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My previous question about equal temperament was answered with a flourish, so here's the follow-up one:

Because math and harmony don't seem to mix very well, piano tuners have to cheat a little bit, pulling down the fifths, nudging the fourths a bit higher, and so forth (no pun intended).

But there's the string section in the orchestra that is not constrained by anything: no frets, no keys. They can be as "just" as they like.

Question: when accompanying the piano (in Chopin's or Grieg's or Mozart's or Tchaikovsky's piano concerto), do they have to "play along," i.e. do they have to abide by the piano's equal temperament lest the whole structure starts bouncing all over the place, way out of tune?

And what about the woodwinds that have fixed holes - are they made to imitate equal temperament?

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  • 4
    Piano tuners are also not tuning exactly the same for every piano. At the beginner level they can use a tuner to practice, but at higher levels, they tune according to harmonics. The harmonics of each piano, even of the same make, will change depending on the environment (humidity, elevation, temperature, etc.), the wear-and-tear of the instrument, the quality and age of the strings, and all sorts of variables. In short, piano tuners will tune so the entire instrument sounds good, not to some mathematically derived frequency.
    – Nelson
    Feb 11 at 7:15
  • 1
    The piano's sound involve harmonics across all 88 keys and all the strings. A good piano tuner will know how to listen to those. They tune using harmonics that are multiple octaves higher than the actual note they're tuning, because discrepancies are amplified at higher harmonics.
    – Nelson
    Feb 11 at 7:19
  • 7
    Vibrato solves (almost) everything...
    – fraxinus
    Feb 11 at 10:40
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    Woodwinds can vary the pitch up or down a bit by various means, over/under blowing, partial or alternate fingerings, etc. Brass can do the same and some have a pinky-slide for more precise intonation (obviously trombones can easily adjust). Even fretted strings can micro-adjust based on distance between finger and fret. Pianos are in fact one of the only instruments that can't microtune on the fly (though there's some experimental instruments that have that feature). Organs and other keyboards would be some others, though many synthesizers can. Feb 11 at 15:20
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    I see three inaccuracies in your question: 1. Of course math and harmony mix well. We compromise (in part) because equal temperament (irrational math, but still math) is not always a great approximator of just intonation (rational math). 2. The more significant "cheat" in piano tuning is due to the inharmonicity as @Nelson explained. Even JI can clash when the partials are not harmonic. 3. Violins are constrained: When played open (unless you re-tune on the fly).
    – Theodore
    Feb 12 at 15:09
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Tuning in an ensemble is a skill in relative pitch, not absolute pitch. Players will hear what others are doing, and the group will come to a consensus organically. With instruments that are capable of microtuning adjustments, this will also lead to more just-tuned intervals.

But what's important to note is that these tuning decisions happen on the fly, and subconsciously by an experienced player. It's not as if the orchestra makes an overt declaration about the tuning system they plan to use.

So that means that when playing along with a fixed-pitch instrument, a violinist will simply play in tune with it like they always do. They don't consciously decide to use equal temperament, they simply match pitch.

Woodwind construction is a massively complicated topic. The Fundamental Problem of Woodwinds (as I like to call it) is that you need 12 unique fingerings to cover an octave and bridge the first and second harmonics in order to make a fully chromatic instrument (and the problem is worse for clarinet), yet we only have 10 fingers, and usually at least one is used only for holding the instrument, or else the "nothing pressed" fingering would result in us dropping the instrument. What that means is that some fingers/holes have to do double duty and thus there will always be some notes with imperfect tuning.

I'm not a woodwind maker, but I assume that modern instruments are made with equal temperament as a goal. But these construction issues cause tuning discrepancies that are vastly greater than the difference between tuning systems. While a well-constructed instrument should play as close to 12TET as possible, playing in tune is still the player's responsibility.

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    Why does fingers/holes doing double duty have anything to do with being unable to create 12 pitch classes? All you need is for (say) position X + finger Y plays 1 and position X + finger Y plays 2 to both be correctly tuned to different notes in temperament. You have the two requisite degrees of freedom, namely how much to change the pitch when 1 or 2 is played. Besides, in saxophone for example, the left thumb cluster has keys that chain-activate other keys in the same cluster, so it's not even true that double duty implies playing only one of a set of holes.
    – obscurans
    Feb 11 at 5:21
  • ^ I meant pinky cluster. Left index also specifically has two keys, one chain-activates the other.
    – obscurans
    Feb 11 at 5:29
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    Re woodwinds, fingering, «just some holes»: sometimes you spot the difference quite quickly (sequence of the diameter of the holes), sometimes to recognize it when playing together with other instruments (e.g., harpsichord) when it comes to either Baroque or German fingering and forks on a recorder. (And I not refer to «will we tune a = 440 Hz».) Team recorder presented this once in action.
    – Buttonwood
    Feb 11 at 8:44
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    What you say about woodwinds is simply false. All modern woodwinds are built with (at least) one hole for every one of the 12 tones, so they can theoretically be perfectly in tune to any given temperament, at least in the first octave. And you ignore the fact that woodwinds can easily tongue up or down enough to play in any temperament required, or in just intonation. Feb 11 at 10:21
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    People in the commens seem to miss "playing in tune is still the player's responsibility". Yes a woodwind or brass can be played perfectly in tune, but not in the automatic way that e.g. a piano can.
    – OrangeDog
    Feb 11 at 15:55
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The temperament in a professional orchestra tends to vary between just intonation and equal temperament. If they are playing with a piano the strings will tend to adjust to the piano temperament where the difference would be audible and tend towards just intonation when they are playing alone.
Woodwind instruments are built with the aim of playing in tune with equal temperament, but the differences to just intonation are well within the adjustment range of a good player. They will have no problems adjusting to the strings or to the piano.
Amateur orchestras tend be out of tune enough for the difference to be irrelevant.

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    That final line made me (sadly) laugh ;-) Feb 10 at 20:05
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In general? No, because it feels unnatural, similar to how singing does. However, you'll surely find places in music where the natural pitch against the bass note will not feel like the right thing.

You can test this with your voice if you are a resanable singer: Play the E tone and sing it. Next, play the C major chord without this tone (so C+G) and sing the tone -- your voice will go slightly down. However, if you play C major including the tone (so C+E+G), you'll keep your voice down.

This is even more significant if you sing the major 3rd in C major (C+E+G) vs. the minor 3rd in C# minor (C#+E+G#). The one in C major will be much lower than the one in C# minor.

The difference comes from the tuning obviously (f denotes the frequency of C):

  • (major 3rd) If the base note is C with frequency f, the natural frequency of E is f * 5/4 = 1.250 f.
  • (pure 1st) The frequency of the E key itself is f * 2^(4/12) = 1.260 f.
  • (pure 5th) If the base note is A with frequency f * 2^(-3/12), the natural frequency of E is f * 2^(-3/12) * 3/2 = 1.261 f.
  • (minor 3rd) If the base note is C# with frequency f * 2^(1/12), the natural frequency of E is f * 2^(1/12) * 6/5 = 1.271 f.
  • (dominant 7th) If the base note is F# with frequency f * 2^(-6/12), the natural frequency of E is f * 2^(-6/12) * 9/5 = 1.273 f.

So the change in pitch is most significant between E in C major chord (that's the flattest) and E in F#7 chord (that's the sharpest) -- there it will be the easiest to try when singing. However, the original exampe with C major vs. C# minor works pretty well too.

Our ears likes to tune purely, so when given a chance, they will force signers (and players equally) to correct the pitch if it's within their technical ability. (On the violin, this comes as a natural, but experienced woodwind and other players will do it as well, possibly without realizing they're doing it. Most signers in choirs probably have no idea they're doing this.)

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  • The 'signers' being deaf mutes anyhow! Only jesting! Great answer, which explains much more than yes/no. This shows how much 12tet is out from j.i. +1 at very least.
    – Tim
    Feb 11 at 7:29
  • I realized I also forgot to mention the importance of pure 3rds in Western music, I shall add that later
    – yo'
    Feb 11 at 11:48
  • @Tim Yeah, tell me that... (being one of the 'singers' myself probably lots of time)
    – yo'
    Feb 11 at 11:57
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To supplement on the woodwind perspective: the fixed position of wholes does not nearly fix the pitch. Each instrument has its individual deviations from in-tune (however small), which can/need to be compensated, and there are lots of other variables like

  • embouchure,
  • modification of volume of mouth/throat
  • speed of air
  • pressure of air
  • angle of air stream to flute head

as well as alternative fingerings, which are continuously chosen or adjusted by the player in a way quite similar to that of a player of e.g. a violin.


Update: I missed to mention the next step, which is already in MattPutnams answer: All of those variable are used to match, whatever is there: a fixed-pitch instrument (which might even no longer be perfectly tuned) or the most experienced or leading player.

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  • Whilst not particularly addressing the question, whatever instrument is playing with 12tet tuned piano, it makes sense that the player will attempt, skill permitting, to keep with said piano. No piano, anything goes.
    – Tim
    Feb 10 at 20:50
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As a violinist and a pianist, I have over 10 years of semi-professional experience in both instruments (currently not a professional), and have previously played in various performance roles in both instruments.

Violins have different sets of temperaments to work with, and change based on skill level and who they play with.

https://www.violinschool.com/knowledgebase/violin-tuners-intonation/

Equal Temperaments will cause the violin to sound bland, because the notes no longer cause harmonic vibrations with the other strings. Expressive Intonation is the opposite, where the violinist plays according to the harmonics of their instrument, but this pushes the intervals of the scale outside of equal temperaments.

A high level violin soloist will play Expressive Intonation when appropriate. The semi-tone between 3rd note and 4th note will be wider than the semi-tone between the 7th note and 8th note, creating harmonics with the other strings and making it sound much richer. If you have a chance to listen to this, a scale sounds noticeably better when done in Expressive Intonation, but by and large very few modern instrument can do this, and it's simply impractical to bring back instruments that are "perfectly" tuned.

With regards to adjusting their pitch, a violinist does this by ear. Given that a violinist would know how to adjust their notes as necessary based on what key the piece in, it is part of a violinist's skill set to adjust his pitch, literally, on the fly.

A violinist will also change the fingering to avoid open strings, since you can't adjust an open string's pitch, and also will shift to higher positions as needed. There are no hard and fast rule, but it generally has to do with making the piece easier to play and to sound better.

A violinist that lacks experience with accompaniment will incorrectly play Expressive Intonation with other instruments, causing rather unpleasant sounds. It is primarily the violinist's job to learn how to play "in tune" with other instruments because very few instruments can adjust their pitch as readily as a string instrument.

So to answer your question, "Yes, the violin will imitate equal temperament", but the process is largely organic and does not follow hard-and-fast rules. The process of the violinist imitating equal temperament should be a subset of the regularly used skill of adjusting their pitch according to the key they are in.

You won't find any violinist sitting with a pitch machine and practicing with that, because that's not how music works.

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  • Given that your answer is based heavily on the concept of perfect temperament, a definition (is the 2nd paragraph intended as such?) or even better: a reference would improve it.
    – guidot
    Feb 11 at 8:17
  • I've never learned the actual term for it, but I looked it up and matched it up to what the reference uses.
    – Nelson
    Feb 11 at 8:50
  • Harmonic vibrations with other strings being matched? That will only work when the note played has some match with any other open string. And depending on key, and the number of times this is a relevant point that must be quite low. Call me sceptical.
    – Tim
    Feb 11 at 14:49
  • "it's simply impractical to bring back instruments that are "perfectly" tuned": there never were such instruments. While a major scale may sound better (to some) with the half step between the 3rd and 4th being sized differently from the one between the seventh and the octave, that is only one context in which those four notes are played. Another context is harmonic, and if you want the two fifths 3-7 and 4-8 to be in tune, the half steps need to be the same size. In practice this means that the specific frequency selected for certain notes will vary within a single piece.
    – phoog
    Feb 12 at 0:02
  • @Tim - you've heard of the A-432ers, yes? They believe 440 is evil. But they have a similar problem: how often do A's come up in a given piece, say, in Ab major? You'd think our average exposure to 432 would be pretty small, even in that tuning. Feb 12 at 11:11
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Of course any instrument that can play 'in the cracks' will do so when not playing alongside a 12tet tuned instrument. At which point, playng with such an instrument, it makes sense that any decent player will try to be more in tune with said instument, rather than pursuing the 'right path'. Why would anything differ from that scenario?

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  • Just making sure.
    – Ricky
    Feb 10 at 22:02
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String quartet players - ie strings playing with other strings - have to adapt their intonation the whole time, depending on the ensemble, the piece, the key, the period of the music, the context within the music. Generally in quartet playing - and counter-intuitively - major thirds and sevenths are played flat, minor thirds sharp. In an orchestra with mixed instrumentation it tends to be the opposite. This, as you can imagine, is extremely skilled work but gets easier as ensembles play together over months and years and grow accustomed to each other, developing their own sound as a group.

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The question is based on a misconception that singers or players of stringed instruments will naturally play in just intonation, and that therefore if they're playing with a piano they will have to adapt themselves. This just isn't true. For some real-world data, see Deutsch, The psychology of music, 3rd ed., p. 97. Professional musicians generally judge notes to be in tune if they fall within a band of about +-7 cents. The only way the human ear can hear differences in pitch smaller than about 5 cents is when the conditions make it possible to hear beats (which used to be the only technique available to piano tuners), but this isn't normally possible in real-world musical conditions. The difference between an equal-tempered fifth and a just fifth is only 2 cents. Expressive intonation and vibrato are often on the order of 50 to 100 cents.

What equal temperament allows us to do is to modulate between distant keys without a train wreck and have some way of defining what pitches are even intended when we notate a song in a key like F#. This kind of thing becomes an issue only because you can get wildly wrong results if you keep on stacking just fifths on top of just fifths (the Pythagorean comma). That is, the cumulative effect can be large.

Normal practices in composing and arranging don't actually tend to create situations in which it matters whether a violinist exercises freedom in intonation. For example, the difference between a just and an equal-tempered M3 is 14 cents, which is audible under the right musical conditions. But composers usually don't double the third in a chord, so in a piano-violin duet, if the violin has the third, they have freedom to use whatever kind of vibrato, portamento, or expressiveness in pitch they prefer.

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  • Thirds are out of tune in equal temperament by about 13 cents. That's enough to be a problem.
    – PiedPiper
    Feb 11 at 17:09
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    100 cents is a trill, not vibrato.
    – PiedPiper
    Feb 11 at 17:16
  • @PiedPiper - don't tell that to an opera singer. Feb 11 at 20:34
  • This answer is making some very good points, but the first paragraph is highly misleading as it stands. Fifths are largely irrelevant – as you say they're practically identical between JI and 12-edo – but thirds very much aren't, and short of vibrato excesses (opera singers shut up please) those will be naturally corrected to 5-limit JI. Feb 12 at 15:24

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