2

With my layman's knowledge of Western music theory, I understand that all intervals are built up of octaves and fifths. (Or major and minor seconds, or some other pair of independent 'unit vectors'.) For example,

  • 1 (major) third = 4 fifths - 2 octaves = 2 major seconds + 0 minor seconds
  • 1 diminished fourth = -8 fifths + 5 octaves = 1 major second + 2 minor seconds

And then it is simply a matter of choosing an 'origin', like the middle c', and choosing appropriate names (like d𝄪 = d + 2♯, where 1♯ = 7 fifths - 4 octaves = 1 major second - 1 minor second).

(Note that I'm explicitly not talking about tuning or temperaments or intonation or frequencies here. I'm talking about 'logical' notes and intervals, if that is the appropriate adjective.)

The way I see it, this leads to an essentially two-dimensional systems of notes and intervals.

This can be seen visually for example in the Wicki-Hayden layout. Or the following table, where the x-axis 'unit vector' is a minor second, and the y-axis is an augmented first.

 
 
 
 
 
...
...
...
...
 
 
 
 
 
...
B♯♯
B♯
B
B♭
...
 
 
 
 
...
c♯♯
c♯
c
c♭
...
 
 
 
...
d♯♯
d♯
d
d♭
d♭♭
...
 
 
...
e♯♯
e♯
e
e♭
e♭♭
...
 
 
 
...
f♯♯
f♯
f
f♭
f♭♭
...
 
 
...
g♯♯
g♯
g
g♭
g♭♭
...
 
 
...
a♯♯
a♯
a
a♭
a♭♭
...
 
 
 
...
b♯
b
b♭
b♭♭
...
 
 
 
 
...
c'♯
c'
c'♭
c'♭♭
...
 
 
 
 
 
...
...
...
...
 
 
 
 
 

(Apologies for the <kdb>-misusing table formatting.)

(Note how the usual 'augmented first = minor second' or 'B♯ = c' enharmonic is visible in the above table: every 1-down-1-right diagonal represents a single piano or MIDI key, with 12 keys per octave. And using different "diagonals" lead to different keyboards, e.g., 1-down-2-right or 'augmented first = 2 minor seconds' or 'B♯ = d♭' gives us 17 keys per octave.)

My question: Is the Western system of notes and intervals indeed essentially two-dimensional?

Background. I assumed "notes ignoring tuning/temperament are a 2D structure" as an obvious starting point for another question, but that point was not shared by people commenting on that (currently closed/deleted) question. Hence this question.

closed as unclear what you're asking by Dom Apr 19 '18 at 19:48

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    I don't really understand why you start with the idea that "all intervals are built up of octaves and fifths. " That seems to deliberately, but unnecessarily, create a second dimension. You can create any interval just from semitones - then there's no need for the second 'dimension'. Or have I misunderstood? – topo morto Apr 19 '18 at 6:40
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    I don't know where you're getting the idea that "there are two kinds of semitones" - if we are ignoring temperament, I think there's only one, and if we're taking account of temperament, there are potentially many different sizes of semitone. I also don't understand what you mean by "a sharp or augmented first c-cis, and a minor second c-des" but that might just be my ignorance. – topo morto Apr 19 '18 at 6:58
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    @topomorto -- OP is taking intervals to be vectors, and notes that any interval can be constructed from octaves and fifths, "or major and minor seconds, or some other pair of independent 'unit vectors'." But in order to form a basis for a 2-dimensional vector space these vectors need to be linearly independent, which they are not: any interval is a scalar multiple of any other interval, so only one vector is needed. Further, any interval can easily be constructed by adding minor seconds. – David Bowling Apr 19 '18 at 10:42
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    @MarnixKlooster -- I think that you are confusing note names and intervals. Your diagram is only note names, and it could be viewed as having dimensions of pitch class on one axis and enharmonic spelling on the other. If you want to see things this way that is fine, but it doesn't seem meaningful. – David Bowling Apr 19 '18 at 12:03
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    I reread your note. Here is quite a mathematical comment. Dimension is a concept of vector spaces. However, a vector space only has a dimension relative to a specific field. The same vector space could have different dimensions over different fields. E.g. the real numbers are one dimensional over themselves. This would correspond to referring to notes by its frequency: a single real number specifies the note. The real numbers have infinite dimension over the rational numbers. You may be able to regard some just temperaments as finite dimension subspaces. I don't see this as useful. – badjohn Apr 20 '18 at 14:19
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I know you said "ignoring temperament". But I'm not very good at following instructions, so we're going to start with temperament. I think it is key to answering your question. Hopefully that will be clear as we go on.


If we have equal temperament, we (by definition) have one fundamental interval - the semitone. All other intervals can be defined in terms of the semitone.

In other words, a minor second and an augmented first are the same interval.

So, for Western twelve-tone equal temperament (12-TET), your diagram collapses to:

G##/A/B♭♭ A#/B♭/C♭♭ A##/B/C♭ B#/C/D♭♭ B##/C#/D♭ C##/D/E♭♭ D#/E♭/F♭♭ D##/E/F♭ E#/F/G♭♭ E##/F#/G♭ F##/G/A♭♭ G#/A♭

You'll have to imagine it's all on one line.


So, where does that leave us? Is pitch fundamentally two dimensional? We don't know yet, because the single-dimensional 12-TET might be a specialisation of some more general two-dimensional system.

Let's talk about another temperament. Or specifically, the lack of a temperament; that is, just intonation.

First problem is that there is no official 'just intonation'. So, let's pick one. I think the most common is the amusingly named Ptolemy's Intense Diatonic Scale (a type of five-limit just intonation).

It uses not two, but three basis vectors: the minor tone (ratio 10:9), the major tone (ratio 9:8), and the semitone (ratio 16:15). All the intervals are constructed from these three ratios.

This raises a number of interesting points. For a start, the specific frequency of any named pitch depends not only on the name, but the key as well. And now we need three dimensions to represent all possible pitches.

Maybe pitch is three dimensional?


Let's find a system with even more dimensions. Perhaps 7-limit tuning?

Ok, perhaps not. What have we learnt so far? If you have a tuning system with a single basis vector (like 12-TET, or any TET, for that matter), you end up with a single dimension. If you have three basis vectors, you end up with a three-dimensional map of pitch. Is any of these representations more fundamental than the other? I'm not convinced.

Let's go back to your two-dimensional example. A system based on octaves and fifths is a form of Pythagorean tuning. If you use this temperament, you end up with a two-dimensional map of pitch. But I don't think this means that pitch is fundamentally two-dimensional. It just means that if we use a temperament with two basis vectors, we get two-dimensional pitch.


So, back to the question:

Is the Western system of notes and intervals indeed essentially two-dimensional?

From my understanding:

  • You can't really talk about pitches and their relationships without having a defined temperament. Unless specified, this is generally assumed to be 12-TET.
  • There is no fundamentally right temperament (or lack thereof). Many people who are much smarter than me have spent time inventing new ones. They all have advantages and disadvantages.
  • Temperament defines the basis vectors, which construct intervals, which construct a scale
  • The number of basis vectors corresponds with the number of dimensions of your pitch map (by definition)

Therefore, I cannot conclude that the Western system of notes and intervals is essentially two-dimensional. The number of pitch dimensions is simply a byproduct of the temperament that you choose. As as 12-TET is the most common Western temperament at the moment, I think one-dimension is usually a safe assumption for most discussions.

  • You seem to be saying, "If you work in 12TET, then every down-and-right diagonal in the table becomes a single entry, and the 2 dimensions collapse to one dimension." Completely agreed! But I don't see how that answers my question, which was: are notes and intervals (leaving tuning and frequencies out of the discussion) a 2-d system? You seem to be agreeing with me that the answer is, "Yes." Or are you? – Marnix Klooster Apr 19 '18 at 12:15
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    @MarnixKlooster I've significantly expanded my answer, but in short: 1. You can't leave tuning out of this discussion. 2. I can't see any support for 2 dimensions being more fundamental than any other number of dimensions, because the number of dimensions is a property of the tuning system. All that being said, I may be completely wrong here. An interesting question, for sure. – endorph Apr 19 '18 at 13:49
  • This seems a very good answer to a tricky-to-answer question and I also find it an oddly compelling bit of writing! – topo morto Apr 19 '18 at 22:04
  • This is probably the closest to a real answer that is possible without more clarification of terms and concepts from the asker. – Todd Wilcox Apr 27 '18 at 17:40
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I understand that all intervals are built up of octaves and fifths....

Why is any of that necessary? The system for naming intervals is quite simple, and nothing more than the most rudimentary arithmetic is necessary to understand it.

Quoting myself in a different answer:

The Basics: Interval Quality and Quantity:

Every interval has a quality-type and a quantity-size: For example, a Major 2nd interval. Its quantity is a 2nd, its quality is major.

Interval Quantity: The interval quantity: 2nd, 3rd, 4th etc - is dependent on its spelling in the scale you are working with. Every complete scale in the western tonal system is comprised of 7 notes: A-B-C-D-E-F-G. You can start with any note and follow that same order to make a complete scale: C-D-E-F-G-A-B, etc. (Sharped or flatted notes still have the same names - Ab,Bb, etc -the # or b is simply an adjective - a modifier specifying a particular characteristic of that note name.)

The interval quantity is determined by its spelling in the scale - If you are going from A to C, that is a 3rd: Reckoning A as 1, C is the 3rd note you spell in the scale. If you are going from C to G, that is a 5th:Reckoning C as 1, G is the 5th note you spell in the scale. When naming an interval, that is first thing you must determine: its quantity - its position in the spelling of the scale you're dealing with.

Once you know its quantity - 2nd, 3rd, 4th etc - you can begin to determine an interval's quality - major, minor, etc.

Interval Quality: The first thing we need to know with respect to interval quality - the key to it all: Major in the musical context means large, not important. Minor means small, not unimportant. Likewise, Diminished and Augmented, as will be explained.

The interval quality - Major, Minor, Diminished, Augmented, Perfect - is determined by the number of chromatic steps (commonly called half-steps), that comprise the interval.


You are putting "the cart before the horse". You are attempting to use your extra-musical knowledge to analyze music. Why not learn the language of music instead? It's not very difficult - far more simple than the math you are using.

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    Why is any of that necessary?. For what it's worth, octaves and fifths is exactly how some tuning systems are built. It's no more right or wrong than using the 12-TET semitone. In fact, once you leave 12-TET-land, a load of our "normal" assumptions go out the window. Which is why I think we are finding this question a bit hard to size up. – endorph Apr 20 '18 at 0:03
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    How can you define an interval without some reference to a temperament? The distance between two pitches? What is a pitch? I'm not being facetious; just trying to see how you can usefully separate these concepts. If you don't know the temperament you are in, you have no concept of the musical usefulness of any interval, except possibly the octave. Of course, you can ignore all of this almost all the time, because 12-TET is ubiquitous. And none of it is necessary to make good music. You can just use your ears, like I normally do. But it is still interesting. – endorph Apr 20 '18 at 5:34
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    I tend to find everything interesting. Except football. I don't want to bore you further, but I see the linked answer assumes that a semitone has one defined size. That might be fine inside the all-enveloping cocoon of 12-TET, but it's not a fundamental property of music. You're not avoiding temperament; you're assuming it. When someone asks a question, which implicitly uses Pythagorean temperament, that 12-TET assumption is misleading. I'm not trying to be annoying, even though I might be succeeding. But I think it's worth observing that music is a much broader world than 12-TET. – endorph Apr 20 '18 at 10:46
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    I'm not an expert on the development of temperaments, but it seems to me that if a bunch of musicians hadn't done the musical maths to work out the very clever system that is equal temperament, we might not actually have that system in the first place... – topo morto Apr 22 '18 at 9:19
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    Generally, I don't find most people get into mathematics for the fun of it (I certainly don't!) - they do so when they feel it makes a certain job easier, or communicating something more straightforward. I guess there are ways to understand temperament other than reaching for the numbers, but on a rather basic text based site like this, some might feel that the numbers aren't such a bad choice... – topo morto Apr 22 '18 at 21:51
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I think you have certainly presented a coherent and plausible thought, that western music tonality is two-dimensional. Just look at any discussion about music theory: there's always the note and whatever the tonic is right now; it's never good enough just to know the note.

I personally hypothesize that the tonic-right-now is fundamentally a reference to natural objects, like a wood block or glass, and especially a person's voice, that have the natural sequence of harmonics, more or less. When instruments play harmonies or melodies that remind us of a natural object, that's a tonic.

So when we listen to music as it changes tonics (as with I-IV-V) we are somehow reminded of three natural objects represented by I, IV, and V. The note-right-now has meaning in reference to the tonic in addition to its significance from 1 Hz to 20,000 Hz. Two dimensions!

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