IRRATIONAL TIME SIGNATURES
BASICS
In a 4/4 measure, the quarternote (the denominator) gets the pulse, and there are four of them in the measure.
In a 3/4 measure, the quarternote still gets the pulse, but there are three in the measure.
Western music conventionally uses time signatures in multiples of two in relation to a whole note (2/4, 8/8, 3/16).
If the denominator of the time signature is a 6 (i.e. 4/6), it means that the whole note is divided into six parts.
The value of the whole note remains constant (4/4 = 6/6). The six sixthnotes (triplets) are played in the same space of time it takes to play four quarternotes.
At a tempo of 60 BPM, in 4/4 each quarternote is played at a tempo of 60.
In a measure of 6/6, each quarternote is played at a tempo of 90, with the ratio between the two tempi being 3:2.
The relation between the two time signatures is 6:4, or 3:2 twice over.
By extension, a bar of 4/6 would mean four quarternote triplets (three in the time of two, plus one more tuplet note at the same speed).
5/6 would mean five quarternote triplets (three in the time of two, plus two more at the same speed), etc. with each of the 6th notes played at a speed of 90 BPM.
Viewed differently, a whole note in 4/4 equals 15, because there are four quarternotes in a 4/4 bar, and 60/4=15. Similarly, in 8/8, 60/8=7.5 (7.5 being the basic unit)
A time signature of 1/10 would mean that there are ten units per bar. With a basic measurement of four beats one whole note, the subdivision into ten units means there are ten eighthnote quintuplets in a 4/4 measure. one eighthnote quintuplet. The ratio is 5:4, meaning the performance speed of each eighthnote quintuplet is 75 BPM (60/8=7.5*10=75). A time signature of 2/12 means there are 12 divisions in the 4/4 bar, each one being the same as an eighthnote triplet. The ratio is 3:2, with each eighthnote played at 180 BPM (or in eighths at a tempo of 90 BPM). The most important aspect of this is the ratio between time signatures, then playing the correct number of notes at that tempo (i.e. the # of notes is the numerator, and the ratio is determined by the denominator).
EQUATION
The simple equation for determining the speed of notes is:
Denominator/ Tempo (BPM) = Basic_unit * Irrational_subdivision = Tempo_of_irrational_notes (BPM)
OR
Ratio_numerator * Tempo / Ratio_denominator = Tempo_of_irrational_notes (BPM)
(where ratio is the tuplet ratio in a common 4/4 or 8/8 measure. A 2/5 measure is quintuplet quarternotes, with a ratio of 5:4)
BASICS
In a 4/4 measure, the quarternote (the denominator) gets the pulse, and there are four of them in the measure.
In a 3/4 measure, the quarternote still gets the pulse, but there are three in the measure.
Western music conventionally uses time signatures in multiples of two in relation to a whole note (2/4, 8/8, 3/16).
If the denominator of the time signature is a 6 (i.e. 4/6), it means that the whole note is divided into six parts.
The value of the whole note remains constant (4/4 = 6/6). The six sixthnotes (triplets) are played in the same space of time it takes to play four quarternotes.
At a tempo of 60 BPM, in 4/4 each quarternote is played at a tempo of 60.
In a measure of 6/6, each quarternote is played at a tempo of 90, with the ratio between the two tempi being 3:2.
The relation between the two time signatures is 6:4, or 3:2 twice over.
By extension, a bar of 4/6 would mean four quarternote triplets (three in the time of two, plus one more tuplet note at the same speed).
5/6 would mean five quarternote triplets (three in the time of two, plus two more at the same speed), etc. with each of the 6th notes played at a speed of 90 BPM.
Viewed differently, a whole note in 4/4 equals 15, because there are four quarternotes in a 4/4 bar, and 60/4=15. Similarly, in 8/8, 60/8=7.5 (7.5 being the basic unit)
A time signature of 1/10 would mean that there are ten units per bar. With a basic measurement of four beats one whole note, the subdivision into ten units means there are ten eighthnote quintuplets in a 4/4 measure. one eighthnote quintuplet. The ratio is 5:4, meaning the performance speed of each eighthnote quintuplet is 75 BPM (60/8=7.5*10=75). A time signature of 2/12 means there are 12 divisions in the 4/4 bar, each one being the same as an eighthnote triplet. The ratio is 3:2, with each eighthnote played at 180 BPM (or in eighths at a tempo of 90 BPM). The most important aspect of this is the ratio between time signatures, then playing the correct number of notes at that tempo (i.e. the # of notes is the numerator, and the ratio is determined by the denominator).
EQUATION
The simple equation for determining the speed of notes is:
Denominator/ Tempo (BPM) = Basic_unit * Irrational_subdivision = Tempo_of_irrational_notes (BPM)
OR
Ratio_numerator * Tempo / Ratio_denominator = Tempo_of_irrational_notes (BPM)
(where ratio is the tuplet ratio in a common 4/4 or 8/8 measure. A 2/5 measure is quintuplet quarternotes, with a ratio of 5:4)
EXAMPLES
4/4 = four quarternotes 4/6 = four quarternote triplets 5/6 = five quarternote triplets 3/5 = three quarternote quintuplets 3/10 = three eighthnote quintuplets 4/12 = four eighthnote triplets 5/12 = five eighthnote triplets 3/7 = three quarternote septuplets 4/14 = four eighthnote septuplets BASIC IRRATIONAL DIVISIONS x/5 = quintuplet quarters x/10 = quintuplet eighths x/20 = quintuplet sixteenths x/6 = triplet quarters x/12 = triplet eighths x/24 = triplet sixteenths x/7 = septuplet quarters x/14 = septuplet eighths 
SPEED OF EACH NOTE AT BASIC TEMPO OF 60
60 90 90 75 150 180 180 105 210 VISUALIZATIONS A measure of 4/6 is arithmetically equal to: A measure of 4/10 is arithmetically equal to:
2/7 is equal to:

While notated as: 
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