TRAPEZIUM/TRAPEZOID.

Yet another Yank/Brit difference I never knew about. It suddenly occurred to me to wonder why a cross-bar suspended by ropes for acrobatic purposes was called a trapeze. I went to the OED, which said “Prob. orig. applied to a kind in which the ropes formed a trapezium (in sense 1b) with the roof and cross-bar.” So I went to trapezium and found:

1. Geom. a. Any four-sided plane rectilineal figure that is not a parallelogram; any irregular quadrilateral. (The Euclidean sense.)
b. spec. A quadrilateral having only one pair of its opposite sides parallel. (The specific sense to which the term was restricted by Proclus.)
  The specific sense in Eng. in 17th and 18th c., and again the prevalent one in recent use.
c. An irregular quadrilateral having neither pair of opposite sides parallel. (The usual sense in England from c1800 to c1875. Now rare. This sense is the one that is standard in the U.S., but in practice quadrilateral is used rather than trapezium.)
  This is the trapezoid (τραπεζοειδές) of Proclus: see TRAPEZOID A. 1a.

What a mess! The etymology, after explaining that the Greek etymon trapezion is a diminutive of the word for ‘table,’ trapeza, has a long small-type paragraph that describes the shift in meaning from Euclid’s (a above) to Proclus’s (b) and then adds:

This nomenclature is retained in all the continental languages, and was universal in England till late in the 18th century, when the application of the terms was transposed, so that the figure which Proclus and modern geometers of other nations call specifically a trapezium (F. trapèze, Ger. trapez, Du. trapezium, It. trapezio) became with most English writers a trapezoid, and the trapezoid of Proclus and other nations a trapezium. This changed sense of trapezoid is given in Hutton’s Mathematical Dictionary, 1795, as ‘sometimes’ used—he does not say by whom; but he himself unfortunately adopted and used it, and his Dictionary was doubtless the chief agent in its diffusion. Some geometers however continued to use the terms in their original senses, and since c 1875 this is the prevalent use.

I’d sure like to know what Hutton was up to. Was he mischievous, or just dyslexic? The only saving grace is that we don’t have much occasion to talk about trapezoids and trapezia, so not too much confusion results. Still, what a mess.

Comments

  1. Fascinating. :) Now I want to see an article on the origins and development of the -azoid suffix in English.

  2. Paul D, I don’t have an article for you but there was some discussion of -(a)zoid on alt.usage.english a few years ago.

  3. John Emerson says:

    I’m trying to figure out how a murder mystery could hinge on this difference. “Yes, my dear Dr. Watson, Herr Doktor Krautkopf did describe a trapezoidal object….”

  4. I don’t see how this becomes much of a problem for English-speakers: in any of the combinations described above trapeze and trapezoids sound different, written differently, mean different things and appear in circles that do not overlap (except the 4th-graders who study geometry in class might then visit circus…aw well, there is always tried and true response “you’ll understand it when you grow up”)
    The only people who might get confused by the word are French, since they use one word trapeze to mean all of the above. But I guess they are used to it; confusion is their natural state, they are good at it – why fix something that isn’t broken?

  5. This one is particularly stupid, like factoid, because it distorts the general meaning of -oid indicating something which does not quite fit the definition.

  6. Tatyana: The confusion is not between trapeze and trapezoid but between trapezoid and trapezium. Trapeze only comes into it because that’s how I discovered the messy relationship between the other two.

  7. Then I can only offer my deepest condolencies to geometers of out two great nations for their immense miscommunication problem that keeps them from working in unison (is that a word in English?). Suppose it can’t be resolved. How sad.

  8. Ah yes, the old trapez* ambiguities. I wrestled with these some years ago, in correspondence across the Americoid–Britlich divide. But now I must ask how people are pronouncing trapezoid. Accent on which syllable?
    And then, if commenters here have the interest and the energy, we could consider the coiled mess of uncertainties that the word diatonic brings in its train (along with its junior colleague in mischief, chromatic).

  9. Diatonic? OED sez: “denoting the scale which in any key proceeds by the notes proper to that key without chromatic alteration; hence, applied to melodies and harmonies constructed from such a scale.” What’s the coiled mess?

  10. Let’s start this way. Is a piece in C minor (three flats: B♭, E♭, A♭) that uses the note B♮ diatonic? When it uses B♮, is that a chromatic feature of the piece? Is the major chord G-B♮-D, as it occurs in a passage in C minor, a matter of chromatic harmony?
    The following are useful, by the way:
    Flat: &9837; ♭
    Natural: &9838; ♮
    Sharp: &9839; ♯
    (For some these may not display properly, without changing browser settings.)

  11. I never knew this! How weird. Noetica, I am British and place the accent in the same place in both words: on the “e”.

  12. B harmonic minor, with an A flat and a B natural, is chromatic, so the G dominant seventh in a C minor scale is chromatic too.
    The harmonic minor is sort of a messy compromise. The natural minor without the B natural sounds too modal and medieval, and the dominant seventh to tonic progression (G7 to C, whether C major or C minor) had a defining structural role in music from about 1700 to 1900 or later.
    When I studied piano we were taught to play G-A-B-C (all natural) ascending and C-Bflat-Aflat-G descending.

  13. I (a Yank) say tra-PEE-zium but TRAP-ezoid.

  14. You think that’s confusing? How about this: positive and positif mean different things in formal mathematical writing. Mathematicians mostly stopped talking about quadrilaterals about a thousand years ago, but they still talk a lot about positive numbers.

  15. zuzentzailea says:

    I did an entire degree in mathematics and probably never encountered a trapezium, much less a trapezoid, much less either of the words. (Though I knew of the terms and confusion.) There’s just no call for talking specifically about quadrilaterals that aren’t parallelograms.

  16. TRAP-ezoid? Interesting…you might want to ask your shrink, LH, why an innocent geometric form make you think of traps.
    It’s trap-e-zOid for me. Same as octOid, humanOid and factOid, etc.
    Hope you’re not annoyed with me now.

  17. B [read C] harmonic minor, with an A flat and a B natural, is chromatic, so the G dominant seventh in a C minor scale is chromatic too.
    Ah, John Emerson. That would follow from a natural and rational interpretation of the definition of diatonic in OED, conjoined with an acceptance of chromatic as meaning straightforwardly non-diatonic (in appropriate contexts). I take it you mean to include the chord G-B♮-D as chromatic “in” C minor, also.
    But I challenge you to find any published source in musicology or music theory that agrees with you. In fact, you’d be scratching hard to uncover one even on the web.
    Incidentally, if as you say “the dominant seventh to tonic progression (G7 to C, whether C major or C minor) had a defining structural role” (which I agree with), it is odd that a chord foreign to (≈ chromatic to) C minor should have any role in settling that C minor is the key (in modulation from, say, E♭ major).
    The scale that you mention as varying in its ascending and descending versions is the melodic minor. Encyclopedia Britannica has the ascending form, along with the harmonic minor, as non-diatonic (not chromatic, though), and the descending form as diatonic.
    And that’s just the beginning. But the matter, seeming simple at first, may become too convoluted to be dealt with here.

  18. Trapezius or trapezoid, large shoulder muscles. Trapezoid, small wrist bone. Of appropriate shapes. I’ve heard them always pronouced like you, Hat.

  19. Encyclopedia Britannica has the ascending form, along with the harmonic minor, as non-diatonic (not chromatic, though), and the descending form as diatonic.
    Sorry, that should be The New Grove Dictionary of Music and Musicians, along with many others, has…
    Encyclopedia Britannica has the harmonic minor as non-diatonic. But, isolated among sources accepting that allocation, Britannica has both forms of the melodic minor as diatonic; staying with C minor, however, the ascending form has notes that are not “proper to that key” (A♮ and B♮).
    Almost all sources have only the scale that proceeds exclusively by semitones as chromatic. That is, the scale you get if you play in either direction on the piano keyboard and hit every note along the way: white and black.

  20. marie-lucie says:

    Tatyana, you may not realize the number of French speakers (both first and second language) posting to this blog – all hopelessly confused, do you say? that could be, but have we been taking potshots at other ethnic groups?
    John Emerson, I think the scale you mention (different going up and down) is the baroque scale, as for instance in Mozart, and later a uniform scale was adopted, which agrees with the notes of the 7th chord but sounds (and plays) awkward with the large interval between the 6th and 7th notes. So there is not just one type of scale, but which one(s) you need to learn depends on the composer and the era.
    Noetica, I am ever more amazed at all your accomplishments!
    I learned that diatonic refers to the scale combining both tones and semitones (so that a basic tone can be identified), and chromatic to the one using all the semitones, but I am not familiar with the use of these words for chords or other elements besides scales.

  21. Marie-Lucie, you are too kind.
    The scale that John Emerson mentions is certainly not specific to the Baroque. I have never heard or read of a “baroque scale” (nor do I think there is an échelle baroque); I do now find an isolated and I would say erroneous reference here.
    Mozart and others of the so-called Classical Period, like the composers of the Baroque Period, used the melodic minor scale (the one that is different ascending and descending) extensively. Not surprisingly, they used it mostly melodically (that is, “horizontally” in time – not “vertically” in a chord). And yes, they did this to avoid the large step that you mention, which is an augmented 2nd (e.g., A♭-B♮, in the harmonic minor scale in the key of C minor). That interval was considered awkward and unmelodious. Sometimes, though, the harmonic minor scale was used melodically too (in certain of Beethoven’s sonatas, say). Much the same could be said about the Romantic Period, though things got more variable and complex then.
    You write:
    I learned that diatonic refers to the scale combining both tones and semitones…
    One problem is that this characterisation applies also to the ascending melodic minor scale, unless we restrict things further. Britannica does not restrict things further. New Grove does, but is vague on other fronts. Many sources, mainly non-American, want to include all standard non-chromatic scales as diatonic: major, harmonic minor, melodic (both forms). Other sources are just confused, or refuse to deliver judgement on the matter.
    The divergent, diffident, or plain dippy usages of the term ”diatonic” as applied to scales ramify into a veritable semantic hydra once we move on to apply the term to intervals, chords, harmony, genres, styles, and so on. I have found, on a conservative reckoning, seven incompatible meanings for ”diatonic interval” alone.

  22. Noetica, as I said (more or less) terminology in this area is a hodgepodge of compromises. The harmonic minor scale (or the melodic minor, with its different rising and descending forms; above I confused the harmonic and melodic minor) is really a the natural minor scale with the dominant seventh jammed in.
    It’s chromatic, but the word “chromatic” is usually reserved for more extreme forms of chromaticism (borrowed chords, unprepared key changes, etc.) The 12-tone “chromatic scale” is taught to piano students but doesn’t really define a style of music, as the major/minor scales do. (“Chromaticism” is music that tends to use all 12 tones, but “12 tone music” is much more extreme since it uses them in equal proportion.)
    I remember as a piano student being taught to play the harmonic minor scale as a scale, even though it is really almost never used in melodies because of the awkward 1 1/2 step interval between its 6th and 7th tones. It’s a theoretical an pedagogical artifact, I think.

  23. John Emerson:
    …as I said (more or less) terminology in this area is a hodgepodge of compromises.
    Agreed! And I observe that several of those compromises issue in dreadful muddles in the literature, and (worse) in pedagogy.
    The harmonic minor scale is really a the natural minor scale with the dominant seventh jammed in. / It’s chromatic [...]
    Find me one printed source, or one even half-respectable web source, that classifies the harmonic minor scale as “chromatic” and I’ll be mighty surprised, and even more mighty grateful.
    The idea of “jamming” the dominant seventh into the natural minor scale is interesting but under-articulated, I think. (I get the general idea.)
    It’s a theoretical an pedagogical artifact, I think.
    Arguably all of these scales are artefacts [sic; British spelling] in some perfectly reasonable sense. In fact, the harmonic minor has often been used melodically, as I have pointed out. And of course it is present in abstraction harmonically much more than the so-called natural minor – itself an artefact if ever there was one (in the context of the Common Practice Period, roughly 1600–1910, and in most contemporary music other than “art” music).
    The chromatic scale doesn’t directly define a style. But the sense in which the term chromatic is employed in chromatic scale is similar to that in which the term is applied in the broad style of tonal music sometimes called chromaticism.

  24. marie-lucie says:

    When I wrote “baroque scale” I was not quoting an accepted technical term (I am not very familiar with those as John and Noetic seem to be), but I meant that it is used a lot in baroque music (and of course later too but then the other, fixed scale takes over).
    The French word for “scale” as a musical term is “une gamme” – “une échelle” refers to a technical measurement such as for temperature, pressure or similar phenomena. (In the Tintin album “Les bijoux de la Castafiore”, that lady is constantly nagging her accompanist with “Faites vos gammes!” – an injunction which he evades by tape-recording them).

  25. Siganus Sutor says:

    Marie-Lucie, does it have something to do with the fact that in modern music the standard frequency for the note la — or A — is fixed at 440 Hz while the “la baroque” is set more around 415 Hz?

  26. “Baroque scale” is pretty reasonable, because it was in the baroque period that the major-minor system supplanted the modal system. In the major minor system you have only two modes, and the minor (natural minor in the modal system) is rather violently adapted so that you still have a dominant seventh. And there’s no getting around the fact that the progression G dom 7 –> Cm is chromatic (with the B-natural / E-flat interval).
    Even modal music would sometimes have a teentsyy bit of chromaticism when it introduced accidental notes in order to avoid tritones (e.g. using a B-flat or an F-sharp to avoid the B–>F tritone interval).
    By calling the harmonic minor scale treated as a scale an “artefact” I just meant something like “a product of teaching conventions and theoretical expedients which is not very descriptive of actual music”, rather than “something old and obsolete”.
    I could be wrong, but I don’t think that the 1 1/2-step leap (e.g. A-flat –> B-natural in C) is used melodically in baroque and classical music; if it is used, I’d say you have indeed gone into chromaticism.
    I don’t think that “chromaticism” defines a specific harmonic style the way “major-minor” or “modal” do. It’s just everything that uses a lot of chromatic steps and progressions.
    Even for someone like me, the major minor system organizes everything else around it. Alas. Some of the composers who made escape from it possible (Debussy or Stravinsky, for example) had new theories, but the real pioneers (Satie and Musorgsky) seemed to have just ignored the rules, and they were often thought of as naive and incompetent, rather than as heroes. Before them people were moving away from the major-minor system in rather timid steps.

  27. *marie-lucie: au contraire, I’m well aware of the French presence here. The number makes little difference, though. Unless you’re going to vote me out; do you? That must be entertaining, why don’t you try it?
    Thank you for confirming my opinion.

  28. Marie-Lucide, notre frangin Robert dit:
    échelle [...] Mus. L’échelle des sons. Échelle diatonique, chromatique, harmonique. → gamme.
    There seems to be a systematic difference, and I fear that it may be, trapezoidoidally, the inverse of the Anglophone difference between our cognates gamut and scale. If so, how neat for this thread.
    French Wikipedia has articles called Gamme musicale and Échelle musicale. I’ll analyse them and report back.
    Tisane tuteur, I don’t think the pitch difference you allude to is especially germane.
    John Emerson:
    And there’s no getting around the fact that the progression G dom 7 –> Cm is chromatic (with the B-natural / E-flat interval).
    Please cite a published source that supports the contention that the progression in question is ever called chromatic. I understand why you say it, but I have not seen it written anywhere. And believe me, I have surveyed a great deal of material.
    By calling the harmonic minor scale treated as a scale an “artefact” I just meant something like “a product of teaching conventions and theoretical expedients which is not very descriptive of actual music”, rather than “something old and obsolete”.
    I did not think that you meant “old and obsolete”. But I’m curious about this, because the harmonic minor scale is, I believe, quite a good construct for explaining music of the last few centuries. What seems to me artificial is the pretence that the so-called natural minor has much relevance in that period. It seems to me to be a product of largely American musicology, as a manifestation along with the major of “the” diatonic scale. But it is much rarer to see music of the period (from Buxtehude and Bach, through Mozart, Beethoven, and Schubert, to Brahms, Bruckner, and even Reger) using this natural minor.
    Once more, yes: there is in fact an American–British divide. The British are far more likely to include the harmonic and ascending melodic as diatonic, and to be silent about this “natural” minor.
    I could be wrong, but I don’t think that the 1 1/2-step leap (e.g. A-flat –> B-natural in C) is used melodically in baroque and classical music; if it is used, I’d say you have indeed gone into chromaticism.
    Look in Beethoven’s piano sonata Opus 2 no. 1, last movement; and Opus 13 (the Pathétique), last movement. You’ll find descending harmonic scales stretching over more than two octaves. And fragments of harmonic scale (with augmented seconds) elsewhere in those movements. That’s what I discovered at a casual perusal. It would be a matter of definition whether these are to be called chromaticism, but I don’t think you’ll find sources that do so. Nor do those passages at all resemble the more accepted chromaticism of the late nineteenth century. Beethoven is, for what it is worth, often said to be far less “chromatic” than Mozart.

  29. Siganus Sutor says:

    Unless you’re going to vote me out; do you?
    No, probably not, especially if you show us a photograph of your mellow self wearing your cloche at Yanny’s.

  30. OK, by now I’m not sure what we’re arguing about. The “natural minor” is important in modal music, but not in baroque or classical music. By talking of the natural minor (Aeolian) and the major (Ionian) you put major-minor harmony in the same framework as modal harmony, which has some historical or analytic value even though the other modes are rarely used in baroque or classical music.
    You could skip the natural minor entirely, but nothing much would be gained by doing so, and you’d lose the conceptual link to the earlier music and would also have the harmonic/melodic minor with its three forms sitting there on your hands as a sort of unexplainable monstrosity.
    As for why no one ever calls the harmonic minor “chromatic”, I have no idea. Based on my own education it evidently is chromatic; as I learned it, anything using more than seven scale tones in the standard succession (M3M3m3M3M3M3m3) is chromatic. When music is described as “chromatic” I think of late nineteenth century music (and some pre-baroque music such as Gesualdo) which is even more chromatic than just the harmonic minor: frequent and sudden key changes, borrowed chords, augmented sixth and diminished seventh chords, etc.
    The “feeling” of chromaticism we get comes from the augmented and diminished intervals, and what chromaticism there was in Renaissance music was meant to mask these intervals (e.g. using B-flat instead of B-natural in the neighborhood of an F-natural, or using F-sharp instead of F-natural in the neighborhood of B-natural). Once you get away from the pentatonic to the seven-tone scale you have an essentially unstable system where the tritone (diabolus) keeps popping up.

  31. OK, by now I’m not sure what we’re arguing about.
    I haven’t had the faintest idea for days now—I just watch the ball fly back and forth over the net while sipping my tisane. (My wife and I have both gotten a bad cold somehow.)

  32. SS: that’s a blackmail.
    You know I’m willing to suffer..er…anyone, if only given a chance to show off my cloche!

  33. marie-lucie says:

    Noetica: gamme and échelle:
    Well, if Robert says it, you must be right! But I don’t think I have never run into the word échelle applied to music. No doubt this is because my knowledge of music is practical, with just enough theory to get by but no in-depth study such as as you and John Emerson demonstrate.

  34. John Emerson:
    OK, by now I’m not sure what we’re arguing about. The “natural minor” is important in modal music, but not in baroque or classical music.
    To what extent are we arguing at all? I’m not sure. We agree that certain terms are used confusingly; I go further and say that this confusion proliferates when the terms are applied beyond talk of scales, and has dreadful effects in theory and pedagogy. So far you seem not to have disputed that.
    We certainly disagree about details. How, I ask, can the “natural minor” be important in modal music? The natural minor does not even arise as an analytical construct for modal music. Modal music has its modes as analytical constructs. The term is only pressed into service for music after the decline of the modes: roughly, we might say, for music of the Common Practice Period. But in this later period the “natural minor” is not in standard use, for real music-making. Apart the major and occasionally the harmonic minor, melody is constructed from the melodic minor, which has an ascending and a descending form (though in various contexts either form can be used ascending, and either descending). The descending form is formally identical to the “natural minor”; but this term is not usually employed specifically to refer to that descending form, but fictionally – as somehow of fundamental importance, representing along with the major scale some Platonic and eternal “diatonic scale”, rescued as a diamond from the medieval mud of misread Greek theory. An academickal imposition to be deplored.
    A coiled mess. Even without talk of “diatonic intervals”.
    Tat:
    You know I’m willing to suffer… er…anyone,
    I’m sure… how shall I put this?… everyone understands what you mean.

  35. Basically the natural minor allows you to see the harmonic/melodic mess as a transformation of something very analogous to the major scale, starting a minor third lower with the same notes. It’s identical to the Aeolian mode of pre-baroque music. To me it’s useful pedagogically to talk this way if the student is ever going to study theory, music history, or composition. Even for a pure performer it might be useful if they venture into early music. But as I said, the whole harmonic-melodic-natural-minor thing is a pedagogical and-or theoretical construct. construct.

  36. Now that we speak of diatonic and chromatic, I have a technical question for the Unicode mavens who hover hereabouts.
    Editing in Wikipedia I wanted to supply a polytonic transcription for Greek χρῶμα. I could do χρωματικός without any problem: khrōmatikós. But I could not find a way to combine diacriticals for χρῶμα, and had to resort to a monotonic form: khrṓma. Is there a polytonic Unicode transription for ῶ, or is this one of the lacunae?

  37. I don’t know much about transcription of polytonic Greek, Noetica, so I’m not entirely certain I know what you want to write. But if it’s an “o” with a macron over it and a circumflex above that, such a combination does not appear as a precomposed character in Unicode, but you can use ō + U+0302 COMBINING CIRCUMFLEX ACCENT, ō̂ (which may or may not display correctly depending on your font and software).

  38. Thanks Tim. Yes, we’d want a lower case “o” with a macron, and a circumflex above that. The solution you have offered doesn’t display like that for me (in Mozilla), because the circumflex and the macrom are superimposed. Same for everyone, I suspect.

  39. My browser (Konqueror) actually gets the positioning correct, but doesn’t substitute for the lack of a combining circumflex in this font, so I get a box over the macron.

  40. David Marjanović says:

    TRAP-ezoid? Interesting… [...]
    It’s trap-e-zOid for me. Same as octOid, humanOid and factOid, etc.

    What? Bizarre. I guess I’ll never understand English word stress — I mean, I get most of it right, but the rest is ineffable.
    I’d have pronounced all of them on the 3rd-to-last syllable, which seems to be the standard English approach to Latin-looking loans, and which I used to think was the way to pronounce the coracoid bone. I can almost see how that doesn’t work with “trapezoid”, though (from looking at the word, the probability that the e is supposed to be /ɛ/ is slightly greather than that it’s supposed to be /iː/).
    In German, this kind of word is treated as French and accordingly stressed on the last syllable, which is long for the same reason. I think it’s the same in Russian. (But perhaps I’m grasping for straws of predictable stress in Russian!)
    ——————
    Here in Internet Explorer 7, the o with macron and accent displays fine, but the circumflex appears as a square behind an o with macron. That’s because IE uses the font you tell it to use; someone has to tell it to use Arial Unicode MS or (maybe) one of the Lucida or Tahoma abominations.

  41. Tatyana says:

    David, Russian is my native language, and that’s how all the words I listed are pronounced in Russian. With the stress on O.

  42. Here in Internet Explorer 7, the o with macron and accent displays fine, but the circumflex appears as a square behind an o with macron.
    Hmmm, David. How could that count as displaying fine? I’m trying to visualise what you describe. A square “behind an o with macron”? Ne razumem.
    I am right now using Explorer 7, set to Arial Unicode MS, and what I see in Tim May’s post is still an “o” over which there is a circumflex superimposed on a macron. Same as in Mozilla.

  43. Here is an exchange from earlier in this thread:
    John Emerson:
    C harmonic minor, with an A flat and a B natural, is chromatic, so the G dominant seventh in a C minor scale is chromatic too.
    Me:
    …But I challenge you to find any published source in musicology or music theory that agrees with you. In fact, you’d be scratching hard to uncover one even on the web.
    I am delighted now to be able to report one such occurrence, from 1830:
    In modern music, the seventh note Si is often made one semitone higher, and then the scale of the minor key becomes chromatic. [...] The sixth and seventh notes are both occasionally altered at the same time, and then also the scale is chromatic. [...] This is the usual method of ascending the minor key, but in descending, the ancient diatonic scale is commonly used.
    (Elements of Musical Composition, Crotch, William, 1830 [reproduced 1991, Boethius Press, Aberystwyth, Wales], pp. 21-22)
    Many rebel spies died to bring you this intelligence. [A general pause, during which heads are bowed in mute homage.] Well, it took an hour in the library, at least. If anyone can find another like this in print, please let me know!

  44. You know, when I started this blog, little did I know it would wind up at the cutting edge of the history of musical theory.

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