बासर/ वारचनेचे गDणती अिध%ान ूकरण चौथे बासर/ वारचनेचे गDणती अअअिधअिधिधिध%ान%ान%ान%ान MATHEMATICAL CALCULATIONS FOR TRADITIONAL HINDUSTANI BANSURI OF SIX TONE HOLES Mathematical Calculations for Traditional Hindustani 6 Tone Hole Flute* Determining approximate finger hole locations for a traditional 6 tone hole flute (i.e. 1 blow hole and 6 tone holes total 7 holes) is somewhat complicated. The wavelength of sound produced is determined by the flute tube and tone holes. For a pipe of length L, open at both ends, and ignoring end effects, the wavelength of sound is twice the length of the tube. The frequency produced is given by dividing the speed of sound (345 m/s) at 25 C by the wavelength. The method described here is to estimate an effective length for a real (cylindrical) tube taking into account end effects, the size of tone holes, etc. Let’s discuss the basic anatomy of a primitive flute, the geometries that are involved, and name and number the holes. १८५ The basic flute is constructed of a single tube with a stopper at one end (typically cork.) This discussion will refer to this end as the “Close” end. The close end is also the end which will have the blow hole. The blow hole is the hole the instrumentalist will blow across to generate the sound (it’s the hole the player blows across.) The blow hole is located just beside the stopper. The other end of the flute is open and will be called the “Open” end. This discussion will consider a traditional Hindustani flute of 6 tone holes. The tone holes are the holes in the flute which the player will open or close using his fingers. The order that the holes are numbered for this discussion will be from 1 to 6, where 1 is the hole closest to the open end and 6 is the hole closest to the * In illustrating examples, the traditional Hindustani flute we are considering is a Safed 4 (F) Basuri of 6 tone holes. * Throughout this discussion, we will be considering the multiplication symbol “x” as “*”. close end. Other conventions or definitions may apply when learning how to play such an instrument but we’re talking about this in a way to keep it as simple as possible for the following mathematical calculations. Here’s a summary table . १८६ TRADITIONAL HINDUSTANI 6 TONE HOLE FLUTE FLUTE TERMINOLOGIES • Tube – The body of the flute • Stopper – The cork stopper which is closing one end of the tube. Tuning the instrument can be done by moving the stopper. • Close End – The end of the tube which has been closed by the stopper in it. • Open End – The end of the tube which is open. • Blow Hole – The hole the player will blow across to make the flute play. • Tone Hole – Holes that the user will “open” or “close” with his fingers when he plays the flute. There are many dimensions involved with the flute. Here’s a list of geometric parameters that will be used. १८७ TRADITIONAL HINDUSTANI 6 TONE HOLE FLUTE VARIABLE PARAMETERS OF TRADITIONAL FLUTE • t - wall thickness of the tube. • Lo - Length of the tube from the center of the blow hole to the open end. • Les - Effective length of the tube at the close end. • Dt - Inside diameter of the tube • Rt - Inside radius of the tube • Lbs - Distance from center of blow hole to stopper • L0 - Actual real distance from center of blow hole to open end. • Le0 - Effective distance from center of blow hole to open end. • Lbt1 - Distance between the blow hole and tone hole 1. • Lbt2 - Distance between the blow hole and tone hole 2. • Lbt3 - Distance between the blow hole and tone hole 3. • Lbt4 - Distance between the blow hole and tone hole 4. • Lbt5 - Distance between the blow hole and tone hole 5. • Lbt6 - Distance between the blow hole and tone hole 6. • Dbh - Diameter of blow hole. • Dt1 - Diameter of tone hole 1. १८८ • Dt2 - Diameter of tone hole 2. • Dt3 - Diameter of tone hole 3. • Dt4 - Diameter of tone hole 4. • Dt5 - Diameter of tone hole 5. • Dt6 - Diameter of tone hole 6. • Rbh - Radius of the blowhole. If the fundamental (all tone holes closed) corresponds to an effective length, then the desired effective lengths for a six (or fewer) hole flute to produce notes of the Hindustani (Indian) flute are given by the following table: Note Note Note actually Ratio, i.e. Number being played length units of Le0 0 Sa Pa 1 1 Re Dha 8/9 2 Ga Ni 4/5 3 Ma(Sudha) Sa 3/ 4 4 Pa Re 2/3 5 Dha Ga 16/27 6 Ni(Teevra) Ma(Teevra) 128/243 The actual physical length of the flute will be shorter than Le0 and the distance to the tone holes (from the close end) will be shorter than the values calculated. The end effects (at both the blow hole and at the first open hole) act (approximately) as an additional length which must be subtracted from the calculated effective lengths to get the physical lengths desired. As long as you don’t deviate too much from typical flute proportions, these approximate calculations should get you pretty close. Step One १८९ First, make your flute with no tone holes and adjust the length to match the desired lowest note. You can do this by estimating what the tube length should be by using the following formula. Tube Length = the wavelength / 2 = (Speed of Sound / Frequency) / 2 For a flute which you would like to be of note ‘Sa’ of safed 4 (F) (Frequency 349.22 Hz) i.e., the open end note is ‘Kharja Pa’ (frequency 261.92 Hz) the calculations are as follows. If sound travels at 345 * 0.904 meters per second we can estimate the length you should start your tube at. For our example, 345*0.904 = 311.88 Note: 0.904 is the multiplying factor for L0 open end length corrections at 25 C. Frequency of Pa (safed 4) = 523.84 Hz Frequency of Kharja Pa (open end note) = 523.84 / 2 = 261.92 Hz Tube Length = 311.88 / (2 * 261.92) = 593 mm You want to add about 80 mm* for the stopper and adjustment of the blow hole. Therefore you should cut your initial tube to 593 + 80 = 673 mm, rounded to 675 mm. Now you want to drill the blowhole. With reference to page no of part “Basic Principles of Flute” point number 15, the appropriate blow hole diameter for effortless playing/blowing for a Hindustani flute is 11 to 11.5 १९० mm. This enhances the flute’s properties, such as, sound quality, tunefulness and octave range. If the blow hole is of a bigger diameter, the tone is deeper. However, the power and stamina which needs to be exerted for playing also increases. Therefore, flutes with bigger diameters are not recommended for elaborate musical *The additional 80 mm length is to keep safe distance of the stopper cork from the end of the tube to avoid easy tampering after setting the same for final tuning. performances lasting for more than 3 hours. Also, octave range reduces to 2 or even more so. Oval shaped blow holes are good for flute playing. For our example, We are going to choose a Bamboo or PVC tube of Inner diameter 25 mm. As such, we have taken the values of Example Variable Results Dt 25 mm Dbh 11.5 mm Step Two Cut some material (cork or rubber) with a diameter which will snugly fit into the tube. We now have to calculate where the “stopper” will need to be placed. In reality it doesn’t matter too much as long as the inner edge is pretty close to the blowhole. १९१ We’ll also need to know the thickness of the tube wall. You will have to find this using calipers or use the specifications of the tube. Example Variable Results t 2 mm We need to calculate temporary value for further calculations called Heff. Heff = t + 0.85 * Dbh For our example, Heff = 2 + (0.85 * 11.5) = 11.775 mm Now we’ll define a parameter called e where e = fraction of blow hole covered by the player’s lip and has a value typically in the range of 0 to 0.15 and in some special cases even negative. In this example, we’ll use 0.1 for a value of e. With this value now defined, it’s possible to calculate Les. Les = Heff * (1-e) * (Dt / Dbh)^2 For our example, Les = 11.775 * (1-0.1) * (25/11.5) * (25/11.5) = 11.775 * 0.9 * 2.174 * 2.174 = 50.0827 mm Using a single Les for all notes is a reasonable approximation IF Lbs is close to 0.37 Les. We want to know Lbs so that we know १९२ where to put the stopper that we made therefore we’ll use the following expression to calculate Lbs. Lbs = 0.37 * Les For our example, Lbs = 0.37 * 50.0827 = 18.531 mm Step Three Now you have to adjust the length of the flute by blowing across the blow hole and shortening the length of the tube (cutting or sanding off lengths of the open end) until the tone you get out of the flute corresponds with the note (in this case a Kharja Pancham of F. F Shadja = 349.22 Hz.