
Well vs. Equal Temperament (2000) - Phithagoras
http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html
======
pierrec
I appreciate a good attack on 12-tone Equal Temperament from time to time (and
more importantly a good defense of natural intervals), but this article really
goes over the top in being extremely opinionated while forcibly trying to
sound factual.

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.

~~~
Omnus
In addition, the article puts forth a commonly held (but false) notion that
equal temperament destroys any sense of key color or identity, apart from
pitch. That would be true if every instrument or voice sounded exactly the
same throughout its entire range, but that is never the case.

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.

~~~
vitd
Yeah, I had to laugh at that, too. I remember in my music theory class, one
time the professor played something in D-minor, and someone in the class
pulled out that line from Spinal Tap, "D minor is the saddest of keys." We
chuckled, but the prof said, "Yeah, it really is!" and we had an interesting
discussion about it.

------
codezero
Does anyone have links to audio samples where these differences are prominent?

In addition to comments, this video has a decent comparison using sawtooth
tones: [https://www.youtube.com/watch?v=VRlp-
OH0OEA](https://www.youtube.com/watch?v=VRlp-OH0OEA)

~~~
acjohnson55
Wouldn't that be nice, rather than whining about numbers?

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.wikipedia.org/wiki/Stretched_tuning)

~~~
baddox
It's probably true that 99.9% of music listeners don't know, don't care, and
can't tell the difference. But a few of us care a lot, and it's sometimes very
important. I've been recently obsessed with barbershop quartet music, where
the tuning of chords is absolutely pivotal. The alignment of overtones from
justly tuned chords is, if you ask me, the entire point of the genre. The
"ringing" produced from the aligned overtones is, I would guess, noticeable to
a large portion of people with rudimentary music education.

[https://en.m.wikipedia.org/wiki/Barbershop_music#Ringing_cho...](https://en.m.wikipedia.org/wiki/Barbershop_music#Ringing_chords)

Two examples of fairly audible ringing (probably start a little before my
timestamps to get a little context):

[http://youtu.be/KpBRetTqnqQ](http://youtu.be/KpBRetTqnqQ) at 3:07

[http://youtu.be/z9L4QEptVRo](http://youtu.be/z9L4QEptVRo) at 4:23

~~~
acjohnson55
You make a good counterpoint, for barbershop. It's definitely a form designed
to bring out the effect of these microintervalic adjustments. Of course, those
cords are sung in just intonation, not well temperament :)

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.

------
haberman
Aside from temperament, I had my piano tuner claim to me that tuning pianos is
actually more complicated than that. He said that the high end of the piano
had to be tuned different from the low end, with some explanation like how the
hammer strike bends the pitch of low vs high strings (I wish I could remember
the specifics of what he said, but this was many years ago).

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.

~~~
gtani
legit, she/he was talking inharmonicity and the resulting octave stretching
that makes octaves sound truest.

[https://en.wikipedia.org/wiki/Inharmonicity](https://en.wikipedia.org/wiki/Inharmonicity)

~~~
elihu
[https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_...](https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_curve)

------
xpda
There is some debate as to whether Bach intended equal-temperament or well-
temperament. These terms were not well-defined in his day.

[https://books.google.com/books?id=yZ95L8Xohs0C&pg=PA4](https://books.google.com/books?id=yZ95L8Xohs0C&pg=PA4)

~~~
plaguuuuuu
There's one guy on youtube who claims Bach's squiggles underneath each piece
title in the original WTC manuscript actually represents an ideal tuning
system for that piece.

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!

~~~
lamby
Bach seems to attract all sorts of bizarre numerology, alas.

~~~
plaguuuuuu
I agree, but when it actually _works_.... shrug

~~~
lamby
Well, The Bible Code "works" too.

------
Cshelton
So this is interesting, I know a piano tuner who has tuned for countless
famous artist. He was working on a project to record songs, like Beethoven, as
someone would have heard it in his time.

~~~
c3534l
I actually found it a little odd that Beethoven was mentioned, since it he was
a big fan of equal temperment. His 5th symphony was largely written to explore
the idea. In well tempered tunings, you have to play around a central "root
note" in order to get a harmonic melody. But in equal temperment, the
dissonance is equally spread out across the notes. His 5th symphony was able
to take advantage of that in two ways: by being an unusually dissonant
symphony to begin with and by jumping all over the place, going from very high
to very low, playing the same note repeatedly and yet having it sound the same
no matter what key he went to. We're so used to it, however, I doubt most
people could really hear the difference.

~~~
shadgregory
I had no idea that Beethoven used equal temperament. Do you have a source for
that?

~~~
c3534l
Nah, just from learning about music years ago. Not everything he wrote was
equal temperment, though.

------
analog31
In addition to the sonic characteristics of the earlier tunings, an important
practical factor was that the tuning procedure could be carried out
successfully by the keyboardist prior to every performance, since instruments
went out of tune quickly before the modern piano.

------
graycat
The OP and some of the links in this thread have a lot of terminology about
_tuning_ that is a bit short on precise definitions.

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?

------
arntatis
Can this be applied to an electric piano to get a similar type of colour in
the sound?

~~~
mootothemax
>Can this be applied to an electric piano to get a similar type of colour in
the sound?

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.

------
JoachimS
There is a company in Sweden that builds necks for guitars with true
temperament frets:

[http://www.truetemperament.com/](http://www.truetemperament.com/)

Looks weird, but is actually ok to play on. Steve Vai has an EVO with TT
frets.

[http://www.truetemperament.com/videos/](http://www.truetemperament.com/videos/)

------
Zuider
Related link posted in a previous discussion:

[https://news.ycombinator.com/item?id=10120648](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.

------
abannin
It's all fun and games until you start trying to tune a guitar. Pianos have it
easy.

~~~
plaguuuuuu
Remove all frets and tune to 5ths :|

~~~
Zuider
Q: What's the definition of a semitone? A: Two fret-less bass guitars playing
in unison.

~~~
plaguuuuuu
:D

