37

We can tune each string/pipe to a given frequency as accurately as we need to for musical purposes. We can't do it so that they collectively satisfy several musically desirable properties, because it turns out our definition of those properties is logically inconsistent. The best technology in the world cannot fulfill a requirement that contradicts itself. ...


28

I want to make an addition to all these excellent answers. With just intonation, it's not possible to make all the chords just. Not even in a single key. Let's look at the common just major scale based on I, IV and V just major triads: C 1:1 D 9:8 E 5:4 F 4:3 G 3:2 A 5:3 B 15:8 In this scale, I, IV, V major triads (4:5:6) and iii and vi minor triads (10:...


24

In principle, the answer is yes, with software instruments it is feasible to (re-)set the tuning so that you can realize music with modulation that stays in just intonation across these changes. The frequencies are directly accessible in sound synthesis environments like PureData or Overtone, and even just by setting the tuning information in a set of MIDI ...


21

Why can't notes be tuned according to a defined frequency? They can. But what we can't do is tune them to "the correct" frequency, because there are different ways in which the 'correct' frequency could be specified. You've mentioned two of them in your question - just intonation, and equal temperament. As Kilian Foth's answer explains, both of those ways ...


19

Every note has a pitch, determined by the fundamental frequency of the sound wave that produces it. When you have two different notes, you have two different pitches, caused by two different frequencies. The distance between those pitches is called an interval, and corresponds to the ratio of the note's frequencies. For example, if one note is an octave ...


17

You cannot even realize "just temperament" reliably when you are working with continuous-tone instruments like singers and trombones. Take a look at even something as old as J.S. Bach's mass in B minor, like the "Confiteor" which goes off-tonality somewhere after 2:30 (in this recording) and loses tonal center rather thoroughly between 3:00 and 4:00. The ...


14

Modern harmony revolves around tonic-dominant relationships, and part of what makes a V-I or V-i sound convincing to our ears is that the leading tone goes up a half-step to the tonic. In the common-practice era, this is why we see so many examples in minor that seem to be well described by the idea of a melodic minor scale. On the way up, we want to hear ...


13

It's a bit more complicated than may appear at first glance. Within a single key, if Just Intonation makes the I,IV, and V chords all (4,5,6) ratios, the ii chord will be off. The other question is what note to play as a melody note. Often, melodies are somewhat independent of the underlying chords (at least in CPP if not in Jazz and other Pop theories). I ...


13

The earliest use of equal temperament was on fretted instruments with fixed frets. The ratio of 17:18 for the string length for successive frets is a good approximation to equal temperament. The errors were well within the tolerance of other intonation issues such as non-uniform gut strings, and the different amounts of string bending on different frets ...


11

"just intonation better than equal temperament" Judgement call there. When instruments are slightly off perfect ratios, there can be very appealing beating and chorus effects. Piano strings are intentionally mistuned from each other by slight amounts. Nothing but perfect ratios can sometimes lead to a very thin sound. Depends on context. "instruments which ...


10

I think there are actually two separate (though interrelated) issues that you're asking about here. On the one hand, there's a question about music with pitches tuned to the overtone series of a particular fundamental ("should I base the tuning on the same tonal center?"), and on the other hand there's tuning based on simple ratios between integers ("how ...


10

If you're using a tuner, then you can safely use the octave harmonic to tune. She is wrong in saying blanketly that "the harmonics are slightly flat". Some are flat, some are sharp, some match equal temperament exactly*. This page has a figure that shows the relative sharpness and flatness of the first several harmonics. Often people use "...


9

I admire a lot of Adam's work, but I think he's exaggerating a bit about the reasons why choirs get off pitch. (Though he's stating a commonly held belief -- or perhaps common excuse.) Yes, most choirs tend to drift in pitch when singing a cappella, but I guarantee you that at least 99% of the time, it's not due to "comma drift." Instead, non-professional ...


8

Just intonation does produce harmonic sounds; perhaps the most harmonic sounds possible. You are correct that for a Justly tuned system to work, then each of the tones that you use will need to be adjusted relative to the current tonic. Because of this, you are correct to think that there will need to be many different 'flavors' of each note, depending on ...


8

The Pythagorean chromatic scale uses 3-limit just intonation to get 17 pitches in the octave, with no notes between B and C or between E and F. Ptolemy's intense chromatic scale uses 5-limit just intonation to get 19 pitches in the octave, including B♯/C♭ between B and C, E♯/F♭ between E and F. You can read a little bit about them at the Wikipedia entry for ...


7

Using a keyed instrument with Just Intonation creates a bunch of puzzles that need to be solved. You are either faced with observing limits on navigating from place to place, or doing "comma pumps" (equating near by intervals, or bend/vibrato between them because they are close enough). The problem isn't really Just Intonation though. It's caused by ...


7

Unfortunately, it's a bit more complicated than that. There is no single "Just Intonation System"; instead, there are multiple systems which can be said to be just, by virtue of the fact that they use just intervals (i.e. integer frequency ratios). The problem is in determining which ratios you want to use. One such Just system is the Pythagorean system, ...


7

Your diagram indeed shows a 19-pitch Pythagorean scale. 19 just perfect fifths exceed 11 octaves by an interval of frequency ratio 319:230. This interval is about 137.145 cents. If you want to divide the octave into 19 and have a closed circle of 19 fifths, then your 19 fifths, however you temper them, will add up to exactly 11 octaves. So the question is ...


7

Even without the piano (or even keyboard instruments), there were other forces pushing toward something close to equal temperament. The common narrative is that chromatic music was instrumental in the use of temperaments that were more equal. While that is a factor, it's important to look at how chromatic music could be even in the 1600s. Frescobaldi's ...


7

With a bit of training, a good musician can hear differences of 2 cents, and with significant talent and/or a lot of practice, 1 cent. I base the above statement on my personal experience with developing ear training software for musicians. For example, I have been working on an app for training musicians to tune a guitar or a piano, purely by ear. This ...


6

The first thing to consider for 13-limit is the octave-reduce thirteenth harmonic, 13/8. It is the first sixth that occurs in the harmonic series and comes in at about 840.53 cents. It's pretty close to being smack dab in the middle of the 12tet minor sixth and major sixth. So, like 11-limit, this limit is going to contain some neutral intervals. In fact, ...


6

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 ...


6

The biggest issue here is that computer based virtual instruments just don't sound right. Even virtual versions of analog synthesizers don't sound quite like the real thing. Plus, the feeling and method of play usually can't be reproduced at all, as in the case of the violin or French horn or clarinet, say. In addition, I'm not aware of an algorithm ...


6

Just Intonation is a tuning system; that is, it defines the tuning of a scale. We commonly use equal temperament, which is a compromise (or temperament) that allows us to play in all keys with a limited number of notes (without constant retuning). Just Intonation defines the actual size of the intervals, or the tuning of the notes with respect to the tonic ...


6

To the best of my knowledge there was not a specific known frequency of one pitch that was compared to a specific known frequency of another pitch. However, it was relatively simple to pluck a string, hear the pitch. Then pluck the string at half of the length (possibly by depressing that string at that point, or touching the string at the half-way point),...


6

The reason doesn't seem to be a technical advantage but the favored traditional sound of the just tuning that bagpipers describe as more colorful and warmer: Patrick McLaurin writes in his bagpipe blog: If bagpipers used the equal temperament scale the drones would sound out of tune for every note but the As, although B, D, and E might be close enough. C#, ...


6

One more problem is that piano strings are under far more tension than those in other instruments. On average, each string is under 200-300 pounds of tension. Unlike the violin or guitar or harpsichord and their close relatives, piano strings are anharmonic vibrators. The frequency of the first overtone is more than 2/1 and the second is higher than 3/1. The ...


5

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 ...


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