This way, sequential notes will sound consistent and steps are clear, all whole steps are much larger than the semitones. 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. To put this in tabular form, including additional pitches for each chord:īut 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. 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. The E that is a just major third above it is 325 25⁄27 Hz or 325.93 Hz. 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. 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. The first progression, G-C-Am, starts with the same root, G at 195 5⁄9 Hz. 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.Īs 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. Starting with the second progression, G-D-Am, we take the same D as the root of the second chord. 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. We'll start with G as 195 5⁄9 Hz or 195.56 Hz. Since the question is expressed in terms of specific frequencies, it might be helpful to perform this analysis with specific frequencies. 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. This in turn pushes the A down in the next chord. 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. 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. I the A and the E will both be lower by a syntonic comma in the first progression as compared to the second. 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)? Therefore, in the keys where the open strings tend to be the thirds, you'll want to avoid the open strings. In general, the thirds have more room for flexibility, and open strings are rather less flexible with respect to tuning. 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. 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. 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. 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. But that advice is useful when you want to play in just intonation, where the thirds are critical. Yes, you just tuned your strings in Pythagorean fifths (which, it must be noted, are also just intonation fifths). 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.) 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). Your E can have two different frequencies depending on the context. One fact about just intonation that many people miss is that it isn't a fixed scale. The reference tone can be the previous note, the tonic pitch, or a note that someone else is playing, such as your cellist.