There is some basic information that is very wrong in this article. For example:
"My track is tuned in a system called five-limit just intonation via the magic of MTS-ESP. It’s the basis for all the tuning systems used in Western Europe between about 1500 and 1900."
No - at no point in the last 500 years was 5-limit just intonation ever the predominant tuning system used anywhere in Western Europe. The real predominant tuning system was called "meantone temperament," to which this article sadly devotes only about 3 words - and those words are only about 12 tone meantone keyboard layouts, not about the bigger, abstract idea of meantone temperament in general as it was understood and taught by practitioners of the day.
There is a very important difference between meantone and just intonation. The goal of meantone was to have four tempered perfect fifths (approximately a 3/2 frequency ratio) add together to approximate the fifth harmonic (or a 5/1 frequency ratio). Thus, the major third from the circle of fifths would approximate a 5/4 ratio with the tonic, and the major chord would approximate a very crunchy sounding 4:5:6 ratio. In order to do this, fifths are all flattened slightly to make the tradeoff - flattening the fifths by 1/4 of a "syntonic comma" was typical, or "quarter-comma meantone". Even though keyboard instruments evolved in a more well-tempered direction, meantone was the way that teachers of the common practice era (such as Leopold Mozart) still taught and thought about this stuff.
Meantone sounds noticeably different from just intonation, where the major third from the circle of fifths is a syntonic comma sharp of a 5/4 ratio (about 22 cents). In just intonation, if you want your major chords to be 4:5:6, you need to bring in this other, different, independent 5/4 major third that is not on the circle of fifths. As a result, certain chord progressions that are common in Western music will tend to exhibit strange sounding "comma drifts" if you play them in just intonation. Adam Neely has a good video on "Benedetti's Puzzle" about this for those who are interested.
Of course, there is nothing wrong with just intonation, and comma shifts can sound interesting if you want to deliberately use them in some kind of modern microtonal setting, but it simply isn't the tuning historically used in common practice Western music.
Anyway, though, if you go through the article with a marker and replace all instances of "just intonation" with "meantone," the general idea is mostly correct.
Hi, I'm the author of the blog post. I said that 5-limit is the basis for systems like meantone, which is true. Meantone systems take 5-limit as their starting point and then modify it. I deliberately skated over the specifics of how meantone works on purpose, because I have too much experience watching my students' eyes glaze over when I talk about this kind of thing. I'm trying to strike a balance between giving correct information and not turning people away.
I can't comment regarding how you think is best to teach your students. This is now a popular blog post that has gone viral on HackerNews to a much wider audience of well-educated people, so you should expect people will clarify these things on here. I'm talking mostly about stuff like this:
> Five hundred years ago, however, it would have made a very big difference. Before the advent of temperament systems, D-sharp and E-flat were two different notes. They weren’t just written differently; they sounded different. You can compare the historical versions of these notes yourself in this track I made... My track is tuned in a system called five-limit just intonation
^ These are not the historical versions of those notes. 5-limit just intonation was not in widespread use in Western Europe 500 years ago. 500 years is not before the advent of temperament systems. And so on. Teach this to your students however you think is best, but people on here may be interested to know that.
I know that pure five-limit just wasn't ever a prevalent tuning system, but all the temperament systems are based on it, so it seems reasonable to present it as a kind of baseline, the "ground truth" from which the meantone and well temperaments are departing to a greater and lesser degree. As to the dates, I have read a bunch of histories of tuning, and the main conclusion they present is that there was no uniform (or even predominant) tuning standard until 12-TET, so simplification is necessary. I am definitely open to the idea that 1500 isn't close enough to the "real" widespread advent of temperament to use as an approximation and that I should instead be saying 1400 or whenever.
"My track is tuned in a system called five-limit just intonation via the magic of MTS-ESP. It’s the basis for all the tuning systems used in Western Europe between about 1500 and 1900."
No - at no point in the last 500 years was 5-limit just intonation ever the predominant tuning system used anywhere in Western Europe. The real predominant tuning system was called "meantone temperament," to which this article sadly devotes only about 3 words - and those words are only about 12 tone meantone keyboard layouts, not about the bigger, abstract idea of meantone temperament in general as it was understood and taught by practitioners of the day.
There is a very important difference between meantone and just intonation. The goal of meantone was to have four tempered perfect fifths (approximately a 3/2 frequency ratio) add together to approximate the fifth harmonic (or a 5/1 frequency ratio). Thus, the major third from the circle of fifths would approximate a 5/4 ratio with the tonic, and the major chord would approximate a very crunchy sounding 4:5:6 ratio. In order to do this, fifths are all flattened slightly to make the tradeoff - flattening the fifths by 1/4 of a "syntonic comma" was typical, or "quarter-comma meantone". Even though keyboard instruments evolved in a more well-tempered direction, meantone was the way that teachers of the common practice era (such as Leopold Mozart) still taught and thought about this stuff.
Meantone sounds noticeably different from just intonation, where the major third from the circle of fifths is a syntonic comma sharp of a 5/4 ratio (about 22 cents). In just intonation, if you want your major chords to be 4:5:6, you need to bring in this other, different, independent 5/4 major third that is not on the circle of fifths. As a result, certain chord progressions that are common in Western music will tend to exhibit strange sounding "comma drifts" if you play them in just intonation. Adam Neely has a good video on "Benedetti's Puzzle" about this for those who are interested.
Of course, there is nothing wrong with just intonation, and comma shifts can sound interesting if you want to deliberately use them in some kind of modern microtonal setting, but it simply isn't the tuning historically used in common practice Western music.
Anyway, though, if you go through the article with a marker and replace all instances of "just intonation" with "meantone," the general idea is mostly correct.