but most of them are within almost tolerable limits: It's a compromise; there is no way of having the perfectly tuned just chords and freedom to modulate to any key you want with a reasonable amount of pitches (e.g. piano keys).

To hear the difference, listen to a good barbershop quartet and then play the same chords on an ET instrument like a piano. The nice, ringing qualities of the chords will be gone.

Anyway, ET divides the octave equally into 12, so the ratio between adjacent notes (like C and C#) is the twelfth root of 2 (2^(1/12) ≈ 1.05946309436). You start from your reference frequency (say, A=440), and multiply it by this number for every consecutive note. Here's a chart.

Answer 2

Short answer, these days, for finding a tonic, you probably want to use ET (specifically 12-TET) because that's what everyone else uses. However it may depend on who else you're playing with, and what they're using, so tune to them.

First, none of the tuning systems you mention specify an inherent reference pitch, so you'll need that as well. Sounds like you're going with the modern standard A4=440. Secondly, strictly speaking, JI is not actually a single tuning system, but rather a specification that two (or more) intervals must be small ratios of each other. I'll focus on the other two.

Pythagorean Tuning uses justly-intonated fifths and octaves (but no other intervals) to generate all of it's notes. Because a justly-tuned series of fifths will never close the octave (as cyco mentions), PT specifies pitches for an entire line-of-fifths. Only 12 of these actually appear on a keyboard, and there will be ugly "wolf fifth" where they don't match up. The formula is:

f = f0 * (3/2)^x

where f0 is the frequency of your reference pitch and x is the number of fifths you are away from the reference note (negative x is OK). Note that you may need to multiply or divide your answer by 2 several times to get the frequency into a usable range. (f0 < f < 2 f0 is in the octave above f0, while f0/2 < f < f0 is in the octave below, etc...).

For 12-tone Equal Temperament (12TET) each fifth is squeezed so that the octaves line up, and the circle is closed. There is no ugly wolf fifth, but no intervals (besides octaves) are justly-intonated. The formula is:

f = f0 * 2^(x/12)

where f0 is again the frequency of your reference pitch, but here x is the number of half steps above (positive) or below (negative) your reference pitch. Note that for x=12, you just get the octave doubling.

Answer 3

Well, it depends.

When playing with 12-edo tuned instruments, basically you have no option but to adapt to their root: that tone needs to be spot on, regardless of whether you render third / leading tones in just / Pythagorean intonation in a way those instruments can't.

Even when playing in a string ensemble or alone, 12-edo is quite a reasonable option for the root, being such a good "one size fits all" compromise. However, it's not really practical to pick up a 12-edo reference before you start playing. Also, it remains just that: a good compromise!

A far more natural way of seeking a reference is in the open strings. Two obvious ways to do it:

• Stick to Pythagorean. You can brachiate down the circle of fifths, i.e. G-c, C-f, f-B♭. That brings you to the "perfect Pythagorean B♭" (the frequencies given in dan04's answer). However, apart from being awkward to find, this is not really useful for playing music in B♭ major. The main problem: a double-stop between that B♭ and the open D string is a Pythagorean major third. Fine if you like Gregorian chants, but for anything post-1600 this is pretty horrible. And there's no way to correct an open string downwards.
  You can live with that if you avoid the open strings, at least the low ones. Common enough thing to do since the Romantic period. The open A-string actually makes for a usable leading note in this tuning.

• But much preferrable IMO is to tune that very double stop B♭-d to just intonation (5-limit). This is quickly done and fits naturally with double-stops actually found in many pieces. You can still play the melodic aspects in Pythagorean, in particular the leading note A should be a good bit higher than the empty string. But for harmony, the empty strings are much harder to avoid, so you should adapt the tuning to go naturally with this.
  A perhaps more convincing argument for this strategy is to think you're actually playing in g-minor. Tuning the g-minor triad G-D-b♭ gives you that same high¹ b♭. Then you just omit the G root.
  The frequency of this JI-B♭4 is
  
                         440 Hz ⋅ ⅔ ⋅ ⅘ ⋅ 2 ≈ 469.3 Hz
  
  Notably higher than in both Pythagorean and 12-edo.

So what's the standard? I don't think it's possible to say. The diapason actually sways quite a bit, even (particularly?) in good performances. In this rendition of the 6th string quartet in the end of the exposition we can hear an extremely high C double-dominant, followed by a rather lower (comparing WRT 12-edo) B♭ in the repetition. But I wouldn't conclude from this that they tune so extremely high (would be about 447 Hz) and then use a Pythagorean B♭, rather I reckon they use substantially "narrowed" tuning so a very high cello's C-string² matches a rather more ordinary, perhaps 444 1st violin e-string to JI. I'm pretty sure the open D (prominent in the 2nd violin's arpeggio at the beginning) is even lower than that standard, to make a nice just major sound together with the viola's B♭.

The overall tuning is then always somewhere in between JI and Pythagorean, but not in the same sense as with 12-edo: notes that would nominally be the same may be intonated to different pitches, depending on the musical context.

Kurt Sassmannshaus has made a couple of nice videos discussing overall string intonation.

---

¹Note that this tone is actually often replaced with the lower Pythagorean b♭ whe playing downwards-motifs in g minor, but not within vertical harmony.

²More so because the cello open C has quite a tendency to go up when played loudly.

Answer 4

Assuming A = 440 Hz, the octave starting on Middle C has the frequencies (in Hz):

• C♭ = 244.1687412149232
• C = 260.74074074074076
• (B♯₃ = 264.298095703125)
• D♭ = 274.6898338667886
• C♯ = 278.4375
• D = 293.3333333333333
• E♭ = 309.02606310013715
• D♯ = 313.2421875
• F♭ = 325.5583216198976
• E = 330.0
• F = 347.65432098765433
• E♯ = 352.3974609375
• G♭ = 366.2531118223848
• F♯ = 371.25
• G = 391.1111111111111
• A♭ = 412.0347508001829
• G♯ = 417.65625
• A = 440.0
• B♭ = 463.53909465020575
• A♯ = 469.86328125
• B = 495.0
• (C₅ = 521.4814814814815)
• B♯ = 528.59619140625

This is obtained from the formula f = f0 * (3/2)^n, where f0 is the reference frequency and n is the number of steps along the Circle of Fifths. Then multiply or divide by 2 as necessary to get all the notes into the same octave.

Answer 5

Use the frequencies of the open strings for the notes of the scale played on them, and calculate the tonic from there.

You probably do this automatically when you play, so that a high G excites sympathetic vibrations on the G string and makes the tone sound richer.