Well vs. Equal Temperament (2000)
math.uwaterloo.ca
math.uwaterloo.ca
His point is not helped by making false claims such as "Equal temperament actually did not come into use until the 20th century."
More importantly, the idea that Bach's WTC was intended to "demostrate the varying key colors in well tempered tuning as one progresses around the circle of fifths" sounds pretty uncanny when you know some context. As an organist, Bach was used to being heavily restrained in which keys and intervals could be used while dodging the many obstacles that were dissonances in the instrument's tuning.
The purpose of Well temperament, used on the harpsichord, was to get rid of those limitations and gain the ability to transpose everything freely while remaining consonant and in-tune. In this way, the Well temperament can be seen as an approximation of equal temperament in a time when the mathematical tools to achieve it were not really known.
To me, it's pretty clear that the WTC is Bach gleefully exploring the freedom of transposition-invariance and free modulation. In other words: Bach would have loved the equal temperament, and this article would likely give him a hearty chuckle.
I like natural intervals, but dividing the octave in 12 tones is a rather inappropriate way of using them. European classical music adopted this system early on, and therefore was doomed to end up on the equal temperament at some point. It's not the ultimate system nor the end of temperaments, but it's been really, really useful.
Even in equal temperament a G minor chord sounds different than a D minor chord because they fall on different ranges of the instruments and the human auditory system. If you try transposing a piece in G minor to D minor, you'll notice a clear difference. You could either go up a 5th or down a 4th. If you choose to go up a 5th, things will sound quite a bit brighter. Conversely, going down a 4th could very well cause the piece to sound muddier, depending on the orchestration. At any rate, the other forms of tuning have other effects, of course, and are worthwhile to explore.
Or shrill.
> muddier
Or warmer.
(Just a nit, but I think it's important to present these qualities in both their positive and negative forms.)
Is there any evidence for this narrative? Like anyone from the time writing about this who said "the keys are not all identical, but this well-tempered tuning is the best we are capable of doing."
> The purpose of Well temperament, used on the harpsichord, was to get rid of those limitations and gain the ability to transpose everything freely while remaining consonant and in-tune.
It was certainly designed to make all keys usable, but it doesn't necessarily follow that all keys were intended to be identical.
Actually, it is important to note that the problem of temperament is generally related to keyboard instruments, which have a fixed intonation, and 12 notes to the octave. Not so on wind or string instruments.
> I like natural intervals, but dividing the octave in 12 tones is a rather inappropriate way of using them.
This is a gross misunderstanding of how we came to have 12 notes to the octave. Actually, depending on how far back you wanna go, we started with 6 notes (the hexachordal system), ut (C) to la (A), to which were added the mi and the fa of the hard and the soft hexachords, which became B and B flat. The rest of the black keys on the keyboard came from further modulating the hexachords.
In the 16th century, there were actually numerous keyboard instruments built with more than 12 notes to the octave, reaching 19 notes and sometimes even 31 notes to the octave. Having separate keys for e flat and d sharp meant being able keep using meantone tuning while playing in such remote keys as b major and d flat major.
> It's not the ultimate system nor the end of temperaments, but it's been really, really useful.
That's true, it's a practical system which makes instruments more approachable and makes it easy to play together, but there are downsides: ET removes a lot of the charm of harmony (all keys sound the same and all chords are more or less discordant). Equal-tempered thirds are actually very discordant (that is, out of tune), but people have become so used to ET, that you play them something in meantone and they'll claim the pure thirds are out of tune!
> To me, it's pretty clear that the WTC is Bach gleefully exploring the freedom of transposition-invariance and free modulation. In other words: Bach would have loved the equal temperament, and this article would likely give him a hearty chuckle.
While Bach was known to have been critical of mean-tone tuning, he was, according to evidence, a practical guy who employed the practices of his time.
Yes, one can imagine that he would have preferred equal temperament to well-temperament. One can also argue, and many do, that Bach would have preferred the piano to the harpsichord. Maybe he would have also preferred a Yamaha D7 to a pipe organ. One can imagine...
As for the discordance of equal-tempered thirds against the perfection of pure thirds, this may be true in a mathematical sense, but not necessarily in a human sense. Our brains are logarithmic in many regards, and a lot of research argues that a logarithmic scale is something for which we have affinity.
Here are a few starters to go through the recent evolution of this debate:
Just Intonation Confuted, J. Murray Barbour, 1938, and all the subsequent works that cite it
Violin intonation, PC Greene, 1937, and the many works that cite it
In addition to comments, this video has a decent comparison using sawtooth tones: https://www.youtube.com/watch?v=VRlp-OH0OEA
For me, the closest analogy would be watching Japanese movies and TV shows being able to understand their original language as opposed to relying on subtitles or dubs; there's a whole dimension of nuance I was missing before I learned the language.
http://contentspiano.blogspot.com/2006/02/pythagorean-equal-...
https://news.ycombinator.com/item?id=10145785
And another clip I uploaded of using pitch-detecting software on a studio recording (notice it detects a lot more than 4 simultaneous notes, because it's "hearing" overtones):
The first two groups I just listed are in the Bay Area, btw, so concerts are readily accessible to readers in Silicon Valley.
It's an informative article, but the reality is that nobody cares. And I mean "nobody" in the approximate sense of 99.9% of music listeners. The reality is that our brains are more than happy to glue these nasty thirds to their hypothetical frequencies. Sure, they have a little roughness to them, but it's not a big deal, especially considering that most piano music from the 18th and 19th century is composed not to highlight the weaknesses of temperaments.
Also, equal temperament isn't quite the final word on piano tuning: https://en.wikipedia.org/wiki/Stretched_tuning
https://en.m.wikipedia.org/wiki/Barbershop_music#Ringing_cho...
Two examples of fairly audible ringing (probably start a little before my timestamps to get a little context):
http://youtu.be/KpBRetTqnqQ at 3:07
http://youtu.be/z9L4QEptVRo at 4:23
I probably could have phrased my comment a little bit more delicately, but my point was in reaction to the article, not the overall concept of tunings and temperaments. What drives me nuts are these "you're doing it all wrong" types articles about something that really makes very little difference to most people. Especially this one, in which the author really didn't bother to provide any auditory demonstration of a purely auditory phenomenon.
in the approximate sense of 99.9% of music listeners.
Seriously though, it would be amazing to have mainstream acceptance and performance of classical works in their intended tunings. That would be cool. Half the stuff just doesn't sound right in equal temperament.
The article, like everything else in life, is for those who do care.
- pure major/minor
- pythagorean
- mean tone
- werckmeister/kirnberger, which i've heard mentioned as a good option for acoustic pianos a few times (i suppose i should try it)
For any besides ET, you have to specify the root/base note.
I think my synths have these options as well, so if you know anybody that has one, you can try it yourself. I think for string instruments, a few cents difference is pretty audible, but not so much for, say clarinet which doesn't have the sharp transients, unless you're playing above the treble clef staff. and then when you hit a track with overdrive/distortion, reverb and delay, it becomes very hard to discern differences.
I couldn't decide at the time if what he was saying was legit or if he was just trying to convince me that I needed to hire him again instead of just buying an electronic tuner and a wrench.
The tuning described by the above beating plan provides a good approximation of equal temperament across the range of the temperament octave. If extended further, however, the actual tuning of the instrument becomes increasingly inaccurate because of inharmonicity, which causes harmonics to run slightly sharp, as increasingly higher tones in the harmonic series are reached. This problem is mitigated by "stretching" the octaves as one tunes above (and to an extent below) the temperament region. When octaves are stretched, they are tuned, not to the lowest coincidental overtone (second partial) of the note below, but to a higher one (often the 4th partial). This widens all intervals equally, thereby maintaining intervallic and tonal consistency.
Also, piano tuners can create just the right amount of movement in the sound by detuning the strings under each note.
You're not going to create that effect if you have no idea what you're doing.
I thought that was a crackpot theory, but then he played some pieces in their respective individual tuning systems on a harpsichord, and they sounded absolutely amazing, better than well tempered (or equal obviously). So who knows!
As I discovered just now, the whole thing is available on YouTube [https://www.youtube.com/watch?v=TxEcifeMf08, only 1.5 hours, not 3 hours as the video incorrectly suggests].
Okay, since I made some progress with violin, I understand that the intervals of a minor third (three semi-tones), major third (four semi-tones), a fourth (five semi-tones), a fifth (seven semi-tones), a sixth (nine semi-tones), and an octave (12 semi-tones) consist of two frequencies that are ratios of small whole numbers and where the interval of a semi-tone is the ratio of two frequencies of approximately 2^(1/12). Commonly violinists call these intervals, especially the major third, fifth, and octave, perfect.
And I understand that setting the frequency of the first A above middle C to 440 Hz and all the semi-tone frequency ratios to 2^(1/12) is tempered tuning.
Ah, now I see in the OP:
"Equal temperament, the modern and usually inappropriate system of tuning used in western music, is based on the twelfth root of 2. The ratio of frequencies for each semi tone is equal to the twelfth root of two."
So, what I called tempered tuning the OP calls "equal temperament".
Is that the same as well temperament?
For all the other terminology about approaches to tuning, e.g., mean tuning, I have no clear definitions.
E.g., it appears that the OP is still vague on just what equal tuning is.
Is there a source with clear definitions?
Might be a little tricky to set up, though, since only a few DAWs and plugins support it.
Every electronic "stage" piano I've owned has had an option to modify the tuning, at minimum allowing you to choose between equal, well, meantone, pythagorean.
I remember one Yamaha I owned even had the option to customise the tuning yourself. Can't remember the model number, it cost a decent-ish chunk of change, around 800 GBP about 15 or so years ago.
http://www.truetemperament.com/
Looks weird, but is actually ok to play on. Steve Vai has an EVO with TT frets.
https://news.ycombinator.com/item?id=10120648
A web-demo of the Tune.js library which offers a small, but instructive selection of tunings and temperaments which can be controlled by the alphanumeric keyboard, or by pointing and clicking.