Time signature...Irrational meters: the use of these signatures makes metric relationships clearer or more obscure to the musician.

Time SignaturesThe method behind the Music and notes fly to an ideal world...Notes Written on the Staff...Ledger Lines...Note Durations...Time Signatures and no-Photoshop in this picture just F22.(Bold denotes a stressed beat):
one two three (as in a waltz
Example of an irrational 4
3 time signature: here there are four (4) third notes (3) per measure. A "third note" would be one third of a whole note, and thus is a half-note triplet. The second measure of 4
2 presents the same notes, so the 4
3 time signature serves to indicate the precise speed relationship between the notes in the two measures.
These are time signatures, used for so-called irrational bar lengths,[16] that have a denominator that is not a power of two (1, 2, 4, 8, 16, 32, etc.) (or, mathematically speaking, is not a dyadic rational). These are based on beats expressed in terms of fractions of full beats in the prevailing tempo—for example 3
10 or 5
24.[16] For example, where 4
4 implies a bar construction of four quarter-parts of a whole note (i.e., four quarter notes), 4
3 implies a bar construction of four third-parts of it. These signatures are only of utility when juxtaposed with other signatures with varying denominators; a piece written entirely in 4
3, say, could be more legibly written out in 4
4.

{
\time 4/2
c''2 d'' e'' f'' |
c''^\markup {
\note #"1." #1
=
\note #"1" #1
} d'' e'' f''
}
The same example written using metric modulation instead of irrational time signatures. Three half notes in the first measure (making up a dotted whole note) are equal in duration to two half notes in the second (making up a whole note).

{
\time 4/2
c''2 d'' e'' f'' |
\time 12/4
c''2. d'' e'' f''
}
The same example written using a change in time signature.
Metric modulation is "a somewhat distant analogy".[16] It is arguable whether the use of these signatures makes metric relationships clearer or more obscure to the musician; it is always possible to write a passage using non-irrational signatures by specifying a relationship between some note length in the previous bar and some other in the succeeding one. Sometimes, successive metric relationships between bars are so convoluted that the pure use of irrational signatures would quickly render the notation extremely hard to penetrate. Good examples, written entirely in conventional signatures with the aid of between-bar specified metric relationships, occur a number of times in John Adams' opera Nixon in China (1987), where the sole use of irrational signatures would quickly produce massive numerators and denominators.[citation needed]
Historically, this device has been prefigured wherever composers wrote tuplets. For example, a 2
4 bar of 3 triplet crotchets could arguably be written as a bar of 3
6. Henry Cowell's piano piece Fabric (1920) employs separate divisions of the bar (anything from 1 to 9) for the three contrapuntal parts, using a scheme of shaped note heads to visually clarify the differences, but the pioneering of these signatures is largely due to Brian Ferneyhough, who says that he "find[s] that such 'irrational' measures serve as a useful buffer between local changes of event density and actual changes of base tempo.[16] Thomas Adès has also used them extensively—for example in Traced Overhead (1996), the second movement of which contains, among more conventional meters, bars in such signatures as 2
6, 9
14 and 5
24.
A gradual process of diffusion into less rarefied musical circles seems underway.[citation needed] For example, John Pickard's Eden, commissioned for the 2005 finals of the National Brass Band Championships of Great Britain contains bars of 3
10 and 7
12.[17]
Notationally, rather than using Cowell's elaborate series of notehead shapes, the same convention has been invoked as when normal tuplets are written; for example, one beat in 4
5 is written as a normal quarter note, four quarter notes complete the bar, but the whole bar lasts only 4⁄5 of a reference whole note, and a beat 1⁄5 of one (or 4⁄5 of a normal quarter note). This is notated in exactly the same way that one would write if one were writing the first four quarter notes of five quintuplet quarter notes.
This article uses irrational in the music theory sense, not the mathematical sense, where an irrational number is one that cannot be written as a ratio of whole numbers. However, at least one composition—Conlon Nancarrow's Studies for Player Piano—uses a time signature that is irrational in the mathematical sense. The piece contains a canon with a part augmented in the ratio √42:1 (approximately 6.48:1).
Now that you have an idea of basic rhythmic values and notation used in music, you need to learn a little about time signatures.A time signature tells you how the music is to be counted. The time signature is written at the beginning of the staff after the clef and key signature. Time signatures consist of two numbers written like a fraction.The top number of the time signature tells you how many beats to count. This could be any number. Most often the number of beats will fall between 2 and 12.The bottom number tells you what kind of note to count. That is, whether to count the beats as quarter notes, eighth notes, or sixteenth notes. So the only numbers you will see as the bottom number (the denominator) will correspond to note values:
1 = whole note (you’ll never see this)
2 = half note
4 = quarter note
8 = eighth note
16 = sixteenth note
You could continue on with 32, 64, but you will hopefully never encounter them! After a while it gets a bit unwieldy. The most common bottom numbers are 4, 8 and 16.Let me give you some examples so you better understand the concept...
4/4 Time Signature Example:
A time signature of 4/4 means count 4 (top number) quarter notes (bottom number) to each bar. So the pulse, or beat, is counted 1, 2, 3, 4, 1, 2, 3, 4, and so on.That means all the notes in each bar must add up to 4 quarter notes. Any combination of rhythms can be used as long as they add up to 4 quarter notes. For instance, a bar could contain 1 half note, 1 quarter note rest and 2 eighth notes. (See diagram.) Summed together they add to 4 quarter notes total. You can never have more than or less than the sum total of the number of beats in the time signature.
3/4 Time Signature Example:
A time signature of 3/4 means count 3 quarter notes to each bar. This is an often-used time signature giving you a waltz feel. 1, 2, 3, 1, 2, 3, 1, 2, 3…
Again, the rhythms in each bar can be anything as long as they add to 3 quarter notes. This is where time signatures start to seem illogical and students often get confused. How can 3 quarter notes add up to a whole measure? You have to remember that all of our rhythmic terminology is based on 4/4 time since it is the most common. You’ll just have to accept the fact that music has some weird conventions just as any language. Think of all the illogical ways similarly spelled English words are pronounced.
6/8 Time Signature Example:
A time signature of 6/8 means count 6 eighth notes to each bar. This is also a very often-used time signature. You would count the beat: 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, and so on…
Now you will wonder why can’t you just reduce 6/8 to 3/4? After all, they add up to the same amount. One reason you might pick one time signature versus the other is how the music is organized. 6/8 is grouped into 2 groups of 3 eighth notes. 3/4 time would be grouped into 3 groups of 2 eighth notes. Depending on the structure of the bassline or song, it may make sense to group it one way instead of the other. So 6/8 feels more like two, while 3/4 feels more like three.
Time Signature Abbreviations
A few other time signatures you may see use special abbreviations instead of numbers. 4/4 is called common time since it is so common. 4/4 time is often marked with a C instead of 4/4. It means the same thing.
[Completely unimportant historical note: the C is not actually short for the word common. It is actually an incomplete circle from an older form of notation called mensural notation.]
Another common abbreviation is for cut time meaning 2/2 time. Cut time is usually written as a C with a slash through it.
Time Signature Summary
This was just a brief guide to what time signatures mean and their notation in written music. In later lessons I will explain meter and time signatures in much more detail.
The main thing to remember is a time signature tells you: How many of what kind.
That’s it. A time signature is the number of beats and the type of note the beat is.
www.studybass.com/lessons/reading-music/time-signatures/
The time signature (also known as meter signature,[1] metre signature,[2] or measure signature[3]) is a notational convention used in Western musical notation to specify how many beats (pulses) are to be contained in each bar and which note value is to be given one beat. In a musical score, the time signature appears at the beginning of the piece, as a time symbol or stacked numerals, such as Commontime inline.png or 3
4 (read common time and three-four time, respectively), immediately following the key signature or immediately following the clef symbol if the key signature is empty. A mid-score time signature, usually immediately following a barline, indicates a change of meter.
There are various types of time signatures, depending on whether the music follows simple rhythms or involves unusual shifting tempos, including: simple (such as 3
4 or 4
4), compound (e.g., 9
8 or 12
8), complex (e.g., 5
4 or 7
8), mixed (e.g., 5
8 & 3
8 or 6
8 & 3
4), additive (e.g., 3+2+3
8), fractional (e.g., 2½
4), and irrational meters (e.g., 3
10 or 5
24).
Simple time signatures[edit]
Basic time signatures: 4
4, also known as common time (Commontime inline.png); 2
2, alla breve, also known as cut time or cut-common time (cut time); plus 2
4; 3
4; and 6
8
Simple time signatures consist of two numerals, one stacked above the other:
The lower numeral indicates the note value that represents one beat (the beat unit).
The upper numeral indicates how many such beats there are grouped together in a bar.
For instance, 2
4 means two quarter-note (crotchet) beats per bar—3
8 means three eighth-note (quaver) beats per bar.
The most common simple time signatures are 2
4, 3
4, and 4
4.
Notational variations in simple time[edit]
The symbol Commontime inline.png is sometimes used for 4
4 time, also called common time or imperfect time. The symbol is derived from a broken circle used in music notation from the 14th through 16th centuries, where a full circle represented what today would be written in 3
2 or 3
4 time, and was called tempus perfectum (perfect time).[4] The symbol cut time is also a carry-over from the notational practice of late-Medieval and Renaissance music, where it signified tempus imperfectum diminutum (diminished imperfect time)—more precisely, a doubling of the speed, or proportio dupla, in duple meter.[5] In modern notation, it is used in place of 2
2 and is called alla breve or, colloquially, cut time or cut common time.
Compound time signatures[edit]
Main article: Compound meter (music)
In compound meter, subdivisions (which are what the upper number represents in these meters) of the main beat are in three equal parts, so that a dotted note (half again longer than a regular note) becomes the beat unit. Compound time signatures are named as if they were simple time signatures, in which the one-third part of the beat unit is the beat, so the top number is commonly 6, 9 or 12 (multiples of 3). The lower number is most commonly an 8 (an eighth-note): as in 9
8 or 12
8.
An example
3
4 is a simple signature that represents three quarter notes. It has a basic feel of (Bold denotes a stressed beat):
one two three (as in a waltz)
Each quarter note might comprise two eighth-notes (quavers) giving a total of six such notes, but it still retains that three-in-a-bar feel:
one and two and three and
6
8: Theoretically, this can be thought of as the same as the six-quaver form of 3
4 above with the only difference being that the eighth note is selected as the one-beat unit. But whereas the six quavers in 3
4 had been in three groups of two, 6
8 is practically understood to mean that they are in two groups of three, with a two-in-a-bar feel (Bold denotes a stressed beat):

one and a, two and a
or

one two three, four five six
Beat and time[edit]
Time signatures indicating two beats per bar (whether it is simple or compound) are called duple time; those with three beats to the bar are triple time. To the ear, a bar may seem like one singular beat. For example, a fast waltz, notated in 3
4 time, may be described as being one in a bar. Terms such as quadruple (4), quintuple (5), and so on are also occasionally used.

Actual beat divisions[edit]
As mentioned above, though the score indicates a 3
4 time, the actual beat division can be the whole bar, particularly at faster tempos. Correspondingly, at slow tempos the beat indicated by the time signature could in actual performance be divided into smaller units.

Interchangeability, rewriting meters[edit]

3
4 equals 3
8 time at a different tempo About this sound Play (help·info)
On a formal mathematical level the time signatures of, e.g., 3
4 and 3
8 are interchangeable. In a sense, all simple triple time signatures, such as 3
8, 3
4, 3
2, etc.—and all compound duple times, such as 6
8, 6
16 and so on, are equivalent. A piece in 3
4 can be easily rewritten in 3
8, simply by halving the length of the notes. Other time signature rewritings are possible: most commonly a simple time signature with triplets translates into a compound meter.

12
8 equals 4
4 time at a different tempo and requires the use of tuplets About this sound Play (help·info)
Though formally interchangeable, for a composer or performing musician, different time signatures often have different connotations. First, a smaller note value in the beat unit implies a more complex notation, which can affect ease of performance. Second, beaming affects the choice of actual beat divisions. It is, for example, more natural to use the quarter note/crotchet as a beat unit in 6
4 or 2
2 than the eight/quaver in 6
8 or 2
4.[citation needed] Third, time signatures are traditionally associated with different music styles—it might seem strange to notate a rock tune in 4
8 or 4
2.

Stress and meter[edit]
For all meters, the first beat (the downbeat, ignoring any anacrusis) is usually stressed (though not always, for example in reggae where the offbeats are stressed); in time signatures with four groups in the bar (such as 4
4 and 12
8), the third beat is often also stressed, though to a lesser degree. This gives a regular pattern of stressed and unstressed beats, though notes on stressed beats are not necessarily louder or more important.
Complex time signatures[edit]

19
16 Time Drum Beat
MENU0:00
Problems playing this file? See media help.
See also: List of musical works in unusual time signatures, Quintuple meter, and Septuple meter
Signatures that do not fit the usual duple or triple categories are called complex, asymmetric, irregular, unusual, or odd—though these are broad terms, and usually a more specific description is appropriate.[citation needed] The term odd meter, however, sometimes describes time signatures in which the upper number is simply odd rather than even, including 3
4 and 9
8.[8] The irregular meters (not fitting duple or triple categories) are common in some non-Western music, but rarely appeared in formal written Western music until the 19th century. The first deliberate quintuple meter pieces were apparently published in Spain between 1516 and 1520,[8] though other authorities reckon that the Delphic Hymns to Apollo (one by Athenaeus is entirely in quintuple meter, the other by Limenius predominantly so), carved on the exterior walls of the Athenian Treasury at Delphi in 128 BC, are probably earlier.[9] The third movement (Larghetto) of Chopin's Piano Sonata No. 1 (1828) is an early, but by no means the earliest, example of 5
4 time in solo piano music. Reicha's Fugue 20 from his Thirty-six Fugues, published in 1803, is also for piano and is in 5
8. The waltz-like second movement of Tchaikovsky's Pathétique Symphony, often described as a limping waltz,[10] is a notable example of 5
4 time in orchestral music. Examples from the 20th century include Holst's Mars, the Bringer of War and Neptune, the Mystic (both in 5
4) from the orchestral suite The Planets, Paul Hindemith's Fugue Secunda in G,(5
8) from Ludus Tonalis, the ending of Stravinsky's Firebird (7
4), the fugue from Heitor Villa-Lobos's Bachianas Brasileiras No. 9 (11
8) and the themes for the Mission Impossible television series by Lalo Schifrin (in 5
4) and Jerry Goldsmith's theme for Room 222 (in 7
4).

In the Western popular music tradition, unusual time signatures occur as well, with progressive rock in particular making frequent use of them. The use of shifting meters in The Beatles' "Strawberry Fields Forever" (1967) and the use of quintuple meter in their "Within You, Without You" (1967) are well-known examples,[11] as is Radiohead's "Paranoid Android" (includes 7
8).[12]

Paul Desmond's jazz composition Take Five, in 5
4 time, was one of a number of irregular-meter compositions that The Dave Brubeck Quartet played. They played other compositions in 11
4 (Eleven Four), 7
4 (Unsquare Dance)—and 9
8 (Blue Rondo à la Turk), expressed as 2+2+2+3
8. This last is an example of a work in a signature that, despite appearing merely compound triple, is actually more complex.

However, such time signatures are only unusual in most Western music. Traditional music of the Balkans uses such meters extensively. Bulgarian dances, for example, include forms with 5, 7, 9, 11, 13, 15, 22, 25 and other numbers of beats per measure. These rhythms are notated as additive rhythms based on simple units, usually 2, 3 and 4 beats, though the notation fails to describe the metric "time bending" taking place, or compound meters. For example, the Bulgarian Sedi Donka consists of 25 beats divided 7+7+11, where 7 is subdivided 3+2+2 and 11 is subdivided 2+2+3+2+2 or 4+3+4.[citation needed] See Variants below..
While time signatures usually express a regular pattern of beat stresses continuing through a piece (or at least a section), sometimes composers place a different time signature at the beginning of each bar, resulting in music with an extremely irregular rhythmic feel. In this case the time signatures are an aid to the performers, and not necessarily an indication of meter. The Promenade from Mussorgsky's Pictures at an Exhibition (1874) is a good example:
In such cases, a convention that some composers follow (e.g., Olivier Messiaen, in his La Nativité du Seigneur and Quatuor pour la fin du temps) is to simply omit the time signature. Charles Ives's Concord Sonata has measure bars for select passages, but the majority of the work is unbarred.

Some pieces have no time signature, as there is no discernible meter. This is commonly known as free time. Sometimes one is provided (usually 4
4) so that the performer finds the piece easier to read, and simply has 'free time' written as a direction. Sometimes the word FREE is written downwards on the staff to indicate the piece is in free time. Erik Satie wrote many compositions that are ostensibly in free time, but actually follow an unstated and unchanging simple time signature. Later composers used this device more effectively, writing music almost devoid of a discernibly regular pulse.
Early music usage[edit]
Mensural time signatures[edit]
In the 14th, 15th and 16th centuries, a period in which mensural notation was used, four basic mensuration signs determined the proportion between the two main units of rhythm. There were no measure or bar lines in music of this period; these signs, the ancestors of modern time signatures, indicate the ratio of duration between different note values. The relation between the breve and the semibreve was called tempus, and the relation between the semibreve and the minim was called prolatio. The breve and the semibreve use roughly the same symbols as our modern double whole note (breve) and whole note (semibreve), but they were not limited to the same proportional values as are in use today. There are complicated rules concerning how a breve is sometimes three and sometimes two semibreves. Unlike modern notation, the duration ratios between these different values was not always 2:1; it could be either 2:1 or 3:1, and that is what, amongst other things, these mensuration signs indicated. A ratio of 3:1 was called complete, perhaps a reference to the Trinity, and a ratio of 2:1 was called incomplete.

A circle used as a mensuration sign indicated tempus perfectum (a circle being a symbol of completeness), while an incomplete circle, resembling a letter C, indicated tempus imperfectum. Assuming the breve is a beat, this corresponds to the modern concepts of triple meter and duple meter, respectively. In either case, a dot in the center indicated prolatio perfecta (compound meter) while the absence of such a dot indicated prolatio imperfecta (simple meter).

A rough equivalence of these signs to modern meters would be:

Mensural time signature 1 (alternative).svg corresponds to 9
8 meter;
Mensural time signature 2 (alternative).svg corresponds to 3
4 meter;
Mensural time signature 3 (alternative).svg corresponds to 6
8 meter;
Mensural time signature 4 (alternative).svg corresponds to 2
4 meter.
N.B.: in modern compound meters the beat is a dotted note value, such as a dotted quarter, because the ratios of the modern note value hierarchy are always 2:1. Dotted notes were never used in this way in the mensural period; the main beat unit was always a simple (undotted) note value.

Proportions[edit]
Another set of signs in mensural notation specified the metric proportions of one section to another, similar to a metric modulation. A few common signs are shown:[18]

Allabreve.svg tempus imperfectum diminutum, 1:2 proportion (twice as fast);
Mensural proportion 2.svg tempus perfectum diminutum, 1:2 proportion (twice as fast);
Mensural proportion 5.svg or just Mensural proportion 4.svg proportio tripla, 1:3 proportion (three times as fast, similar to triplets).
Often the ratio was expressed as two numbers, one above the other,[19] looking similar to a modern time signature, though it could have values such as 4
3, which a conventional time signature could not.

Some proportional signs were not used consistently from one place or century to another. In addition, certain composers delighted in creating "puzzle" compositions that were intentionally difficult to decipher.

In particular, when the sign Allabreve.svg was encountered, the tactus (beat) changed from the usual semibreve to the breve, a circumstance called alla breve. This term has been sustained to the present day, and though now it means the beat is a minim (half note), in contradiction to the literal meaning of the phrase, it still indicates that the beat has changed to a longer note value.
en.wikipedia.org/wiki/Time_signature

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