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In Name and structure of staggering beat in Ramones' “Merry Christmas,” the rhythm in the second measure of the following example is described as a hemiola:

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But to me, these are just triplets and not an example of a hemiola. Hemiolas, in my experience, are more in line with the following two examples:

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Scott Wallace brought this issue up in the comments, and it led me to look up a definition. In the Harvard Dictionary of Music, hemiola is defined as:

the use of three notes of equal value in the time normally occupied by two notes of equal value.

In which case the top example would be a hemiola.

But I have yet to find a resource that explicitly shows something like the top example as a hemiola. The examples cited in the Harvard Dictionary are:

  • the French courante
  • the Viennese waltz
  • music by Schumann and Brahms (e.g., the opening of their third symphonies)

Note that every cited example is either in 3/4 or some other meter with a triple subdivision of the beat and/or meter. And of Wikipedia's eleven notated examples, every single one is also in one of these meters.

Based on these examples, a better definition may be:

within a meter of predominantly triple subdivision, the use of three notes of equal value in the time normally occupied by two notes of equal value.

The Ramones' example, which has a predominantly duple subdivision, would not be a hemiola following this definition.

Is our definition of the rhythmic hemiola lacking? I'm not trying to redefine a term here; I'm curious if either a) I'm being needlessly specific, b) there is a way to differentiate between these two types of hemiolas, and/or c) if other scholars/resources are more specific in their definition of the term.

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    @Aaron It depends on the predominant subdivision of the 2/4. In your stated example, where we've had eighth-note triplets (thus a predominant triple subdivision), it would be a hemiola. But if we've only had eighth notes leading into it, I'd argue it's just a triplet. – Richard♦ Jan 5 at 22:10
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    In my experience hemiola is used to describe regrouping of two groups of three into three groups of two as in your second definition. (I like to say that it is a musical manifestation of the commutative property of multiplication.) But the Oxford English Dictionary says that its plain meaning is simply "one and a half," that is, a synonym of sesquialtera. Both words were used both for the rhythmic proportion and to denote the pitch interval we know as the perfect fifth. The history of its use in later music theory is not covered other than to refer the reader to the 1880 Grove dictionary. – phoog Jan 5 at 22:16
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    Update: the 1880 Grove only mentions the perfect fifth and triplets. It says "three minims, sung against two, are called Hemiolia major; three crotchets (semiminims) against two, Hemiolia minor." I note that in 3/4 time, the regrouping of two measures into three groups of two beats may perhaps be seen as "the same as" three minims against two if you reach back into mensural concepts of perfection and imperfection, but otherwise it would be instructive to find out when the term began to be applied to the pre-cadential regrouping so common in triple meters in the Baroque period. – phoog Jan 5 at 22:28
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    @leftaroundabout It would just be a semantic difference: whether "hemiola" describes a notational or experiential shift in rhythm. – Aaron Jan 5 at 22:38
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    It's funny: the textual part of the definitions seem to be more general, basically saying 3:2, but the notation examples are all triple/compound. – Michael Curtis Jan 5 at 23:28
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Hemiola turns out to be fairly strictly defined (see Sources and Definitions, below). The key distinction from other 3:2 relationships is that it is a metrical event, as opposed to a rhythmic one. That is, hemiola temporarily redefines the meter of a piece by affecting the beat level or without changing the primary subdivision pulse.

Thus, the following would be hemiola

at the beat level

X: 1
T: Hemiola
T: 6/8 to 3/4
M: 6/8
L: 1/8
K: C clef=perc stafflines=1
!>!BBB !>!BBB | !>!BB!>!B B!>!BB |

X: 1
T: Hemiola
T: 2/4 to 3/4
M: 2/4
L: 1/4
K: C clef=perc stafflines=1
!>!B B | !>!B B | B2- | B B- | B2 |

or at the subdivision level

X: 1
T: Hemiola
T: 2/4 to 3/16 (sixteenths to triplets)
M: 2/4
L: 1/16
K: C clef=perc stafflines=1
!>!BBBB !>!BBBB | !>!BBB!>!B BB!>!BB | B!>!BBB !>!BBB!>!B | BB!>!BB B!>!BBB |

Note that in all cases, the predominant rhythmic pulse doesn't change, but the metric accent does.

By contrast, the following are not hemiola, because the 3 and 2 divisions comprise different subdivisions from each other.

X: 1
T: Not Hemiola
T: eighths versus triplets
M: 2/4
L: 1/4
K: C clef=perc stafflines=1
V:V1
V:V2 name="basic subdivisions" 
[V:V1] B/2B/2 B/2B/2 | (3BBB |
[V:V2] B/2B/2 B/2B/2 | (3B/2B/2B/2 (3B/2B/2B/2 |

X: 1
T: Not Hemiola
T: eighths versus triplets
M: 2/4
L: 1/4
K: C clef=perc stafflines=1
[V:V1] B/2B/2 B/2B/2 | B      B      | (3BBB    |
[V:V2] B/2B/2 B/2B/2 | B/2B/2 B/2B/2 | B/2B/2 B/2B/2 |

Because these examples rely on duple versus triple subdivisions of the same basic pulse, they affect the rhythm but not the meter. Thus, by the strict definition, they are not hemiola.

Sources and Definitions

It is not immediately obvious from any individual source the metrical quality which distinguishes hemiola. Some sources describe it as "rhythmic." However, in all cases, it is described as affecting meter, all examples are metrical, and no examples of, say, duplets against triplets are give.

1. The Oxford Companion to Music, edited by Alison Latham (Oxford University Press, 2002), page 579.

A term denoting the ratio 3:2. In modern notation, a hemiola occurs when two bars in triple meter (e.g. 3/2) are performed as if they were notated as three bars in duple meter (6/4), or vice versa.

2. The Norton/Grove Consice Encyclopedia of Music, edited by Stanley Sadie (Macmillan Press, 1994), page 359.

In the modern metrical system it denotes the articulation of two bars in triple meter as if they were three bars in duple.

3. Stephen G. Laitz, The Complete Musician, 2nd ed. (Oxford University Press, 2008), page 63.

Another type of metrical disturbance, closely related to syncopation, is the hemiola. In a hemiola, the established meter temporarily is displaced by a competing meter.

4. Edward Aldwell and Carl Schachter, Harmony and Voice Leading, 2nd ed. (Harcourt Brace Jovanovich, 1989), page 41.

In triple and compound meters, shifted accents sometimes transform two groups of three beats into three groups of two beats. ... The technical name for this rhythmic device is hemiola.

(Note the description as "rhythmic" device, although the example given in the book is metric. That "rhythmic" and "metric" are used interchangeably is, I believe, the source of ambiguity in defining the term.)

5. Richard Hoffman, The Rhythm Book, 2nd ed. (Richard Hoffman, 2009), page 70.

Hemiola is a special form of syncopation describing the use of three even durations in place of two. ... [For example,] regroupings within the measure of triple meter.

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  • "Because these examples rely on duple versus triple subdivisions of the same basic pulse, they affect the rhythm but not the meter." Doesn't either device effect meter? If you kept playing the rhythm of either device you would get the feel of a new meter of three. – Michael Curtis Jan 12 at 14:31
  • @MichaelCurtis My understanding is that the scenario you suggest would be an actual change of meter, as opposed to the perceived, temporary change of meter brought about by hemiola. – Aaron Jan 12 at 16:37
  • but other than notation, how do you say that aurally the same 'perceived, temporary change of meter' isn't happening in duple time with triplets displacing the sense of beat? – Michael Curtis Jan 12 at 17:05
  • @MichaelCurtis One would have to be able to distinguish aurally between, say, a 3/4 meter and a triplet rhythm. However, your question makes clear I should find a better way to express the rhythmic basis of "things that aren't hemiola", since clearly they could still suggest a change in meter. – Aaron Jan 12 at 17:09
  • My point is your first line "Hemiola turns out to be fairly strictly defined" explains things well. (Maybe 'arbitrary' is an appropriate description.) It's only triple/compound changed to duple and not vise versa. It seems there isn't any deeper theory of rhythm/meter at play. Otherwise it seems our reference sources don't provide enough diverse examples. – Michael Curtis Jan 12 at 17:23
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The primary use of hemiola is to describe a sort of aliasing of 3 beats vs 2 beats. There are two examples I remember from early music (actually, I remember seeing, not really being there): one is two-quarter-notes in one voice against a quarter-note triplet in the other. The author called this a "vertical" hemiola. The other example was two quarter-notes in one measure vs a quarter-note triplet in the next measure (called, of course, "horizontal" hemiola).

One modern version of the "horizontal" hemiola is the song "America" from "West Side Story." (The music I've seen just alternated 3/4 with 6/8 time. I've heard some strumming patterns of 6/8 vs 3/4 in the bass. This yields a vertical hemiola.

Sometimes this is done in waltzes over pairs of measures using half notes followed by two quarters tied across the bar line then another half note.

I think the term is used rather broadly for any 3 vs 2 rhythmic organization.

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My Honegger-Massenkeil reference provides a similar dry definition, but adds a consequence, which seems more accessible (my translation):

Such hemiolas, which effect a significant shift of emphasis in the musical flow, are frequently found in the works of Dutch composers since Dufay, but also in the Baroque era, especially preceding cadenzas and also with even shorter note values. The shift is frequently shared by a corresponding sung text.

In the examples I encountered, hemiolas therefore superimpose a structure to at least two bars, so only every second one begins with a strong beat.

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