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moz.ac.at - Time Calculations in Music

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Time Calculations in Music by Archer Endrich

Section 1 - Bringing durations to life

Numbers are the life-blood of music. In computer-based music-making, numbers inevitably play a major role as we strive to instruct the computer with our musical intentions. It is essential for a musician to think of these numbers in a living, feeling way, as a pathway to inner human psychological states, not just as numbers in a scientific, arithmetical way. We need to make a connection between the numbers and inner experience. I suggest that a possible key is to think about durations in terms of meter. A beat is the steady pulse, tempo is the rate per second of that pulse, and meter is the patterning of the beats. Meter is normally written as the number of beats in a group over the note value. E.g., ¾ means 3 beats of quarter notes (crotchets), and each group normally comprises 1 bar. is normally written as the number of beats in a group over the note value. E.g., ¾ means 3 beats of quarter notes (crotchets), and each group normally comprises 1 bar. Tempo is given as beats (upper number) per minute (lower number), such as 60/60, 90/60, 120/60. This is usually written as MM e = 60, where e is the type of note given one beat (it could be any type of note: quavers tend to be used for slow passages (the performer can handle seeing a more complicated looking notation), and minims for fast passages (the performer needs the music to look as simple as possible). Suppose the number is 5: five units (at a tempo of your choice). Count mentally up to 5 at this chosen tempo while sustaining a tone (physically or also mentally) throughout the 5 counts, stopping at the end of the 5th count, just before the 6th would begin. Now you are experiencing the length of 5. In this way, durations move in your experience from being numbers to being feelings, and this brings them to life from a musical point of view.

Try experiencing the following number patterns:

1 - 1 - 1 - 1 - 1 - 1 a steady pulse of units of equal length and strength a steady pulse of units of equal length and strength 2 - 1 - 2 - 1 - 2 - 1 alternating long and short, giving a lilting quality alternating long and short, giving a lilting quality 3 - 2 - 1 - 2 - 3 a symmetry that compresses towards the centre and then expands a symmetry that compresses towards the centre and then expands 5 - 3 - 1 - 6 - 2 - 4 a mixture of lengths with an irregular feel, but also somewhat balanced because all the numbers from 1 to 6 are used a mixture of lengths with an irregular feel, but also somewhat balanced because all the numbers from 1 to 6 are used 6 - 5 - 4 - 3 - 2 - 1 steady compression like a gas coming under increasing pressure, feel the temperature rise, feel the walls closing in, the excitement level rising: what will come next? In reverse, this sequence falls into a deep repose. Another important consideration in the musicalisation of time is rate of information flow. This is the amount (density) of information per temporal unit and is, in fact, independent from tempo. You can have a fast tempo along with an almost unchanging information flow, inducing a sense of suspense, delay, withholding or stasis. Or you could have a slow tempo but a rapid change in information content, inducing a sense of rapid movement, like scenery flashing by outside a train window (as you sit still). Information density can also expand or compress within the tempo, such as the 'divisions' embellishment type in Elizabethan music, or the Ockeghem 'rush to the cadence': information density rate of chord change increases in the approach to the final cadence.

Section 2 - Practical use of SMPTE frames

This section is based on a paper by Michael Graves.

This information is all that is needed when working with SMPTE and without standard musical notations. SMPTE time code (Society of Motion Picture and Television Engineers) is in the form: Reel/cassette.minutes.seconds.frames as in: 01:00:20:00. Note that the first number is not hours, but the number of the reel of film. Music cues are also written in the form 2M8, meaning Reel 2 Music cue 8. The SMPTE cue above is therefore Reel/cassette 1, zero minutes, 20 seconds and zero frames. In computer-based music, we are used to thinking in terms of minutes and seconds, so it is only the frames that we need to convert to find the precise time when we want to place a music cue. (This process is called ‘spotting’). Michael provides a simple conversion from frames to 10 ths of seconds: 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 23-24-0 1-2 3-4-5 6-7 8-9-10 11-12 13-14-15 16-17 18-19-20 21-22 The two-frame ‘advance’ present in the above correlation ensures that the music never comes in too soon. Another consideration is that the perceived entry of some sounds may come a little later than the actual start of the soundfile, just as some instrumentalists need to begin performing their note before the beat so that it is heard on the beat. The placement of the sound against the frame rate may need to take this into account, i.e., the precise point of sync will be when the sound is actually perceived. When a cue is given as start and end SMPTE codes, you can easily calculate the length of the cue by converting everything to seconds (on your calculator). E.g.,

1M1

1:01:19:24

1:02:10:20

Using your calculator, the first code becomes 60+19+1 seconds (24 being a full second) = 80. Put this into Memory. The second code becomes 120+10+.83 = 130.83 seconds (20 frames ÷ 24 frames per second = 0.83 seconds). Now subtract the contents of Memory (80): 130.83 - 80 = 50.83 seconds duration for this cue. Note that the frame rate for film is 24 frames per second (USA) and 25 frames per second (Europe), while the frame rate for video is 30 frames per second. Note that when the music / sound can fade in and out, this eases synchronisation at the start and end of the cue. A more accurate correlation between frames and times is given in Section 3 in order to make a more direct connection with musical notation. Section 5 relates time codes and tempo.

Section 3 - Connecting duration units with standard music notation

Answering the question: ‘What are the numerical divisions of 1 second and how do they relate to standard musical notation and frames at the 24 (25) and 30 frame rates per second?’ The purpose of this section is to connect frame time points with written-out music in standard notation. This section is also useful when entering durations in numerical form, as required by some software, such as CDP’s TEXTURE Set. Section 5 outlines the relationship between seconds and microseconds (division of seconds into 1000 parts). e = 1 sec. (MM e = 60 = 60/60 = 60 beats per min of 1 sec duration each) [Times start at 0.0; 0 frames = 24 - 8 / 8 ]

Division into 2 (quavers)

Time point (sec) 0 0.5 Time point (ms) 0 500 Frame loc. @ 24 0-24 12 (exact) Frame loc. @ 30 0-30 15 (exact) 8th Frames 0-8/8 4/8 (exact)

Division into 3 (quaver triplets)

Time point (sec) 0 .34 .66 (1st note longer) Time point (ms) 0 340 670 Frame loc. @ 24 0-24 8 16 (exact) Frame loc. @ 30 0-30 10 20 (exact) 8th Frames 0-8/8 3/8 5/8 (approx.)

Division into 4 (semiquavers)

Time point (sec) 0 .25 .5 .75 Time point (ms) 0 250 500 750 Frame loc. @ 24 0-24 6 12 18 (exact) Frame loc. @ 30 0-30 7.5 15 22.5 (rounded) 8th Frames 0-8/8 2/8 4/8 6/8 (exact)

Division into 5 (semiquaver quintuplet)

Time point (sec) 0 .2 .4 .6 .8 Time point (ms) 0 200 400 600 800 Frames @ 24 0-24 5 10 14 19 (rounded) Frames @ 30 0-30 6 12 18 24 (exact) 8th Frames 0-8/8 2/8 4/8 6/8 7/8 (approx.)

Division into 6 (semiquaver sextuplet)

Time point (sec) 0 .17 .34 .5 .67 .84 Time point (ms) 0 170 340 500 670 840 Frame loc. @ 24 0-24 4 8 12 16 20 (exact) Frame loc. @ 30 0-30 5 10 15 20 25 (exact) 8th Frames 0-8/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 (approx.)

Division into 8 (hemisemiquavers)

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(Shows accurate timing of the 8 th frame units)

Time point (sec) 0 .125 .25 .375 .5 .625 .75 .825 Time point (ms) 0 125 250 375 500 625 750 825 Frame loc. @ 24 0-24 3 6 9 12 15 18 21 (exact) Frame loc. @ 30 0-30 2.75 7.5 11.25 15 18.75 22.5 26.25 (rounded) 8th Frames 0-8/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 (exact) (Note that these 8 ths match their decimal equivalents)

Division into 10 (2 x hemisemiquaver quintuplet)

(10 note events within a second is very easy!)

Time point (sec) 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Time point (ms) 0 100 200 300 400 500 600 700 800 900 Frame loc. @ 24 0-24 2 5 7 10 12 14 17 19 22 (rounded) Frame loc. @ 30 0-30 3 6 9 12 15 18 21 24 27 (exact) 8th Frames 0-8/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 (approx.)

Section 4 - Durations and tempo

Answering the questions: ‘How long in seconds is a beat at a given tempo?’ and ‘How many beats are there in 1 second at a given tempo?' This is a simple arithmetical calculation. We have seen that tempo refers to the number of beats per second, and that MM e = 60 means 60 beats per minute of 1 second each. This can be written as Number of seconds in a minute 60 (= 1 second duration for each beat) Number of beats per minute 60 This formula gives the duration of one beat for any tempo. Thus MM e = 120 or 60/120 = 0.5 duration for each beat, i.e., each beat lasts for ½ second. For a tempo of MM e = 72, 60/72 = 0.83 sec duration for each beat. The result of this calculation is also a ratio that can be useful if there is no automatic tempo adjuster and you need to work out an exact sequence of note event times at a given tempo. For example, if you have the numerical sequence in seconds (i.e., e = 60) for semiquavers as in Section 3 and you need to change it to e = 84, you calculate the ratio for e = 84 (60/84 = 0.714) and multiply the numerical sequence by this ratio. Thus 0.25 x 0.714 = 0.179, 0.5 x 0.714 = 0.357 and 0.75 x 0.714 becomes 0.536. The new timing for the 4 semiquavers is therefore: 0, 0.179, 0.357, 0.536, each semiquaver being 0.179 sec in duration. Several of the CDP Texture Set programs include a tempo function (mult). The number you enter to adjust the tempo is the ratio as discussed here. The answer to the second question, ‘How many beats are there in 1 second at a given tempo?’ is the inverse of the first question. Thus we invert the formula to read: Number of beats per minute 60 (= 1 beat per second) Number of seconds in a minute 60 At MM e = 120 (i.e., 120 beats per minute), we can confirm how this formula works. Number of beats per minute 120 (= 2 beats per second) Number of seconds in a minute 60

On sensing tempo speeds

These two tempos give a basic mental framework for sensing the speed of a tempo: 1 per second or 2 per second - without using an electronic or mechanical metronome. For e = 60 you count in seconds (‘1000-1, 1000-2’ etc.), thinking one beat per second, and for e = 120, you put 2 beats in the same time-space (1 sec.) and then start counting beats at this new speed. For the tempo midway between them ( e = 90), you put 1½ in the same time-space (1 sec.), which you would probably think of as 3 in place of 2: i.e., you would space 3 beats evenly across 2 seconds. In the latter case, you could also think about the duration of a beat at this tempo, i.e., 60/90 or 0.67: 2/3 of a second, which relates to the 3 beats across 2 seconds and then start counting beats at this new speed. These three tempos provide a mental framework, and with practice and experience, you can get good at visualising many other tempo variations: e = 66 is a ‘little more than e = 60’, e = 72 a little more again; e = 84 is a little less than e = 90, etc. Remember that any type of note (e.g., quavers or minims) can be used as the note-unit for 1 beat.

Section 5 - SMPTE time points and tempo

This section relates only when you are working with standard musical notations.

In Section 2 we correlated frames and parts of a second. In Section 3 we correlated frames and standard music notation. Now we add tempo and first query where in the music at a given tempo a time point will occur so that we can ensure know where the time point is in the music notation, and that the sync point in the music can be achieved by the musicians. This section is based on Appendix E in On the Track (Karlin & Wright. Schirmer Books. 1990). The easiest way to work out music cues when tempo is involved is to look it up in a click book. This may not always be convenient or possible, and the way to calculate it ‘by hand’ is really fairly straightforward. There is also software available to handle these calculations, as well as more complicated matters such as acceleration etc. Here we summarise some of the basic underlying arithmetical procedures, trying especially to formulate what we are trying to find out, the questions we are trying to answer. Answering the question: ‘At what beat or part of a beat in which bar does a min.sec.frames time point (i.e., music cue) occur at this tempo?’

OR,

phrased like this: ‘How many bars and beats in x/x meter @ y tempo do I have to write in order to reach that sync point?’ Let’s first look at the simplest case, where the tempo is e = 60, i.e., identical with a count in seconds. Suppose the time code for a sync point were 01.01.10.18: the 18 th frame of the 71 st second. Here we are using a frame rate of 24 frames per second. Step 1 - calculate the frame rate: the number of frames per second at that tempo (= beat = click): When there are 24 frames per second, there are 60 x 24 frames per minute = 1440 frames. In this case, there are no surprises: 1440 / tempo = 60 = 24 frames per second. This is the click rate in frames. Note that some tempos will have fractional click rates, and 8 th frames are commonly used. Section 3 relates these to their decimal and music notation equivalents. Step 2 - calculate the total time (i.e., hit point) as frames 01.01.10.18: Ignoring the first number, which is the number of the reel, the 01 means 1 minute, the 10 means 10 seconds and the 18 means 18 frames. The total number of frames is therefore (60 x 24 frames)+(10 x 24 frames) + 18 frames: 1440+240+18 = 1698 frames. Step 3 - calculate the number of beats (clicks) by dividing the total time in frames by the click rate in frames: 1698 divided by 24 = 70.75 beats (clicks). We add 1 to account for time 0, the first beat. Note that the beat number here is coming at the end of the beat but when we count in music we start the count at the beginning of the beat. We add 1 to convert to music-think. The 71.75th beat is where the sync point would be in the musical notation. You can count total beats to this point in your music or go on to Step 4 and count in bars and beats. Step 4 - calculate where this is in bars and beats (at this tempo): In a 4/4 tempo, there are 4 beats to the bar. 71.75 divided by 4 gives us 17.94 bars. A tricky bit here is ‘Where is 0.94 of a bar?’ To find out, we multiply the 4 beats of the bar by 0.94m and this gives us 3.76. Section 3 has shown us that 0.75 lines up with the fourth semiquaver of a beat, so we can change 3.76 to 3.75 and know that the hit point will be on the 4th semiquaver of the 4 th beat of the 18 th bar. Note that 17 fills up 17 bars, so the .94 will be in the 18 th bar. Similarly 3 fills up 3 beats, so the .75 will be in the 4 th beat. Let’s now re-do this same example (01.01.10.18) with a different tempo: e = 72

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Step 1 - calculate the frame rate: the number of frames per second at that tempo (= beat = click): Now we have 1440 divided by 72, which gives us a frame rate of 20 frames per second. This is 1 beat in the music, 1 click on the click track. Step 2 - calculate the total time (i.e., hit point) as frames: This is the same as above: (60 x 24 frames) + (10 x 24 frames) + 18 frames = 1440 + 240 + 18 = 1698 frames. Step 3 - calculate the number of beats (clicks) by dividing the total time in frames by the click rate in frames: 1698 divided by 20 frames per beat (click) = 84.9 beats. Adding 1 for the first beat, we now have 85.9 beats. . The 85.9 th beat is where the sync point would be in the musical notation. This may be close enough to round it to 86, or you may want to anticipate the 86 th beat by a fraction for a dead hit. Step 4 - calculate where this is in bars and beats (at this tempo): Still in 4/4, we divide 86 by 4 and get 21.5. 0.5 x 4 beats in a bar = 2. The music cue is therefore at the start of the 3 rd beat of the 22 nd bar. (21 fills the first 21 bars, so the 0.5 relates to the 22 nd bar. 2 full beats of this bar pass before we reach the half-way point of the bar. Therefore the hit point is at the start of the 3 rd beat.

Some related issues

Cues in minutes and seconds only (and rounding)

If we are given a cue in minutes and seconds only, i.e., without frames, the above method is used in just the same way except that the total time is given to us right away, e.g., 10.7 seconds. We convert it to total time in frames by multiplying by the film (24) or video (30) frame rate per second. Thus 10.7 x 24 = 258.8 frames. Karlin & Wright say to round this value to the nearest frame since anything less than a frame is too small to be observed. 258.8 therefore becomes 259 frames. Beat units (clicks) are found by dividing total time in frames by the frame rate per second, e.g., 259 divided by 18, which is 14.38. Now we have a decimal portion, and if we can round to the decimal equivalents of 8ths of frames. The correlation is shown in Section 3 for the division into hemisemiquavers: 1/8 = .125, 2/8 = .250, 3/8 = .375, 4/8 = .50, 5/8 = .625, 6/8 = .75, and 7/8 = .825. We therefore select the decimal equivalent of the nearest eighth of a frame. In this case .38 is closest to .375 ( 3/8 ), so this is the value to which we round. When calculating the number of frames per beat (click) from the musical tempo, therefore, these units round to the nearest eighth of a frame. For example, at a tempo of 84 beats per minute, 1440 divided by 84 = 17.14 frames per beat (click). The decimal 0.14 is closest to 1/8 , = 0.125, so this is the decimal value that we use.

Calculating musical tempo from a given frame rate

We can also consider this problem in terms that relate directly to clicks. Clicks are in fact beats and their rate of flow is tempo. In this case, we are given the number of frames per click (beat). We calculate the tempo from the frame rate (i.e., the number of frames per click) by putting the frame data into minute units in order to match the way tempo is indicated (beats per minute). We need to have both using the same units in order to do the arithmetic. The formula is therefore:

Number of frames per minute Number of frames per click

At 24 frames per second, 1 minute is 60 seconds x 24 frames = 1440 frames per minute. If the click rate is 12 frames per click (beat), 1440/12 = 120 beats per minute: e = 120. (Remember, different types of notes can be used as the music notation unit.). If the click rate is 18 frames per click (beat), 1440/18 = 80 beats per minute: e = 80. We can rearrange our numbers in another way to answer the question: ‘What is the tempo (beats per minute) and the frame rate (number of frames per click) if the beat lasts 0.54 seconds?’

The duration of a beat is

the number of seconds in a minute 60 the number of beats per minute x In this case, 0.54 = 60/x, so x = 60 / 0.54 = 111. The tempo is MM e = 111. The frame rate will be 24 frames per second times the duration of 1 beat: i.e., 24 x 0.54, which is 12.96, rounding to 13 frames. The click track will therefore be working at 13 frames per click and at a tempo of 111 clicks per minute. Note that 8 th parts of frames are also used for more precision. The frame rate in the Click Book by Alexander Brinkman found in Karlin & Wright advances by eighths of frames, giving figures for very small increases in tempo.

Converting microseconds into seconds

Musicians are often not used to dealing with scientific numerical representations. This section offers some guidelines to help ease the process of converting between seconds and microseconds, which is often encountered in computer-based music. The microsecond is 1/1000th of a second. 1000ms = 1 sec. We find the decimal equivalent in seconds of a microsecond by dividing 1 by 1000. An integer has a decimal point after it, whether written down or not. If we move that decimal point 1 place to the left, it is then in front of the integer. 1.0 becomes 0.1 (1 10th). This is the 10th place. The 100th place is achieved by moving it 2 places to the left. 1.0 becomes 0.01 (1 100th). The 1000th place is achieved by moving it 3 places to the left. 1.0 becomes 0.001 (1 1000th). Thus 375 microseconds = 0.375 seconds, 500 microseconds = 0.50 seconds, 12.5 microseconds (the smallest grainsize) = 0.0125 seconds, 15 microseconds (default splice slope) = 0.015 seconds. And 50 microseconds = 0.050 seconds. Converting in the other direction, from seconds to microseconds moves the decimal point to the right 3 places. Thus 0.1 seconds = 100 microseconds, 0.05 seconds = 50 microseconds, and 0.25 seconds = 250 microseconds. The charts in Section 3 provide many examples of note-related times. Durations less than 0.1 second tend to be given in microseconds in computer software. These concern grainsize (smallest is 12.5ms), attack transients (10 or 20ms has little effect, but 50 to 100 starts to soften the attack) and splice slopes (Default is 15ms). Alternative method for calculating sync points from a SMPTE cue and a given musical tempo Step 1 calculate the total time in seconds (including the frame portion): min + sec + frames in seconds (e.g., 01.01.10.18 = 70 seconds + (18 frames =) 0.75 seconds = 70.75 sec.) Step 2 calculate the duration of one beat at that tempo: 60 ÷ number of beats per minute (e.g., 60 ÷ 72 = 0.83 sec) Step 3 calculate the number of beats (clicks): total time ÷ duration of 1 beat (e.g., 70.75 ÷ 0.83333 (use the decimal places!) = 84.9 sec, and add 1 = 85.9 beats) Step 4 calculate the number of bars & beats: (e.g., 85.9 beats ÷ 4 beats in a bar = 21.475 bars rounding to 21.5) Step 5 convert parts of a bar to determine which beat contains the cue: (e.g., 0.5 th of a bar = 0.5 x the number of beats in the bar: 0.5 x 4 = 2.0, so the cue will start on the 3 rd beat of the 22 nd bar (at this tempo). Remember that 21.5 'fills up' 21 complete bars and 2 beats.

Last updated: 2 October 2004

moz.ac.at - Time Calculations in Music

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