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I read that the violin can be tuned to perfect fifths so I was thinking of tuning my violin to G3=195.55Hz, D4=293.33Hz, A4=440Hz, E5=660Hz.

But I don't understand what it means to play single stops with pythagorean intonation and double stops with just intonation, since they seem to be functions of a reference tone. How do I choose the reference tone? I got unsure because I saw a video where they said one avoids open strings in flat keys (and I thought I just tuned the open strings in pythagorean temperament? perhaps there is some conflation of terms I have missed.)

I have seen amongst others

I have read amongst others https://en.m.wikipedia.org/wiki/Just_intonation https://en.m.wikipedia.org/wiki/Pythagorean_interval https://en.m.wikipedia.org/wiki/Wolf_interval

It is probably some kind of rule of thumb I have missed...

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  • It would be very helpful if you would post specifically what you read and/or link(s) to the videos you watched.
    – Aaron
    Oct 18 at 19:49
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    Oof; I'm a violinist (and a baroque violinist at that, so I have to talk about temperaments!) and am already confused. But let's problematize one assumption right off the bat: "I read that the violin resonates more if tuned to perfect fifths..." 1) But does it? and 2) what is meant by "perfect"? In the music theory sense, "perfect fifth" just means a fifth as we know it, and in that sense, any other tuning would be non-standard. But if we're talking about nudging that fifth narrower or wider by a few cents, then "resonance" is not going to be the primary aim. ... Oct 18 at 20:18
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    I think I mean sympathetic vibrations on open strings but not sure. I can remove that word since it doesn't matter for my question.
    – Emil
    Oct 18 at 20:21
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    The resonance of the instrument is affected much more profoundly by many other things. The placement of the sound post most of all—if you want the instrument to resonate more at a specific pitch, you can monkey around with the sound post positioning. The gauge and material of the strings, the height and shape of the bridge, the relative humidity of the room all play more of a role in "resonance" than a temperament. Add to that the fact that the violin is a fretless instrument, so we adjust (or misadjust) the temperament of all non-open notes manually. Oct 18 at 20:21
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    And one footnote to any discussion of temperament and tuning: If you're playing one instrument, solo, you can of course do anything you want. If you're playing with others, you have to ask yourself what works best overall. A string quartet can take principles that work well for violin and spread them over 5 fifths. If you're playing with an equal-tempered piano, you better get on board the ET train. If you're playing with a harpsichord... then they spend 15 minutes explaining the temperament, 2 hours tuning, 30 seconds playing in tune, and the rest of the concert being out of tune. Oct 18 at 20:26
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How do I choose the reference tone?

The reference tone can be the previous note, the tonic pitch, or a note that someone else is playing, such as your cellist.

One fact about just intonation that many people miss is that it isn't a fixed scale. Suppose you're playing in G major. Your E can have two different frequencies depending on the context. If the harmony is G-C-Am, the frequency of that E will be 5/3 times the frequency of G (4:3 times 5:6 times 3:2), but if the harmony is G-D-Am, the frequency of the E will be 27/16 times the frequency of the G (3:2 raised to the third power, divided by 2).

I saw a video where they said one avoids open strings in flat keys (and I thought I just tuned the open strings in pythagorean temperament? perhaps there is some conflation of terms I have missed.)

Yes, you just tuned your strings in Pythagorean fifths (which, it must be noted, are also just intonation fifths). But that advice is useful when you want to play in just intonation, where the thirds are critical.

The open strings are more likely to be major thirds in flat keys, and major thirds are the pitches that one lowers to achieve just intonation. For example, in B♭ major, the G string is the major third of the subdominant chord, D is the major third of the tonic, and A is the major third of the dominant. By contrast, in keys with no flats, the open strings are going to be more stable: they are likely to be the root of the subdominant, dominant, or tonic chord.

If you're truly playing in Pythagorean tuning, it shouldn't make a difference whether you use the open string or not, but people rarely use pure Pythagorean tuning because people rarely play a melody by itself with no harmony, and, if there's harmony, you want to tune the major thirds a bit lower.

In general, the thirds have more room for flexibility, and open strings are rather less flexible with respect to tuning. Therefore, in the keys where the open strings tend to be the thirds, you'll want to avoid the open strings.

In a comment, you ask

In the first part, do I understand you correct that in one case two chords contain E so you use the just E (major sixth of G), and in the other case one chord contains E so you use the pythagorean E (major sixth of G)?

I responded:

@Emil the A and the E will both be lower by a syntonic comma in the first progression as compared to the second. The reason for this is not because of the number of chords containing E but because of the identity of the first chord that contains either A or E. The G-C progression in the first example causes the first A or E that appears to be E as the major third of the C chord. This in turn pushes the A down in the next chord. By contrast, in the second progression, G-D gives us A as the fifth of the D chord, which pushes the E up in the next chord.

Since the question is expressed in terms of specific frequencies, it might be helpful to perform this analysis with specific frequencies. We'll start with G as 195 5⁄9 Hz or 195.56 Hz.

To find D, which is the fifth of the initial G chord, we multiply that frequency by 3:2, giving 293 1⁄3 Hz or 293.33 Hz, the open D string.

Starting with the second progression, G-D-Am, we take the same D as the root of the second chord. The fifth of that chord is A, and, no surprise, multiplying 293 1⁄3 Hz by 3:2 yields 440 Hz for the frequency of A, the open A string.

As we did before, we use that A as the root of the succeeding chord, and, again, to nobody's surprise, the frequency of E works out to 660 Hz.

The first progression, G-C-Am, starts with the same root, G at 195 5⁄9 Hz. Using that G as the fifth of the following C chord, we can find C through multiplication by 2:3, which gives the frequency of the viola's open C string. To keep this in violin range, however, we'll go up a perfect fourth, using 4:3, which gives us a frequency of 260 20⁄27 Hz or 260.74 Hz. The E that is a just major third above it is 325 25⁄27 Hz or 325.93 Hz. Going up a perfect fourth from that E to find the root of the following chord, we get an A of 434 46⁄81 Hz or 434.57 Hz.

To put this in tabular form, including additional pitches for each chord:

Frequencies in Hz G D Am G C Am
E5 660 E5 651 23⁄27 651 23⁄27
C5 528 C5 521 13⁄27 521 13⁄27
A4 440 440 A4 434 46⁄81
G4 391 1⁄9 G4 391 1⁄9 391 1⁄9
F♯4 366 2⁄3 F♯4
E4 E4 325 25⁄27
D4 293 1⁄3 293 1⁄3 D4 293 1⁄3
C4 C4 260 20⁄27
B3 244 4⁄9 B3 244 4⁄9
G3 195 5⁄9 G3 195 5⁄9

Within each half of the table, left and right, the values in any given row are the same. Between the left half and the right half of the table, however, the values for A, C, and E differ by a factor of 81:80, the syntonic comma.

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  • In the first part, do I understand you correct that in one case two chords contain E so you use the just E (major sixth of G), and in the other case one chord contains E so you use the pythagorean E (major sixth of G)?
    – Emil
    Oct 19 at 19:50
  • @Emil the A and the E will both be lower by a syntonic comma in the first progression as compared to the second. The reason for this is not because of the number of chords containing E but because of the identity of the first chord that contains either A or E. The G-C progression in the first example causes the first A or E that appears to be E as the major third of the C chord. This in turn pushes the A down in the next chord. By contrast, in the second progression, G-D gives us A as the fifth of the D chord, which pushes the E up in the next chord.
    – phoog
    Oct 19 at 20:03
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    @Emil I expanded the answer to put some concrete numbers into the analysis.
    – phoog
    Oct 20 at 10:18
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But I don't understand what it means to play single stops with pythagorean intonation and double stops with just intonation, since they seem to be functions of a reference tone.

It basically means that when playing single-note melodies, you should focus on getting the tuning of all whole steps close to 9:8, and any two notes related by a fourth or fifth should be at a ratio of 3:4 or 2:3, respectively. This way, sequential notes will sound consistent and steps are clear, all whole steps are much larger than the semitones.

The flip side is that intervals such as major thirds and sixths come out with complex ratios (81:64, 27:16). That isn't such a noticeable problem when an agile solo melody plays over a background accompaniment, but it's quite notable when two equally-prominent voices are harmonizing, in particular if they're both played on the same instrument. In that case, you'll hear a strong beat between the lower overtones of each tone. E.g. in two voices playing the 81:64 Pythagorean major third of A3 against C♯4, the A3 has its fifth overtone at 1100 Hz, whereas the C♯4 has its fourth overtone at 1114 Hz. The difference of 14 Hz is neither high enough to be really heard as a clear resultant, nor low enough to be a smooth sound-modulation – it just sounds rough.

That can be addressed by tuning the major third narrower, namely to Ptolemaic just intonation. Which sounds great for held chords, but it introduces its own problem: now you have two whole steps of different size, namely 9:8 and 10:9. And that sounds weird in a melody: both steps are similar enough to be classed as the same interval, yet just different enough to seem like something's wrong.

Of course, this issue is resolved by meantone temperaments, including 12-edo, by putting the whole step right in the middle of the major third. Now all whole steps are the same again... but none of them have the wide, confident character of Pythagorean steps!

What your source is suggesting is that you should watch out for all these problems and choose the tuning accordingly, to fix whatever doesn't work in a given piece/segment.

  • A double stop sounds rough? Tune it to JI. This mainly entails making major thirds and sixths a bit narrow.
  • A melodic line sounds uneven or feeble? Widen all the whole steps to Pythagorean.

they said one avoids open strings in flat keys

The reason for this would be: e.g. in the key of F major, the open A is the major third of the tonic. If you're going to have that major third as a JI double stop, the only way to do it with an open A is to raise the F up. But that won't work if e.g. the piano accompaniment already contains a low F bass note: your F would be notably out of tune relative to that.

Instead, the suggestion is to play the A fingered and slightly flat compared to the open string. This way, both your F and your A will sound good over the piano bass note. That A will be 15 ct flatter than an A the piano might play, but (probably not coincidentally) classical counterpoint avoids giving two different voices the third of a chord at the same time, so this isn't a big deal.

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    "yet just different enough to seem like something's wrong": it never sounded wrong to me. "Probably not coincidentally": indeed. But I suspect that this is less about tuning with other instruments (piano accompaniments of solo parts rarely have open fifths when the solo part is on the third) than it is about how raising the F would also imply raising most of the other pitches in the F major scale. Though as a bass I have to say that judicious raising of bass notes can be very effective.
    – phoog
    Oct 19 at 0:25
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FWIW, I think most of us bowed string players tune in mathematically true fifths (or fourths for the bass viol), if for no other reason than to do otherwise leads to out-of-tune open string double stops.
As everyone is pointing out, you can adjust the tuning of every fingered note -- and to some extent even the harmonics -- to match the temperament selected for the music you are playing. We do avoid single-stop open string notes more often in quiet passages; when the music is forte the pitch differences become pretty much undetectable by the human neuroaudiologics.

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  • Also perfect fifths are very close to equal temperament. String players playing with harpsichord or organ in an unequal temperament often do tune open strings separately to the keyboard, because the fifths are often rather farther from perfect than are equal-tempered fifths. In fact, in at least one unequal temperament, the open string fifths are the only fifths that are tempered; the others are tuned pure. I never quite understood why that would be desirable.
    – phoog
    Oct 19 at 20:10

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