Although unusual time signatures represent more information, or a bit more clarity as far as composers and conducters are concerned, one has to balance this with the confusuin that it may bring to a performer.

This has nothing to do with the concept of "irrational numbers" in mathematics (a mathematical irrational number is one that cannot be represented as the ratio of two integers: ex.: π). The term "irrational" here refers to a denominator that is not a power of two. "But you said that the denominator must always be a power of two in the basic lesson," you say. That is correct, I did say that. However, Irrational Time Signatures were invented (fairly recently) to describe other information.

The whole idea of irrational time signatures stems from the idea of tuplets. When a composer wishes to insert three notes in the space of two, for instance, tuplets give a perfect representation of the music:

However, what if a composer wishes to insert, say, more or less notes with a triplet duration than the triplet notation calls for (i.e., not 3, 6, 9, ...)? Then a new notation is required. This is how Irrational Time Signatures were born.

Here is an example of a piece that includes two measures. 4/6 states that a sixth note (which is seen as a note in a quarter note triplet) gets one beat, and that there are four beats to the measure. Thus, the duration of a 4/6 measure is two-thirds of a whole note in 4/4 time, as the mathematical fraction would suggest. We now have the two rules for Irrational Time Signatures:

#### When the time signature's denominator is not a power of two, the beat note is represented by the note with the previous power of two denominator.

In other words, a sixth note is represented by a quarter note. One should remember, however, that a sixth note is only 2/3 the duration of a quarter note.
#### Do not use an irrational time signature if there is a more conventional notation to suit your purpose.

I should make this statement here: in an irrational time signature, only the beat note is changed; the relationships with all other notes remains constant. So, two half notes fill a 4/6 measure, and so does one whole note.

Below is an example of of a piece with 4/6 measures:

Compare the first and second examples. In the first example, there are two "ordinary" beats followed by a triplet. In the second example, the 4/6 beats are __exactly__ the duration of the example 1 triplet notes.

Note also that, in the 4/6 measures, the first 3 beats group together temporally, leaving the 4th beat to dangle. This is because we are used to triplets. However, this example also shows how 4/6 measures can fit in very nicely with 2/4 and 4/4 measures, so that it seems that the tempo temporarily speeds up. By the same token, it could also fit in nicely with 4/8 or 8/8.

This is a very simple example. However, music with irrational time signatures can get very complex very rapidly. This example is based on example 1:

- Measure 1: two quarter notes, 50% of a reference whole note.
- Measure 2: a third note is represented by a half note, and is 33.33% of a reference whole note.
- Measure 3: a seventh note is represented by a quarter note, and is 14.28% of a reference whole note.
- Measure 4: a ninth note is represented by an eighth note, and is 11.11% of a reference whole note. Two ninth notes are represented by a quarter note, and is 22.22% of a reference whole note.
- Measure 5: a whole note in 4/4 time, 100% of the reference whole note.

If this piece sounds familiar yet somehow different, it's because the original was written in 4/4 time, but has now been transposed to 2¼/4, effectively adding a sixteenth note after every two beats (thus adding two quarter notes in eight measures).

A question arises: why would anyone use 2¼/4, when the same measures in 9/16 would be written the exact same way? This example should answer the question: in it, a beat is indicated by a quarter note. There are 2¼ beats to a measure. 9/16 states that a sixteenth note gets one beat; that is not so in this case. A sixteenth note is a fourth of a beat.

Because of the mathematical equivalence of fractions and decimals, most fractional time signatures can be written as decimal time signatures; ex. 2¼/4 can be written as 2.25/4. However, not all fractions can be accurately written this way. (2 1/3)/4 cannot be written as a decimal number since 1/3 is a decimal that does not end.

An example of a song that the songwriters claim has a decimal time signature is *Schism* by the band *Tool*. They claim the song is 6.5/8; in fact, this song has many time signature changes, but I believe none of them are to 6.5/8.

#### Truly Irrational Time Signatures

Some compositions have surfaced with truly irrational time signatures, such as and . I see no reason for these truly irrational time signatures. The square root of 42 is about 6.48; why not just say 6.5/4 for this one? Pi is about 3.14159, so why not just say (3 1/8)/4 for this one? One thing we don't want is for performers to sit down and do math calculations in their head as a prerequisite for a performance.

I have heard of two signatures alternating; I see no reason why more couldn't.

Most time signatures are of the format numerator/denominator, where each term is a single number. This conveys which note gets a beat, and how many beats there are to the measure. However, there is a class of time signatures, such as the one above, where the numerator has multiple terms: in this case 3+2+3. This not only says that an eighth note gets one beat and there are eight beats to the measure, but it also tells how the beats are grouped (in this case, a group of 3, a group of 2, and a group of 3). This particular time signature can be used when transcribing some songs with Basque dance rhythms.

One has to be careful with additive meter, however. Too many terms can confuse more than help; I personally will usually use no more than 3 terms, and on rare occasions I'll use 4. Many software notation programs that accomodate additive meter limit the number of numerator terms to 4.

Music educator Carl Orff has proposed replacing the denominator of a time signature with the actual note that receives one beat. This eliminates the need for compound time signatures, which are confusing to many novices. In the picture, the left time signature is for 3/8 time, while the right signature is for 6/8, in which there are two groups of three eighth notes, and the group, a dotted quarter note, receives one beat. This method is used in many music education books, and is also used in pieces by Carl Orff and many modern American composers.

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