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I still don't grasp the "source of the discrepancy". "the E got counted twice when we went C,D,E and then E,F,G, but only got counted once when we went C,D,E,F,G." — So what? How does this expound the discrepancy? All emboldings are mine.

Impaled on a Fencepost |

The music theorists of the Middle Ages committed a fencepost error that’s too entrenched to dig up now. Consider the chord made of the notes C, E, and G (a C major triad). If music theory nomenclature for intervals made sense, the distance from C up to E plus the distance from E up to G would equal the distance from C up to G. And that’s true if you measure the intervals by counting upward steps. The problem comes when you describe intervals with what I suppose might be termed “ordinal nomenclature”: going from C to E is called going up by a third (because you count 1,2,3 when you play C,D,E) and going from E to G is called going up by a third for the same reason, but going from C to G is called going up by a fifth (because you count 1,2,3,4,5 as you play C,D,E,F,G).

The source of the discrepancy should be clear: the E got counted twice when we went C,D,E and then E,F,G, but only got counted once when we went C,D,E,F,G. So in music theory, when you stack a third on top of a third, you get a fifth. We musicians are stuck with nomenclature that essentially makes us say “3+3=5” so many times that we eventually stop noticing we’re saying it. (Of course, saying “a third plus a third is a fifth” is confusing on a different level, since it sounds like “1/3 + 1/3 = 1/5”. But I digress.)

  • I can only give conjecture. I think the source of this issue could have been the desire to hear the unison (eg: C to C with no movement up or down) as perfectly harmonious, and thus as a kind of singleness or oneness. From that desire spawned the interval of the unison or "oneness" and therefore gets counted as one, not in the sense of moving by one step, but in the sense of being perfectly harmonious; nevertheless it still requires the next step (C to D) to be referred to as a second and so on. – Ootagu Aug 24 '21 at 04:38
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    The music theorists of the Middle Ages obviously weren't programmers, otherwise they would have started counting at zero! Then C would be the base note, E would be the second and G would be the fourth, solving your problem! – CJ Dennis Aug 24 '21 at 22:10
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    I don't know for sure that it's the reason in music, but I do know that inclusive counting (counting an interval by including the first item) was more common formerly than it is now. Seems it may have derived from Latin usage; the Romans had nundinae ("nine days") every eight days. Some Romance languages still use a "fifteen" word for a two-week period. Speculatively, I could link it to the lack of a zero in arithmetic back then. – Tim Pederick Aug 25 '21 at 12:38
  • "E got counted twice when we went C,D,E and then E,F,G" (emphasis mine). I count two E's in "C,D,E and then E,F,G" -- how many do you count? How is this not clear? – David K Aug 25 '21 at 17:25
  • It is the concept in maths known as inclusive counting. – Neil Meyer Aug 25 '21 at 21:13

6 Answers6

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"The source of the discrepancy should be clear"

Well, first things first, the discrepancy in question is that in musical terms, a third plus a third is a fifth. As the article mentions, 3 plus 3 equals 5 in interval speak, which is a discrepancy from the usual arithmetic idea of addition.

Now, the source of that discrepancy is that musical intervals are indexed ordinally - that is, two of the same note are a unison, or a first apart. There is no interval of a zeroth, and this is where the problem begins. Since the unison acts as the additive identity, it is possible to intervallically "add 1" to a note over and over again and still end up on the exact same frequency.

Had the music theorists of yore thought of zero-indexing their intervals, intervals would in fact be additive! Suppose instead that two notes of the same frequency were considered to be a perfect zeroth, and the doubling of frequency a logically-named heptave. In that case, we would be going up a major second from C to E, another (minor) second from E to G, and end up with C-G being a perfect fourth. The shoe fits: 2 + 2 = 4 in this case!

Side note: while intervals cannot be put together arithmetically, there is a way to count intervals that is zero-indexed: by converting frequency ratios into a number of semitones (or, for extra precision, cents), addition is preserved! Observe that C to E is 4 semitones, E to G is 3 semitones, and C to G is 7 semitones. 4 + 3 = 7, and thus the math works out right with half-step counting!

user45266
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  • The arithmetical statement 4 + 3 = 7! is not correct. Don't mix punctuation and formula syntax. – chtenb Aug 25 '21 at 12:15
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    @ChieltenBrinke: The meaning is perfectly clear. Mathematical notation gets much more ambiguous than just exclamation points, and people still understand it. – user2357112 Aug 25 '21 at 15:45
  • The fact that some people use more ambiguous notation than this doesn't mean it is good practice. – chtenb Aug 25 '21 at 20:20
  • @ChieltenBrinke If it bothers you, you have my permission to edit it to something more precise. – user45266 Aug 25 '21 at 22:54
  • @user45266 I lifted the formula into the code span to remove the ambiguity. – chtenb Aug 26 '21 at 08:04
  • @ChieltenBrinke Much appreciated, but generally code blocks on SE should only be used for actual code, since code blocks make it hard for accessibility tools to work properly. Here's my compromise! – user45266 Aug 26 '21 at 09:12
  • This has nothing to do with the use of ordinal numbers and everything to do with inclusive counting. If you count inclusively with cardinal numbers, you get the same result. – phoog Jan 23 '23 at 03:30
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What the quotation is complaining about is that if you count the notes of the third from C to E ...

C = first note
D = second note
E = third note

Thus, three notes in total.

... and then you count the third from E to G ...

E = first note
F = second note
G = third note

Thus, three more notes.

Then the distance from C to G really ought to be a "sixth", because 3 + 3 = 6. However, C to G constitutes only five notes.

C = first note
D = second note
E = third note
F = fourth note
G = fifth note

The "problem" is that the E was counted twice when we added thirds.

C = first note
D = second note
E = third note and fourth note
F = fifth note
G = sixth note

What the quotation is missing is that intervals aren't additive. They only count notes from a given starting point.

Aaron
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    "They only measure distance...." -- this seems to be the problem; distance as we usually think of it is additive, but intervallic distance is not the same thing (it is "counting" distance). If intervals were counted from zero instead of one, then thirds would be called seconds, and fifths would be called fourths, and two seconds would add to produce a fourth. Alternatively, you could think of intervallic distance in terms of difference, which is how we usually think of distance. –  Aug 24 '21 at 04:35
  • Of course, the original quotation's interpretation of why we count this way is mistaken. I mean, just look at your last table. As it demonstrate, if the medieval theorists "counted E twice" they wouldn't have called F the fourth, but they did. And what are they supposed to call E to G by the writer's logic? Presumably a second. But then the same distance would be called a third in one instance (C to E) and a second in the other. No, there's not much logic in it! – Luke Sawczak Aug 24 '21 at 05:39
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    What the quote is missing is that intervals don't properly measure distance. If they did, they would be additive. – phoog Aug 24 '21 at 05:49
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    @LukeSawczak no, both C to E and E to G would be called a second. And if you add them you would get that C to G is a fourth. All of it would be perfectly consistent. – leftaroundabout Aug 24 '21 at 14:51
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    @leftaroundabout Indeed, and from C to D would be a 1st. The writer draws attention to the "twice-counted" E but fails to notice that they have to redefine 1st, 2nd and 3rd for the system to work at all. – Luke Sawczak Aug 24 '21 at 15:37
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    I believe the poster understands the mathematics behind the issue quite clearly. Rather what I understand he is looking for is the "source of the discrepancy." And since he has already shown understanding of the mathematics I believe it's reasonable to read his question as a search for a deeper understanding as to why theorists opted for inclusive counting of intervals. – Ootagu Aug 24 '21 at 18:30
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    @Ootagu Well, some speculation, but not something I'd post as an answer: the term "fourth", for example, goes back at least to ancient Greece, before the acceptance of zero as a number. During the time modern theory was being developed, theorists adopted/adapted much of the terminology of ancient Greece. Perhaps the interval names just stuck (or were even in continuous use). – Aaron Aug 24 '21 at 18:36
  • @Ootagu (Though it's not the poster being discussed here, but the quote and its writer's interpretation of the reason) – Luke Sawczak Aug 24 '21 at 20:38
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Musical intervals count ordinal start and end points not the actual "space" between tones:

ORDINAL: 1       2       3       4       5

TONES  : C       D       E       F       G


RULER  : 0       1       2       3       4
           --1--   --2--   --3--   --4--

If you treat musical intervals as spatial distances, then indeed it doesn't make sense. But, that is not what they are.

It might make more sense if you think of it in terms of ordinal numbers for the scale degrees above the tonic, and then extend that to ordinal numbers above any given tone in lieu of a tonic.

So, in C major the E is the third ordinal position. If you transpose that to start on E going up to G, think of the E as ordinal 1 and the G is the third ordinal above it.

It's kind of like saying in plain English "take a given tone and the third scale step above it" and then condensing it to just "a third" for convenience.

The tip off should be the "-nd", "-rd", "-th" ending of the interval names. Those are ordinal names. We have, for example, fifths, not fives. And of course it isn't a faction like "one fifth."

There are some oddities in the terminology. Musical intervals are very often referred to as distances, but if you really dig into it, we can see they aren't distances. Also, there isn't a musical interval of a first, it's called a unison. Those kinds of terminology inconsistencies aren't unique to music.

The whole 3+3 thing doesn't make sense. If you counted the "spaces", it would be 2+2=4, but because the ordinal numbers start with 1 you add 1 to get the musical interval, 2+2+1=5, a musical fifth. Taking two ordinal threes and then adding them together just doesn't make sense.

Michael Curtis
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The problem is that this isn't a proper counting system starting with 0. So every additional interval only adds "n-1" where "n" is the name of the interval.

An octave means "8," but it's only 7 notes up. It's the 8th note. The next octave is. . . 15ma, the 15th note.

Bennyboy1973
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  • Proper counting systems start with 1. Proper measurements of distance start with zero. – phoog Aug 24 '21 at 05:47
  • Really? I need to dig up Mrs. Henderson from grade one and tell her ghost that she had the number line all wrong. Countdowns should go 3. . . 2. . . 1. . . Potato! :D Anyway, I tested your theory by measuring distance on the piano, but strangely the black keys kept throwing off the measurements. – Bennyboy1973 Aug 24 '21 at 06:54
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    Counting starts with 0. No bananas, one banana, two bananas... Maybe even negatives - I WANT one banana, I WANT two bananas... – Laurence Aug 24 '21 at 10:17
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    @LaurencePayne - counting might start with zero. But I've never heard anyone do it. 'Here's your change - no pounds, one pound, two pounds...' Farmer counting cows 'No cows, one cow, two cows...' Crash survivors, maybe - 'None so far found...' – Tim Aug 24 '21 at 12:09
  • @Bennyboy1973 consider a digital clock counting down from five seconds to zero. It illuminates a couple of (seeming) paradoxes: the seconds digit requires six distinct values for the countdown, not five, and the time expires one second after the seconds digit changes from 1 to 0, not at that moment. – phoog Aug 24 '21 at 12:23
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    Note that the ancient convention of using inclusive counting for intervals wasn't specific to music theory. After the Julian calendar reform of 45 BCE, the Roman pontifices misinterpreted the "add a leap day every four years rule" as adding a day every three years, since years 1-4, 4-7, 7-10, 10-13, etc. were counted as "four-year" intervals. (And Julius Caesar himself was no longer around to check their math.) Augustus had to fix this error decades later. – dan04 Aug 24 '21 at 20:23
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    @Tim Maybe it's my computer programming brain. The first of anything is always 0-- the point of origin in a chart, the first byte of a sequence of memory (i.e. with an offset of 0), etc. Calling an interval a "3rd" which is in fact an upward span of 2 notes is kind of goofy. We're used to it in music, but it's goofy exactly BECAUSE it leads people to misunderstandings-- like 2*8 = 15 for octaves. I remember seeing "16va" in a work by an impressionist-- Ravel maybe? – Bennyboy1973 Aug 24 '21 at 21:47
  • @dan04 - and was there ever a year 0? We all fell for it with the millenium when new years eve 2000 arrived, I think. – Tim Aug 25 '21 at 07:14
  • I just realized we all have too much Corona time on our hands. Oh well, it's all a bit silly, but a fun little diversion. :D – Bennyboy1973 Aug 25 '21 at 09:35
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    @Tim if you paid $10 for a $10 item, here is your change counted back: 0. It depends on what is being counted, but natural numbers can start on either 0 or 1. Time is a good example of zero starting. – Michael Curtis Aug 25 '21 at 11:45
  • @Tim '...then there were NO green bottles, standing on the wall!' – Laurence Aug 25 '21 at 14:32
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    How about this: "Someone gave me a stack of three bottles: one, two. . . done." – Bennyboy1973 Aug 25 '21 at 14:38
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    @MichaelCurtis - and I started to count out the $10 to pay - 1,2,3... and I've never been told 'Here's your change - nothing.' When it happens, the reply will be 'Well, keep the change.' – Tim Aug 25 '21 at 15:41
  • @Tim, I'm trying to point out you're conflating ordinals and natural numbers, but more importantly natural number can start on either 0 or 1 depending on what is counted. Just because money counting starts at 1 doesn't necessarily mean anything about how other things are counted, or whether the situation is actually dealing with ordinal numbers. Flip the situation. Does anyone count money like "first dollar, second dollar..?" Not normally, but that's what's happening with musical intervals. – Michael Curtis Aug 25 '21 at 16:01
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    @MichaelCurtis - I know! Cardinal and ordinal numbers often get conflated, but it's not even that interals are ordinal. True, there's the inervals of thirds, fourths, etc, but the terms aren't really ordinal anyway. that's where the main problem lies. – Tim Aug 25 '21 at 16:09
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I still don't grasp the "source of the discrepancy"

The discrepancy is between the fact that two stacked thirds form a fifth, and the OP's unfulfilled expectation that it should be expressed by addition of corresponding numbers, whereas in fact 3 + 3 ≠ 5. Possibly people who first named the intervals didn't consider performing such mathematical operations.

If music theory nomenclature for intervals made sense...

That's the key sentence. Music notation was first developed, to notate the music. Theoretical analysis came later. It could be compared to languages. For instance, in this site we use language in which the words "red" and "read" can be pronounced the same, while "read" and "read" can be pronounced differently. One can find more examples where conventions used for historical reasons don't follow simple expectations, both in languages, and in music notation.

There are some theoretical developments that fix the alleged "mistake". In particular, music set theory names intervals according to the number of semitones, with the root corresponding to 0, not 1; this allows for addition.

user1079505
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Put simply: C>E is a third (of some sort) as E is the third note from C, counting C as the first. E>G is also a third (of some sort) as G is the third note from E, counting E as the first. And, as we know, C>G is a fifth (of some sort), counting C as the first.

We could go more silly, and say that C>D is a 2nd, D>E is a 2nd, E>F is a 2nd, F>G is a second, G>A is a second, A>B is a second, and B>C is also a 2nd. That makes C>C a 14th, not what we know to be an 8th.

Even sillier, mathematically, we could say 1+2=3, 2+3=5, therefore 1+3= you work it out, I can't.

It's not quite as black and white (or grey!) as that.

We appear to use two distinct counting systems. One starts at O (think measuring with a rule, or timing with a stopwatch), the other at 1 (think counting money, counting sheep, but stay awake...), and I believe the two got interbred.

It all happened so long ago, that it really doesn't make sense to do anything about it. Apparently scientists found that positive and negative (electicity) really are the wrong way round. Should we all change them? Certainly not! It's taken this long for some of us to understand intervals as they are, please don't consider changing, for any reasons, let alone logic or pedancy...

Can't do any more, just been called back to the funny farm, where all is normal.

Tim
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