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The relationship between an A and a B is different in say, just intonation and equal temprement. I understand that the absolute frequencies aren't important but instead the relative difference between them.

If the notes don't have some idealized or 'correct' set of ratios then what do the notes actually represent? How can their use be the same in multiple tuning systems if they don't have the same relationships in those tuning systems?

I am very much a noob to music theory and this is something I do not understand

Edit:

I would like to clarify my question a bit. The notes are in alphabetical order so it seems like you could basically change them out with "1 2 3 4 5 6 7", so as I understand it each note is in some sense 'one more' than the previous. But it seems to me like this 'one more' doesn't have a consistent definition. In 12 TET it's 'times another 12th root of 2' but in just intonation the ratios are different each time, what is the underlying relationship there? And if the meaning of 'one more' is different in each tuning system how does the usage of them in one system transfer to the other?

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  • Btw, within the Western system, this all started with the ancient Greeks. You might get some clarity by looking more deeply into how they divided up the tetrachord and derived tonoi. Oct 9, 2023 at 20:42
  • @Gerald, it looks like there are some fundamental misunderstandings on intervals that are causing your long chains of comments. Instead of doing that (comments are really only for fine tuning content in posts) please go to Music: Practice & Theory Chat as I think you need to be guided through some of the basics in a way that doesn't fit in the Q/A format.
    – Doktor Mayhem
    Oct 12, 2023 at 8:29

6 Answers 6

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I disagree with this:

The relationship between an A and a B is different in say, just intonation and equal temprement.

But A and B have the same relationship in both just intonation and equal temperament (and Pythagorean tuning, meantone tunings, etc.). A and B are a major second apart.

Heck, even on transposing instruments, including badly tuned ones, A and B are still a major second apart.

It doesn't matter all that much whether that major second between A and B is represented by 9/8 (just intonation), something involving the square of the 12th root of 2 (equal temperament), or some botched ratio in between these 2 because your clarinet ended up 5 cents flat when playing written C and some of the clarinet's pieces were pushed too closely together.

Notes represent certain frequency ranges such that two such notes should be a certain interval apart. For example, A and B are a major second apart, while A and C are a minor third apart. About the only thing that can change the interval between two notes is the tuning system, and only if the new tuning system has a different number of notes per octave from the old one. For example, F♯ and G♭ are enharmonically equivalent in 12TET but are a "minor second" apart in 19TET.

Regardless of the tuning system, as long as it has the same number of notes per octave, it's the range of the size of the intervals that is universal across all of them. For example, at least according to the Wikipedia article on wolf intervals, the perfect fifth is between 660-740 cents wide (although 675-725 cents wide produce nicer-sounding results, and 660-675 cents and 725-740 cents are wolf fifth territory - note that the average size of the perfect 5th has to be 700 cents). Any interval that gets too wide gets renamed the next larger interval (e.g. 741 cents wide is an augmented 5th, and great arguments can be made that it's also a minor 6th).

Along the same principles, you can take a number line of cents (extending from some concert pitch - A440 is a very common one, although scientists occasionally use C256, and transposing instruments will try "core pitch"es such as the B♭ clarinet's B440) and divide it into equally sized regions for each note (in the chromatic, 12-tone scale - e.g. E, F, and F♯ get equally sized regions).

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If the notes don't have some idealized or 'correct' set of ratios then what do the notes actually represent?

But they do have an idealized set of ratios. Different tuning systems represent different degrees of compromise to these ratios, but in every tuning system they are more or less close to the idealized ratio.

How can their use be the same in multiple tuning systems if they don't have the same relationships in those tuning systems?

Because these compromise relationships are close enough that the intervals sound more or less the same. If you're playing a piece in A major, you're not going to notice much of a difference if the E above middle C is 330 Hz or 329.63 Hz.

The most noticable variation between different tuning systems is probably that of the major third. If A4 is 440 Hz, the C♯ a half step above middle C can be anywhere from 275 Hz to 278.44 Hz (in 12-tone equal temperament it is 277.18 Hz). So it's a fairly significant difference, but to my ear at least it still sounds like a variant tuning of the same note rather than a distinct pitch with its own distinct identity.

In other words, the variation between tuning systems is less than the precision with which human ears -- at least those trained in traditional European music and its derivatives -- identify tones and intervals.

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Notes do have what could be considered a "correct" set of ratios derived from the harmonic overtone series. Notes also have many "idealized" ratios evidenced by various tuning systems.

You want to understand how the following two things can coexist:

  1. The fact that in different tuning systems and non-diatonic scales, A and B (or the 6th and 7th scale degrees) do not always have the same exact frequency relationship.
  2. Music still works.

If the notes are not always the same, does that mean notes have lost all meaning? Of course not. Try not to overthink it. Also, thinking of notes on a scale as simply "one more" than the last seems partly to be where you are getting stuck, and since that idea lacks musical context, and the whole point of notes in the first place is to make music, I would try not to think of notes that way only. Notes function as a means to an end. You need musical context to understand notes. So here goes:

What are musical notes? Think of it this way, one note by itself has only the natural harmony of its overtones and is somewhat static. Two notes together either create more harmony (if they share overtones) or dissonance. And two notes in a sequence either create tension (a sort of dissonance over time creating anticipation) or lead to resolution. That is what notes are for, they are fundamental building blocks to create harmony/dissonance, tension/resolution in combination with other notes. If A is 440hz, B doesn't have to be 440*(1.12246204831) every day for that to happen.

Interesting ideas to further explore:

-Ancient Greeks and tetrachords.

-The Blue Note (not the jazz club). A note you can't unhear.

-Harmonic/Overtone series: it's not a coincidence that the first 5 harmonics form a major triad that also happens to sound nice.

-Also, the interval of A to B (a major second) sounds different (more or less harmonious) when played with different root notes (context). In other words, when C is the root note, A and B are the 6th and 7th scale degrees. When G is the root, they are the 2nd and 3rd of the scale. They function differently in different contexts.

I also think looking at the larger picture could help. Instead of the question of "what do notes represent", how about "what does music represent"?

-Regards

P.S. I didn't read all the replies (since there were a lot of fractions in some of them) so forgive me if I inadvertently repeated or contradicted anyone. I think user121330 both articulated and answered your question (and even anticipated one about two different people playing together) quite well from a technical standpoint when he mentioned intonation and compromise.

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  • There are even tuning systems in which the relationship between A and B changes within the tuning system (i.e. using 53-tone equal temperament or flexible 5-limit just intonation to realize a piece in traditional European notation).
    – phoog
    Oct 11, 2023 at 19:12
  • Along the same line, there are multiple fingerings on clarinets for some notes (I think written E flat above Middle C is one of them?), and they are varyingly out of tune.
    – Dekkadeci
    Oct 12, 2023 at 12:25
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One thing to add to the responses so far is history.

By the way, I love the fact that this question cuts straight to the root of core definitions, like "What do we mean when we say 'A', anyway?" It reminds me of the book Is Math Real?, which says that asking "Why does 1 + 1 = 2," and not taking "Because it just does" for an answer, is the heart of mathematics.

So this is partly a question about labels. We label notes, and name them, and you're asking what it is that we're naming. Well, keep in mind that Pythagoras didn't have an oscilloscope. Throughout the history of naming notes, measuring the frequency in hertz has only happened since, well, since "the hertz" was named after Heinrich in 1935, I guess. Meanwhile, looking back over history, differing temperaments is the least of your worries. The notion that A should be 440 Hz—the notion that it should even be the exact same frequency, or even close, at all times, in all venues, for all performances, is not an assumption that's been universal in all musical contexts. It still isn't in some.

I highly recommend A History of Performing Pitch: The Story of "A" by Bruce Haynes, for a full attempt to nail down what "A" meant especially in the baroque, to various composers in various cities. Because it often varied; a given cathedral might have its organ at a certain pitch, and if you wanted to play along, by golly you adopted its "A."

Now, that survey is focused on a tradition that we might call "art" music. But there have also been, and still are, situations in which someone might call a pitch "A" and have even less desire to measure or calibrate it. Say, an old-time fiddler takes the fiddle out of the case, tunes up, and starts to play. He tunes without any external reference pitch, just to his own habituation of what seems right. For him, "A" is "A" because that's his "A" string. This is not really far removed from an organ in 17th-century Dresden saying "This is 'A' because it's my 'A' pipe." These all might be 410, 450, or even further afield.

Hopefully this is enough to show that "A" isn't a frequency at all, it's an idea. We're delving into semiotics here; all "notes" are after all just human constructs that we've layered on top of audible wavelengths. And these constructs are much older than the ability to minutely measure those wavelengths. It would be like defining varieties of beer by counting the number of yeast cells in them; people have been brewing for much longer than that's possible.

Which presents you with a much bigger question—too big, honestly, to get a handle on easily: "So where did the 'idea of A' come from?" That's nothing less, really, than attempting a history of the concept of pitch, and it evolves from ancient Greek systems to medieval to modern (and takes other paths in other cultures). That could be an entire field of study, too much even for a single book let alone an SE question. But maybe you can narrow your field of inquiry, choose a slice from that timeline, and interrogate what "A" meant to a specific music-culture at a specific time and why!

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  • Ok but what is that idea? That's my question. I know that it has a history and has been different throughout history and what I want to know is what the underlying idea is. What is the point of labeling the notes "A B C D E F G" if each one isn't in some sense "one more" than the previous.
    – Gerald
    Oct 9, 2023 at 17:33
  • 1
    @Gerald Oh, there is a "one more" component. Even if "A" is on a sliding scale, even the old-time fiddler agrees that B is one note higher than A. And even if "exactly how much higher" varies—between temperaments, or even with human imprecision or a "close enough" attitude to intonation—the interval of a tone is roughly consistent. It's like how, in clothing, a "size 5" might be several centimeters different from one brand to another, but you can count on it to be bigger than a 4. Oct 9, 2023 at 18:04
  • So what does it mean to be one note higher? Is it just to have higher pitch? In that case the only relationship between the notes would be that they are in order from lowest to highest pitch, which doesn't seem very useful to me
    – Gerald
    Oct 10, 2023 at 5:38
  • @Gerald I think you have an underlying assumption that differs from many of the people and times that started organizing pitches: That something imprecise is not useful. In another comment you used the analogy that the meter would not be a useful unit of measure if it weren't standardized—and indeed, for as long as mankind has had units of measure, it has periodically recognized the need to standardize them. But imprecise units are also meaningful. We might describe a distance as a number of city blocks: even though these can vary, they are constant enough to be useful. ... Oct 10, 2023 at 14:27
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    @Gerald Throughout this discussion you've pointed out that the size of a semitone can vary—but the key is that is doesn't vary that much. It's semiotics, a Ship-of-Theseus problem: how much can you change a semitone until it is no longer a semitone? The practical answer is usually "the midpoint"; if your semitone grows past 0.75 of a tone, we'll probably "round up" and hear it as an unusually narrow whole tone. Meanwhile, as harpsichordists tune in temperaments, or as violinists adjust the size of an interval to fit the chord or ... Oct 10, 2023 at 14:32
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If the notes don't have some idealized or 'correct' set of ratios then what do the notes actually represent?

The notes A, B, C, D, E, F, and G represent different degrees of the diatonic scale formed by stacking perfect fourths or fifths (F–C–G–D–A–E–B). Starting from A you get the minor scale or Aeolian mode, while if you start from C you get the major scale or Ionian mode, for example.

How can their use be the same in multiple tuning systems if they don't have the same relationships in those tuning systems?

They do, as long as you're staying within a particular family of tuning such as meantone. But they aren't and can't necessarily do the same if you're moving between families of tunings:

  • In Pythagorean tuning, which tries to preserve as many pure perfect fifths as possible, stacking three fifths results in the minor third being at a 32/27 ratio while the major third is 81/64. These are significantly off from their "ideal" just ratios of 6/5 and 5/4, by an amount known as the syntonic comma, an 81:80 ratio.
  • Meantone tunings tune the fifth narrower than the just ratio of 3/2 to bring the major thirds closer to a 5:4 ratio and minor thirds closer to 6/5. Our current Western notation system is based on meantone, where C–E gives you a major third and C–E♭ gives you a minor third.
    • The entire meantone spectrum, containing all possible sizes of perfect fifths for a meantone tuning between 685.714 ¢ and 700 ¢.
    • The portion between 685.714 ¢ and 694.737 ¢ is known as the "Flattone" range and contains tunings like 33-TET which represents 1/2-comma meantone. These tunings tend to sound very jarring to most people since the diesis (the difference between adjacent accidentals such as C♯ and D♭), is larger than the chromatic semitone (e.g. C–C♯).
    • 19-TET, which represents 1/3-comma meantone, has its fifth at 694.737 ¢. Here, the chromatic semitone is equal in size to the diesis, so C–C♯ and C♯–D♭ are both one step. It forms the boundary between flattone and non-flattone tunings.
    • For most people who are only familiar with 12-TET, the portion of the meantone spectrum that is usable for them lies between 694.737 and 700 ¢, since this is where things start to become usable. It includes 31-TET, which represents the historically-important 1/4-comma meantone, along with the two most common modified meantones (1/5-comma, represented by 43-TET, and 1/6-comma, represented by 55-TET).
    • 12-TET, with its fifths at 700 ¢, lies right on the sharpmost extreme of the meantone spectrum.
  • Superpythagorean tunings, on the other hand, such as 27-TET, widen the fifth from just so that four fifths stack to a septimal supermajor third of 9/7, and three fourths stack to a septimal subminor third of 7/6. C–E gives you a supermajor third instead of a major third, and C–E♭ gives you a subminor third instead of a minor third.
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  • 2
    "The notes A, B, C, D, E, F, and G represent different degrees of the diatonic scale in Pythagorean or meantone tunings": these letters represent the degrees of the diatonic scale in every tuning, including non-meantone just intonation.
    – phoog
    Oct 8, 2023 at 18:45
  • So their use is not the same if you use two tuning systems from different families? What exactly defines a family?
    – Gerald
    Oct 9, 2023 at 5:46
  • @Gerald I've recently updated my reply to address your question.
    – user59346
    Oct 10, 2023 at 15:11
  • I saw that but I decided to just kinda ignore it because you use so many words I don't know that I knew I wouldn't understand even if I tried.
    – Gerald
    Oct 10, 2023 at 15:18
  • "Meantone tunings tune the fifth narrower than the just ratio of 3/2 to bring the major thirds closer to a 5:4 ratio": many if not most have at least one fifth that is wider than 3:2, though, notwithstanding the fact that some people will claim that it's a diminished sixth rather than a fifth.
    – phoog
    Oct 11, 2023 at 18:34
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If the notes don't have some idealized or 'correct' set of ratios then what do the notes actually represent?

Notes are frequencies. Our ears hear much similarity between notes that are doubles - for example, 440 (A) sounds very similar to 220(A an octave lower), 110 (A, one more lower still), 880 (A an octave higher than the first), etc... So, our scale must have room for some number of notes between octaves. The fifth is the first overtone that isn't a doubling, and one can easily populate a scale of 12 notes between doublings by simply adding fifths (and dropping octaves to stay within hearing). The problem is that (3/2)^12 is a small bit higher than 128 (2^7). Thus was born Equal Temperament - the system that uses irrational numbers (2^(k/12)) in the way you want to use a ratio. Now, just because we've defined equal temperament, doesn't mean that everybody uses it. It is usually agreed that an ensemble will choose the same tuning system - often based on the instrument that has the hardest time changing their intonation. In every tuning system, notes do, in fact, have an idealized set of ratios (if you'll permit that the 'ratios' may be irrational).

How can their use be the same in multiple tuning systems if they don't have the same relationships in those tuning systems?

They aren't and they can't. If tuning systems x and y are equivalent, they're the same system, and if they aren't, there's no equivalence map between all of the notes. If you're asking how the same score can inform two performances (one equal, one just), well ... the notation is the same and the notes aren't too different, but people will hear the difference in intonation. If you're really asking how two people on different tuning scales play together, it's like any good relationship - compromise and practice. There are instruments who can't affect their intonation at all - in this case, the tonal compromise is a one-way street. FYI, just tuning does a pretty good job at describing roughly equidistant 12 note octave scales, but there are many other scales with more or less notes in them, and the prominent role of frequency doublings may even play a role them.

As a point of order, the difference between 3/2 and 2^(7/12) is about .173% of an octave which is a remarkably fine distinction to hear. The difference between 5/4 and 2^(4/12) is about 1% which musicians regularly notice. The difference between 7/4 and 2^(10/12) is more than 3%, but the difference between 9/4 and 2^(14/12) is back down to about one half of a percent.

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  • "if they aren't, there's no equivalence map between all of the notes": every 12-tone temperament has an equivalence map to every other, which is the standard 12-tone keyboard. (Of course, there will be pieces for which some temperaments are unusably awful.) As this answer notes, the difference in tuning may be audible, and for certain pairs of temperaments it definitely will be, but people will still recognize the music as being the same music in both temperaments. I assume that this is what Gerald means in saying that their "use" is "the same."
    – phoog
    Oct 19, 2023 at 7:17
  • @phoog Every 12 tone temperament has a map to every other, but it's not necessarily an equivalence map. If it were, the all the tones would have equivalent frequencies.
    – user121330
    Oct 19, 2023 at 12:30

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