Well vs. Equal Temperament (2000)

http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html

by Phithagoras • 11 years ago

99 62 11 years ago

math.uwaterloo.ca

codezero• 11 years ago

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

CarVac2• 11 years ago

I went looking.

https://www.youtube.com/watch?v=TBt6APk21tU

EarthLaunch2• 11 years ago

Thank you. That is astounding. I wonder if there is any way to find music performed using original tunings. It sounds SO much better to me.

abannin2• 11 years ago

Hearing Chopin in Meantone is pretty awesome.

acjohnson552• 11 years ago

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

analog313• 11 years ago

To make matters worse, or maybe better, most non keyboard instruments have no sensible temperament whatsoever. Even fretted instruments are a compromise.

baddox2• 11 years ago

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...

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

plaguuuuuu5• 11 years ago

    in the approximate sense of 99.9% of music listeners.

The best thing about the Internet is that connects the other .1% that actually care, and the rest of humanity that has no appreciation of good music can ignore it. And really, the great composers of the future will be part of that tiny minority.

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.

coldtea• 11 years ago

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 article, like everything else in life, is for those who do care.

smortaz• 11 years ago

Wendy Carlos released a 2nd version of her Bach Brandenberg Concertos that were done using period tuning on synthesizers including modulations. perhaps you can compare the two.

intopieces• 11 years ago

Do you mean "Switched-On Brandenburgs: The Complete Concertos Vol. 1"?

gtani2• 11 years ago

for one of my digital pianos (Yamaha), there's options for these (besides ET):

- 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.

Cshelton• 11 years ago

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.

c3534l4• 11 years ago

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• 11 years ago

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

stevetjoa• 11 years ago

Slightly off-topic, but because I love this topic so much: there is a wonderful documentary called Pianomania about a piano tuner ("piano technician", if you will) to the stars. The story centers around a good-natured piano tuner and his seemingly impossible task of translating the pianists' vaguely expressed desires into a well-tuned piano. As the movie illustrates, piano tuning is just as much about timbre as it is about pitch -- perhaps even more so!

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].

analog313• 11 years ago

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.

haberman2• 11 years ago

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.

gtani2• 11 years ago

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

https://en.wikipedia.org/wiki/Inharmonicity

elihu• 11 years ago

https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_...

vfdfv• 11 years ago

I am certainly no expert either, but maybe your piano tuner was referring to the practice of "stretching" an octave:

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.

https://en.wikipedia.org/wiki/Piano_tuning#Stretch

xpda2• 11 years ago

Traditionally (recent tradition), each note in the upper couple of octaves of a piano have been tuned slightly higher because it's supposed to sound better. Electronic pianos and other instruments don't seem to be tuned this way, so acoustic pianos are now being tuned this way less often.

c3534l4• 11 years ago

Yup. There's even a strange thing where as we get older, our perception of low tones changes so that music that sounds out of tune for an 18 year old sounds great to an 80 year old.

squeaky-clean• 11 years ago

Do you have a source for this? Sounds really fascinating, and I would love to read more about this.

shiro• 11 years ago

High-end electronic tuners actually have settings for stretched tuning.

TheOtherHobbes• 11 years ago

Your piano tuner was completely correct. Piano strings have a non-linear overtone distribution, and if you tune to an electronic tuner without allowing for that they'll sound very bad.

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.

xpda2• 11 years ago

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

plaguuuuuu5• 11 years ago

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!

lamby2• 11 years ago

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

arntatis• 11 years ago

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

archagon• 11 years ago

Yes! With the Scala file format: http://www.huygens-fokker.org/scala/

Might be a little tricky to set up, though, since only a few DAWs and plugins support it.

mootothemax• 11 years ago

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.

abannin2• 11 years ago

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

plaguuuuuu5• 11 years ago

Remove all frets and tune to 5ths :|

Zuider2• 11 years ago

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

Zuider2• 11 years ago

Related link posted in a previous discussion:

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.

JoachimS• 11 years ago

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

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/

pierrec2• 11 years ago

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.

Omnus2• 11 years ago

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.

_greim_• 11 years ago

brighter

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.)

vitd2• 11 years ago

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.

haberman2• 11 years ago

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.

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.

ciconia• 11 years ago

In fact, equal temperament as a concept was known and discussed by theoreticians since at least the 16th century, and was advocated as an ideal solution for tuning and adjusting frets on fretted instruments (namely the lute and the viola da gamba). But equal temperament stayed more or less a theoretical concept, though players of fretted instruments apparently employed an approximation of equal temperament.

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...

pierrec2• 11 years ago

I have a relatively good understanding of how we came to have 12 notes to the octave, you just misinterpret my statement: keyboards already had 12 notes per octave long before ET came into widespread use. And trying to reconcile this division with a tuning system that relies on natural intervals is an awkward exercise, this is the inappropriateness that I speak of.

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

graycat• 11 years ago

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?

Well vs. Equal Temperament (2000)

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