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From a theoretical point of view (let's leave apart for a moment interpretation and writing), are ties necessary to account for note values that notes and dotted notes cannot provide?

Two tied eighths can be written as a quarter, here the tie is unnecessary.

If each eighth is in a different bar, then we can change time signature (remember, interpretation doesn't matter here) as to put them together, and then write as a quarter, so again unnecessary.

If five eighths are tied, that cannot be written as any dotted note. Is it here where it is necessary? Any other situations?

Update: "Ties" only in the sense of altering note values, phrasing is a different use of them.

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    If you're throwing away bar lines and interpretation, why hang on to traditional notation at all? – Laurence Payne Mar 31 '16 at 9:20
  • Because there are sound patterns better suited to the human auditory system processing capabilities than others. – nightcod3r Mar 31 '16 at 10:46
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    Ties are used to indicate a variety of phrasings, not just note length. But in any case, changing a well-understood common language without providing a significant advantage is counterproductive. – Carl Witthoft Mar 31 '16 at 12:19
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    Coming from the hot question list, I really first thought this was a workplace.SE question. And then it took me another second to realize you're not talking about cravats... – Tobias Kienzler Mar 31 '16 at 14:38
  • @CarlWitthoft Absolutely, question updated to focus on this aspect. As for changing languages, they always evolve, keeping what is essential (both to reading and to meaning). – nightcod3r Mar 31 '16 at 15:27
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There are note values not notateable without ties.

For example:

A note the length of a crotchet (quarter note) + a semiquaver (sixteenth note) would need to be written with a tie, as there's no notation which says "add a quarter of the length of the note to its duration". We've got "add half" (dotted notes) and "add three quarters" (double dotted notes), but not "add one quarter". That would most logically look like half a dot, I guess, which could be troublesome to read accurately. This is actually the equivalent problem to your mention of five eighth notes together (which is a minim (half note) + one quaver (eighth note)).

This one's actually fairly common - a crotchet tied to a semiquaver followed by three more semiquavers isn't an unusual pattern by any means.

Therefore, for a notation system which can represent any duration of note you're going to need ties at some point.

Some given piece of music might not have any ties that can't be represented in other ways, and sometimes you find ties which are just to try and make structure more obvious to the player - like notes split where they cross important beats - but in general yes we do require ties.

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    In fact, the majority of possible durations cannot be written without ties (in the sense "pick a random fraction, it's probably not covered"). Just take something oddball like 37/64 and you can be sure that there is no nice way to write that. (AFAIK, tuple notation allows for oddball denominators like 1/37, but for oddball numerators you just have to write it out as ties.) – Mario Carneiro Mar 31 '16 at 12:23
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    This is very true @MarioCarneiro. The vast majority of oddball durations are never used, though. – Matthew Walton Mar 31 '16 at 14:25
  • And some oddball durations can't even be written with any finite number of ties- i.e. pi. But these oddball durations are even less used, I suspect. – Scott Wallace Mar 31 '16 at 14:43
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    If anybody wants me to play a note for pi beats I'll play it for 3 and a bit and they will just have to be satisfied with that. – Matthew Walton Mar 31 '16 at 15:00
  • @MatthewWalton Agreed. Theorists are here for the beauty of maths, not so much for the beauty of sound. – nightcod3r Mar 31 '16 at 15:31
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Yes they are necessary there definitely are note values that cannot be written with mere dots. The one that comes to mind is when you are in 6/8 time and there was an Anacrusis (Upbeat) of a mere quaver.

For you to end the piece or phrase you would have to write a dotted crotchet and a Crotchet as one note. There is no way to write this note value without the use of a tie.

9/8 time with a crotchet upbeat. You want to end the phrase with one note ie seven quavers. You need two dotted crotchets and a quaver all tied together.

12/8 time with a minim upbeat. Here you could just write a semibreve for the note that you end on but for the sake of the grouping it would be better to write two dotted crotchets and a crotchet all tied together.

There is probably the need for ties at every compound time signature where only part of the beat forms part of the anacrusis. Dots work good with multiples of two but when you are dealing with beats with three notes you sometimes do not get a note value that fits with a dot.

  • In compound time signatures, you don't need an anacrusis to make ties necessary. You can't write a note lasting three full beats in 9/8 or 12/8 time without a tie. – user19146 Mar 31 '16 at 22:19
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If performers who are trying to follow a conductor need to perform notes of different durations, it will typically be necessary for them to figure out how the timing of their notes relates to the conductor's motions. If a piece has a consistent time signature (e.g. 4/4), and a performer has a quarter note at the end of a measure tied to a quarter note at the start of the next, the performer will know that the first note should start as the conductor's baton rises at the end of the first measure, and should be held until the baton moves sideways to indicate the second beat of the next measure. Even if the performer misjudges something in the rhythm, each bar line would give another chance to get back in sync. If there were no consistent bar lines, a performer who hadn't counted everything perfectly up to a given point in the music would have no way of knowing what baton motions would indicate what notes, and would thus have no way of getting back in sync.

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You often need ties to have note values add up within and across bars. For example, if you're in 4/4 and the bar already contains, say 3 crotchets (quarter notes) and a quaver (eighth note), then the next note cannot be a crotchet because that would exceed the bar length. Instead it would have to be two tied quavers which are joined with a tie over the bar line.

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