Is there anything universally special about the notes on the musical scale?

Cafe Society
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Backstory: After my guitar was recently “set up” by the teenager at Guitar Center, I noticed that the tuner imbedded in the guitar itself seemed to register all the strings just slightly flat, when compared to the tuner app on my phone. Googling told me to trust the tuner on my guitar, and not the phone app (still not sure if this is true). Anyway this leads to my FQ:

In my search, I came across the following site that lists out the musical notes based on their frequencies: Musical scale

What is special about the frequencies listed? Do they just simply suit the human ear better than those between the notes. Do all mammals that can hear also perceive these frequencies differently than others? What about other animals, or aliens for that matter (if they could hear, or have some other means of sensing sound waves)? Or is the change to the frequency relative to the previous note more important; i.e. it doesn’t matter if we define A to be 440 Hz or 450 Hz, as long as the other notes are played with the same offset?

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The exact frequency does not really matter—it’s just an orchestral standard that is used for manufacturing instruments, but people can and do mess with it to get a brighter sound, for example.

To construct a scale you, at least at first, just worry about the ratios between notes (the most basic intervals are fifths and eighths), rather than the exact fundamental frequencies. So if you change A from 440 to 450 you need to multiply all the other frequencies by 450/440. It may not be so easy to retune an actual piano or clarinet that way, since they are built to a certain size, but we are talking in theory.

ETA guitar, OK, it’s a little complicated and I am not a guitar expert, but you have to worry about inharmonicity of the strings as well as potential deviations from that frequency table, so I would not be surprised if your guitar came with a tuning recommendation (a little flat, and maybe even depending on what keys you want to play in) that is a few cents off from strict equal temperament on the open strings.

ETA2 here is one proposed algorithm for tuning a guitar:

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It’s the ratios of the frequencies that are musically important, but I’ll let a musical theorist describe those. The numerical values of the frequencies are an artifact of our measurement and unit system. One Hertz (Hz) is one “cycle” per second, where a cycle is a complete transition from and back to the same point. Think of the string vibrating up, down, and up again. Counting how many cycles occur within a time interval gives the numerical frequency. The standard time interval is one second, but any could be used and that’ll change the numerical value. For example, 1 Hz is the same as 60 rpm (rotations per minute).

I’m not sure how much the importance of frequency ratios is fundamental and how much is based on our audial perception. For example, we don’t place any value in the frequency ratios of the visible spectrum, but we visually perceive only one octave of electro-magnetic radiation. For whatever reason, human audial perception is good at finding frequency ratios.

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This is a never ending rabbit hole. There is no such thing as a perfect tuning, and there are centuries of efforts to at least get a better tuning for purpose. Fretted instruments have it even worse.

In principle you want the ratios of frequencies in the scale to sit on simple ratios. A fifth is 3:2, that is a fifth is found by multiplying the base note by 3/2. Other intervals can be found by other nice round ratios. But even now you get into trouble. The Pythagorean scale is built out of 2s and 3s.
Just Intonation is generally considered to be the best scale, as it builds the scale out of 2,3,and 5 and their multiples. Even now you get two different tone internals, there is a major tone 9:8 (3\times3:2\times2\times2), and a minor tone, 10:9 (2\times5:3\times3). Keyboards to provide Just Intonation have split keys to get you the extra notes.

Even here you are in deep trouble. Musicians want to transpose and shift keys. On the same scale that you set out a consonant set of intervals, if you move to a different key centre, all the intervals go wrong. Hence equal temperament, where every note is out of tune, but spread evenly so that the result is consistent. The semitone is defined as the 12th root of 2, so that 12 semitones exactly makes an octave. No interval ever lands on a nice integer ratio in equal temperament, but every semitone is identical, so moving key does not lead to problems.
In principle guitars and pianos are tuned to equal temperament. But in reality they are not.

On a guitar intonation is set to take into account the string’s width, which acts to reduce it length slightly, and acts to move the harmonics out of tune with the fundamental. The act of fretting disturbs things as well. Typically the nut is positioned very slightly short to help get the intonation right for open strings, and the bridge set to take account of the differing string widths. But even then you get into trouble. A good guitar tech will set the guitar up so that it play well in one part of the neck, depending upon the player. This is again spreading the pain about. Chords will sound better in one part of the neck to others. But intrinsically it is impossible to set it up to play in tune everywhere, and some chords will sound more consonant than others.

There is the Buzz Feiten system, where he worked out a nut position and precise offsets for string tuning that is supposed to work well for Jazz style chords.

Overall answer. There is lots special about the musical scale. But there is no one true musical scale. Just Intonation is what people like Baroque players try for, but often need to settle for things like 1/3 comma Meantone. Which means - all tones tuned to the same interval, and the semitones no worse than 1/3 of a Syntonic Comma (an error in tuning of 81:80).

You can write, and there are, PhD theses on the subject.

At root, the problem is that the unique prime factorisation theorem tells you, a-priori, that you can never get one true scale. You chose your poison.

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And bear in mind that all of this discussion is based on the classical western music scales. Other cultures have different scales. For example in north India, the scales are quite different. I remember reading an interview with Ravi Shankar a long time ago where he said that the violin was very popular in India because it wasn’t fretted, and therefore could play Indian scales, while a western fretted instrument, like a guitar, was designed to play western scales and therefore didn’t adapt as well to Indian scales.

There are also western scales that don’t match the classical scale, usually tied to folk instruments. The Great Highland Bagpipes are a good example, where the scale does not match classical western scales. Some say it’s close to the Ancient Greek mydloxian mode.

The first time I played the piping scale on my chanter, Piper Dad, who was a talented amateur musician, asked me what I had played. I said « It’s the scale. ». He replied: « Not like any scale I’ve ever heard! »

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Indian music isn’t wildly different to Western scales under the covers. Being almost entirely melodic in nature, with little to no harmony, it is freed to explore the nuances of intonation. So fifths are perfect, and the different widths of tones incorporated. The scale is divided into 22, or even 84, and tones can be 3 or 4 22nd divisions.
These dividing numbers are interesting as they correspond to further divisions that the circle of fifths gives you.
Equal Temperament teaches you that you go up a fifth, that is multiply by 3/2, and if you go out of the octave, divide by 2. Keep doing this and after 12 intervals you get back to the root. Only you don’t. It is impossible. There is a gap, usually called the diatonic comma. It is just that we deem that gap small enough to ignore. But if you keep going, you create further notes, and when you get back to close to the root, you have new scales, depending on how close you allow the gap (the comma).

Indian music is closer to what westerners call microtonal. Blues players on guitar instinctively use a microtonal scale when bending. A one third semitone bend adds a perfect bit of spice to a melody.

But Indian music has huge richness within this wider melodic palate. The above doesn’t even scratch the surface.

Interestingly western tradition understood a lot of this, but equal temperament and a rich history of harmony and counterpoint restricted what was used. Leopold Mozart wrote the first major work on violin technique. He explicitly divided the whole tone into seven, with semitones of three or four divisions wide. Which is fine for a fretless instrument. But not others. By itself a string quartet can play in Just intonation or others of their choosing. But add a piano, most winds, and so on, and it becomes a matter of negotiated compromise.

The idea that modern music is all played in equal temperament is a polite lie. In recent times there has been something of a revival of interest in this, especially in the genre of historically informed practice.

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Since I’m on something of a roll here, I’ll add the complication of intrinsic inharmonicity. Fretted instruments, which includes the piano here, have strings with width. This width makes the length of the string appear different for the different harmonics. So the harmonics are out of tune with the fundamental. The shorter the string the worse it gets. Pianos are tuned so that the octaves are in tune. This requires that the octave is stretched very slightly. Concert sized pianos have less stretch, but it is there.
Concert pianos are often tuned to intonations other than equal. There is a whole family of possibilities. Bach’s Well Tempered Clavier was intended to show off these. Despite what some people are told, Bach did mean equal temperament, he meant a good temperament.
However bowed instruments don’t have problems with inharmonicity. The stick slip of the bow on the string acts to force the harmonics into line, a phenomenon known as mode locking. Flute pipes on a pipe organ are also mode locked. But reed stops are not. This all becomes part of the colour of the stop and how it interacts with other stops.
Again, there is no possible perfect tuning. Most instruments are intrinsically out of tune with themselves. Fretless bowed instruments are about as good as it gets. It is all downhill after that.

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Regarding the definition of A – here is more information than you probably want to know.

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Moderator Action

Since this gets into music theory, temperament, and other things beyond simple physics and frequencies, let’s move this to CS (from FQ).

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And brass instruments can’t be tuned to equal temperament (well, aside from trombones, but they’re not really tuned at all). You play multiple different notes with the same fingering, and hence the same length of tube, and so the ratio of frequencies of those notes is set by the physics (often, you can play the “same note” with more than one different fingering, but those will have slightly different frequencies).

That said, while equal tempering doesn’t get any of the ratios exact (they’re all irrational), it does get pretty close. 2^{4/12} = 1.259921... (instead of the exact 1.25 it’s “supposed” to be), 2^{5/12} = 1.334839... (instead of 1.3333…), and 2^{7/12} = 1.498307... (instead of 1.5), so they’re all within 1% (0.8% for the first, and that’s the worst of them). A trained ear can tell the difference, but an untrained ear probably can’t.

I think you missed a “not”, here.

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Brass instruments are fine for playing contemporary music: that is what the valve slides are for.

Equal temperament is good for jazz or other complex harmony, or chromatic music, or atonal music, or for playing in many different keys. You don’t have to be able to hear the exact frequencies—its character is pretty obvious when playing chords, especially the thirds (and sixths…)

But, just like A = 440 Hz is so universal there is a link explaining how many major orchestras do not use it, you can find composers who specify exact temperaments and microtonal intervals rather than leave it up to interpretation, even in “Western” music.

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Heh, I was returning to this thread to mention trombones. (I play trombone.) They are an excellent resource for teaching the physics of music. The sound is produced on an open-pipe air column. For each note, the player chooses the length of the pipe and the harmonic. The note you get is straight-forward physics from there.

Trombone tuning is done when the slide is fully retracted, and usually on the fourth harmonic. There’s a tuning slide on the back curve that lets you adjust the exact note (the instrument goes flat as it warms up; temperature matters). We usually use the fourth harmonic because the lower ones are more “sloppy” (not sure on the exact physics of that, probably due to being easier to drive the excitation off the resonant frequency).

Similar to fretless strings, the player can adjust the exact note continuously. Beginners learn the standard positions by physical location, but an advanced player uses the sound. It’s required in some keys and some harmonics because the standard positions will be noticeable off. Sometimes the trombones will intentionally “slide” into a note, starting a little above or below the final pitch before settling in. That’s part of the wah-wah trombone effect (the rest from cupping the bell).

(There’s also the opportunity to mess with other sections of the band, by for example, every trombone playing an 1/8-tone off–the trombone parts would all be in tune with each other, but when the trumpets come in, wow are they off pitch. The band teacher tends to figure that out fairly quickly. :wink: )

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Yeah, I always seem to be doing that. At least in this case it is obvious what I meant.

Brass is a mess at the best of times. Then you get instruments where the player has to move their hand in the bell to tweak the note, or make an otherwise unreachable note reachable. The natural horn is a beast by all accounts. But it sounds magnificent in a way other brass just doesn’t seem to manage.

The common complaint you get with equal temperament is that fourths never sound right, and people have never heard how good they can sound. The other complaint is that music composed in times of different temperaments played in equal temperament never quite captures some of the expressive nuances the composer put into the music. (Like it may not be apparent why D minor is the saddest of all keys…). The manner in which nasties appear in certain intervals in certain keys, is lost, which is arguably a good thing, but there are reasons composers chose certain keys that are lost.

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It is still true that instruments are made in different keys. A composer must be aware that a clarinet in B-flat is not identical to a clarinet in A, and a trumpet in B-flat is different to a trumpet in C.

I think the equally-tempered fourths and fifths are not so bad compared to thirds which are noticeably out of tune. In any case, you do run into composers who care about things like that; look at all the accidentals in Ben Johnston’s string quartet:

(e.g. a “+” is 81/80, a “7” is 35/36 (upside-down it is 36/35), and a “13” is 65/64)

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A B flat trumpet can play in any key you like. There is a difference but it has to do with notation and fingering. Nothing to do with which instrument you you would choose for the key you’re in.

You are correct that composers and musicians care about subtleties of intonation. Here’s a performance where by using a number of slight differences between different tuning systems a song transposes into a key halfway between two standard A 440 keys without most listeners even noticing.

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Well, sure. Either of them can play in any key, and they are both trumpets. But they still sound a little different, so there may be reasons to choose one over the other. I did not mean that you would need to swap among 5 different trumpets just because of some key changes in a piece. But, that is a good question: what is the historical reason B♭ trumpets are popular, and if I were designing a completely original Lovecraftian monstrosity of demented trombonology from scratch, how would I decide the “correct” length of tubing and/or natural pitch at which it should blow? Scaling things differently will affect the timbre, so we would need to decide what it should sound like, and there may be other subtle considerations.

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This one time at band camp…

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A propos, a good example is the harp. Harp strings sound better flat. Therefore, you would prefer the key of C-flat major instead of B major—they are not the same, even though that would not be important on the piano, or even when the rest of the orchestra is playing in B major.

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In equal temperament, C-flat major would be exactly the same thing as B-major.

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I was puzzled too so I googled.

I know nothing about harps but I learned that, if I understood right, Harps have 7 strings per octave and each string has a lever that can be set up or down to make that string sharp or flat. (I don’t know if B has a sharp or C has a flat etc.) The flat position loosens the string and listeners prefer the looser string sound. The upshot is–flat keys sound better even if the harp is tuned in perfect even temperament.

That must be why D minor is the saddest of keys.