US3523480A - Longitudinal mode tuning of stringed instruments - Google Patents
Longitudinal mode tuning of stringed instruments Download PDFInfo
- Publication number
- US3523480A US3523480A US689815A US3523480DA US3523480A US 3523480 A US3523480 A US 3523480A US 689815 A US689815 A US 689815A US 3523480D A US3523480D A US 3523480DA US 3523480 A US3523480 A US 3523480A
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- US
- United States
- Prior art keywords
- string
- frequency
- strings
- flexural
- longitudinal
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- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired - Lifetime
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- G—PHYSICS
- G10—MUSICAL INSTRUMENTS; ACOUSTICS
- G10C—PIANOS, HARPSICHORDS, SPINETS OR SIMILAR STRINGED MUSICAL INSTRUMENTS WITH ONE OR MORE KEYBOARDS
- G10C3/00—Details or accessories
- G10C3/10—Tuning pins; Tensioning devices
Definitions
- the nominal frequency of the highest note is 4186 cycles per second and, at the present state of the art, the practical string length (considering suitable safety factors) is limited by the tensile strength of the string material to only a little more than 2 inches even if the strongest steel piano wire is used. This limit is relatively independent of the crosssectional area of the string.
- the increase in mass per unit length is accomplished by progressive increases in the crosssectional area of the string wire.
- to increase the diameter of the strings beyond certain values, which depend upon the lengths of the strings and the materials from which they are formed, produces excessive string stiffness which has an undesirable effect on tone quality. Consequently, at some suitable point, the designer conventionally achieves any further necessary increase in mass per unit length of the strings by loading or wrapping them in helical fashion with a material, such as copper wire, instead of by using a plain string of larger diameter.
- the flexural mode of vibration is characterized by displacement of the string, under conditions of vibration, in a direction transverse to the direction of the string itself when it is at rest.
- the present invention is based on the discovery that the iongitudinal mode of vibration of the strings plays a significant part in determining the characteristic sound of the instrumentv when-it is played in the normal way, and that if the scale of the instrument is designed so as to tune the frequencies of the lon-. gitudinal modes of vibration of the strings to certain optimum values for each string, improved tone quality and greater uniformity of tone quality from note to note over the scale of the instrument can be obtained.
- An object of this invention is the construction of a piano when successive adjacent notes are played on the keyboard the sound of the successive notes will have a similar characteristic sound so as to produce an improved degree of evenness in the scale of the instrument, evenness and uniformity of scale being one criterion for an excellent instrument.
- a further object of the invention is to provide for more precise control of tone color through tuning of the longitudinal mode of string vibration, and to make possible the design of instruments especially suited to certain applications and to particular types or moods of music and to make the sound of the instruments more effective in evoking a variety ofemotional responses in the listener.
- the fundamental frequency of the longitudinal mode of each selected string will be tuned to a frequency closely corresponding to that ofa note of the equally tempered scale, with the longitudinal frequency of each string so tuned being separated from the fundamental flexural frequency of such string by a constant number of semitone intervals. More particularly, in the case of a piano having conventional steel strings, it has been found that tuning of the longitudinal mode to the frequency of a note of the equally tempered scale in the range 4000 5200 cents sharp compared to the fundamental flexural string frequency will produce highly preferred tone qualities. It is to be understood, however, that while the range given is a practical working range for pianos of conventional design and standard piano wire, the practical range may vary depending upon the circumstances. For example, the range will vary depending upo the type of stringed instrument being tuned.
- the preferred tuning of the longitudinal modes will be at integral mutiples of approximately 100 cents sharp with respect to the fundamental flexural frequency of the string which is being tuned.
- the fundamental (lowest possible) resonant frequency of longitudinal vibration of a uniform homogeneous free rod depends upon the length of the rod and upon the elastic modulus and density (mass per unit volume) of the material of the rod.
- Such a rod is v b E 1/2 2L (P) HZ *eycles per second wherein E is the elastic or Youngs modulus of the material, P is the density (mass per unit volume) of the material, L is the length of the rod, and b is a constant whose value depends upon the system of units employed.
- E is the elastic or Youngs modulus of the material
- P is the density (mass per unit volume) of the material
- L is the length of the rod
- b is a constant whose value depends upon the system of units employed.
- Such a rod has other resonant frequencies. their values being, 2, 3, 4, etc., times the fundamental frequency
- a length of piano wire or other string material may be considered a rod in the above context, and therefore will resonate longitudinally at certain frequencies the values of which are basically independent ofthe diameter of the wire.
- the value of the resonant frequency is, however, peculiar to the material of the wire; that is a wire of the same length made of some other material than steel, having a different elastic modulus and density, may have a different fundamental longitudinal resonent frequency. Because both the density and the elastic modulus of steel piano wire normally are held constant within small tolerances, it is therefore possible to predict accurately the longitudinal resonant frequency of a length of wire once the appropriate constants are known.
- the effective length of a string for longitudinal vibrations will vary with the type of string terminations in use. Therefore, the effective length of a given string must be determined by measurement for the par- DESIGNING THE SCALE 'ln'designing that portion of the instrument scale which employs plain wire strings, i.e., strings not loaded or woundwith covering wire, the frequency of the longitudinalmode of each string is tuned to the desired value by adjusting the speaking length of the string in accordance with the formula li -1.
- D f D (4) wherein L is the speaking length of the string, L is the effective longitudinal length, f is the desired longitudinal mode frequency, and D is the difference between the speaking length of the string and the effective length of the string for longitudinal vibrations.
- the value of D depends upon the characteristics of the stringterminations and the string material used and will be determined empirically. The value of E/P normally will remain constant for any specific string material.
- equation (2) cannot be used to predict the fundamental longitudinal mode frequency of a string unless it is suitably modified to take into account the effect of the loading material upon the longitudinal vibration characteristics of the string.
- M is the mass (or weight) per unit length of the wound string and M, is the mass (or weight) per unit length that the string would have if no loading had been'used, i.e., if the string had been made of plain wire of the same size as the core wire actually used in making the wrapped string.
- M is the mass (or weight) per unit length of the wound string
- M is the mass (or weight) per unit length that the string would have if no loading had been'used, i.e., if the string had been made of plain wire of the same size as the core wire actually used in making the wrapped string.
- M and M may be expressed in any system of units as long as both are expressed in the same units, and may either be computed by methods familiar to those skilled in the art or may be obtained from previously prepared wire tables usually available to the designer.
- the fundamental longitudinal mode frequency of a wrapped string may be expressed b E 1/2 b E 1/2 M 1/2 m lm .172) Marat.)-
- f is the fundamental longitudinal mode frequency of a wrapped string of effective length L having a core wire of elastic modulus E,. and density P where M is the mass per lineal inch of the core wire alone and Maw is the mass per lineal inch of the wrapped string. It has been found that, for practical purposes, the addition of the wrapping wire has a negligible effect on the stiffness ofthe string.
- the quantity within the brackets inequation (5) represents the fundamental longitudinal frequency which the string would have ifthere had been no wrapping. If this frequency is previously known, the longitudinal frequency of a wrapped string of the same length can be expressed of the wrapping material necessary to tune the longitudinal mode fi'eq'ue'rrc yofa known string to the desired value.
- I lt shculd also be evidentfrom eq-u ation ('5) that it is possible i to tune the frequency of the longitudinal mode of a wrapped string by varying the mass of the covering wire without changing the actual speaking length of the string. It is thus possible to achieve the correct value of longitudinal mode frequency in the bass section of an instrument even though the length of the strings is limited by the dimensions of the instrument case.
- the required speaking length L for a wrapped string having a fundamental longitudinal mode frequency f is given by the equation b EB 1/2 c 1/2 1.... L... D 1 1) in which E,. and 1 comprise the elastic modulus and density of the core wire, respectively, M is the mass per unit length of the core wire only, M is the massper unit length of the wrapped string, b is a constant which depends upon the system of units employed, and D is the difference between the effective length and the speaking length of the string.
- the length of the wrapped portion of a conventional wrapped string always is made less than the speaking length of the string. If this were not done there would be interference between the wrapped portion of the string and the termination elements. If the wrap were allowed to come into contact with the termination elements, damage to the wrap would occur which would tend to ruin the string. Consequently, in instruments of standard construction, the length of core wire actually wrapped is made about 1 1/4 inches less than the speaking length of the string. It would be expected, therefore, that the longitudinal mode frequency of such a wrapped string would be somewhat greater than it would be if the wrapping extended throughout the full speaking length of the string.
- the fundamental longitudinal frequency of vibration of a particular string is considerably greater than the frequency of the fundamental flexural mode.
- the fundamental longitudinal mode frequency of a particular'string usually will have a value between 10 and 20 times the fundamental frequency of flexural vibration. This occurs for practical reasons, apart from any consideration of the longitudinal mode itself. If the string were to be stretched tightly enough so that the flexural frequency was more than about 1/10 the longitudinal frequency for a given string length, the string would be likely to break due to excessive stress on the wire. If, on the other hand, the string were to be stretched so loosely that the flexural frequency was less than about 1/20 of the longitudinal frequency, the tone quality would tend to be unacceptable by modern piano standards due to the resulting low tension and high inharmonicity.
- the fundamental flexural frequency of any given note of the piano is, within certain tolerances, the same for all properly tuned pianos. This situation exists because in the United States and in most of the world, the frequency 440 Hz. has been established as the standard frequency to which the note A,, the 49th note, counting from the bass end of standard 88-note piano keyboard, should be tuned.
- the fundamental flexural frequencies of all other notes of the keyboard are derived from such a reference frequency according to the well-known relationship of the equally tempered scale in which the design value of frequency for each successive ascending note is determined by multiplying the frequency of the preceding note by the 12th root of the number 2 (which has a'value of approximately 1.05946).
- the tuner by modifying the tension exerted on the strings, adjusts the frequency of each string until the above relation between the flexural frequencies of the notes is approximately reached.
- the factor n is chosen so that the fundamental longitudinal mode of vibration of each string will be tuned to a frequency closely corresponding to that of a note of the stretched equally tempered scale, more particularly, to the fundamental'flexural frequency of a note on the scale or keyboard of the specific instrument itself when properly tuned.
- the useful range of n normally will be between 10 and 20. in terms of the equally tempered scale this range is equivended to tuning the longitudinal mode within the range '4000 5200 cents sharp with reference to the fundamental flexural frequency. (in the musical context a cent is defined as l/ 100 of a semi-tone. 100 cents thus constitutes one semi-tone.
- An octave normally encompasses a 2:] frequency range. Because there are 12 semi-tone intervals in one octave,
- 1200 cents constitutes an octave.
- a ten to one frequency in-- terval is approximately equal to three octaves plus four semitones or 4000 cents.
- a frequency interval is approx'im'ately equal to four-octaves plus four semi-tones or 5200 cents.)
- the preferred tuning of the longitudinal mode will remain at some integral multiple of 100 cents, i.e., at 3600 or at 3700 cents, or at'4000, or at 4100, or at 4200 cents, etc.
- tuning the frequency of the longitu; dina'l mode to certain specific semi-tone or 100 cent intervals within the stated range will produce a tone quaiity more highly preferred than if it is tuned to other semi-tone intervals within a the same range.
- Mosthighly-preferred tunings have been found to be 4300 cents, 4400 cents, 4600 cents, 4700 cents, 4800 cents, and 5200 cents sharp.
- a tuning of 4000 cents is musically acceptable but results in stresses that are considered excessive for standard piano wire.
- tuning of 4700 cents In some pianos it may be desirable to avoid tuning of 4700 cents because if this tuning is used, it may tend to create a confusion of the pitch of the resultant tone because of the fact that the 4700 cents tuning produces a sound only a semi-tone removed (disregarding the octave in which the tone occurs) from the fundamental pitch of the flexural family.
- a tuning of 4800 cents may be preferred for some pianos, on the other hand, because the resultant tone sounds at a frequency four octaves above the fundamantal pitch of the flexural family, tending to reinforce the pitch sensation obtained therefrom.
- a tuning of 4600 cents has been found to produce a particularly satisfying result in certain instruments.
- a tuning of 4300 cents has been found to blend well with the flexural partial series and may be found to be useful in smaller instruments where excessive inharmonicity becomes a problem.
- lnharmonicity is the term given to the characteristic of a flexurally vibrating string that the frequencies of its overtones are not exact integral multiples of its fundamental frequency (as they theoretically would be if the string had no stiffness), but instead are slightly higher in frequency to a degree which depends upon the stiffness of the string and the relationship of the stiffness to other parameters. lnharmonicity influences the final flexural string frequencies of a piano during the tuning process.
- the frequency of one of two notes an octave apart on the keyboard normally is adjusted with respect to the other note until a beatless condition is attained, indicating correct tuning.
- condition of beatlessness normally signifies that the second partial of the lower octave note has exactly the same frequency as the fundamental frequency of the upper note.
- the frequency of the second partial would be exactly twice the fundamental frequency.
- the frequency of the second partial normally is slightly greater than twice the fundamental frequency.
- the longitudinal mode frequency of the string for the note C (flexural fundamental frequency equals 130.81 cycles per second, nominal) to a nominal tuning of 4800 cents sharp
- the frequency of the longitudinal mode at the note C four octaves above the fundamental flexural frequency of the note would be 2093'cycles per second.
- the actual C might be found to be 12 cents sharper than this value, or about 2108 cycles per second. Accordingly, the frequency of the longitudinal mode would be tuned to this latter value.
- the first step is to find the desired value of longitudinal I mode frequency.
- Reference to a standard table ofequally tempered scale frequencies (such as A Table Relating Frequency To Cents, by Robert W. Young, Conn, 1952) indicates that the fundamental frequency of the note number 21 (F is 87.307 Hz.
- the same reference shows that the frequency of a note 4800 cents above this frequency is 1396.9 Hz. (the note F However, this value includes no stretch.
- the tension or pull that this string will have when it is tuned to the correct flexural frequency may be set tothe desired value by proper selection of the diameter of the string wire.
- equation (2) may be used to find f,,,,. Because of the system of units used'to express E and P and because L, has been given in inches, the constant b must be assigned the value 1/2.54, as in the previous example.
- M (1. 98?3) 10 Pd grains/lineal inch .
- the weight of the desired wrapping will therefore be equal to the difference between these two values, or (7.17 2.351) 4.819 grains per lineal inch of core wire.
- the length of wrapping wire required to make one turn around the core wire will be 11(d, +d,,.) and the numer ofturns ofwrap wire per unit length of core wire will be l/d
- the total length of wrap wire per unit length of core wire (assuming no space between individual turns of wrap wire) is thus
- the total weight of this length of wrapping wire, M is given as the product of the weight per unit length of the wrap, M times the total length of the wrap.
- the frequency of the longitudinal mode may be tuned to the frequency ofa partial ofthe flexural mode family. This procedure is less satisfactory because in the frequency range of interest (10th through 20th partials) the frequencies of string partials do not generally coincide with those of equally-tempered scale notes. Tuning to the 14th partial of the flexural mode, for example, might produce a tone Still another method of adjusting the frequency of the longitudinal mode is to tune it according to subjective pitch. This method, while productive of excellent results, presents difficulties in that subjective pitch not only varies from individual to individual but depends upon the sound level of the tone in question.
- the instant invention provides a scale so designed as to provide optimum tuning for the longitudinal mode with reference to the flexural string frequencies over a substantial range of the keyboard. It has been found that proper application of the procedures described, in conjunction with those normally employed in piano scale design, can produce instruments having improved tone quality, more even scales, and more precisely defined pitch. Changing the tuning of the longitudinal mode from one to another of the preferred intervals acts to change subtly the tone color of the instrument. Certain tone colors may be more or less preferred for certain types of music, and it is possible to produce a variety of sounds depending upon the tonal response desired by the designer.
- a method of tuning a substantial number of adjoining strings of a stringed musical instrument which comprises the step of tuning both the flexural and the longitudinal modes of vibration of the said strings, the fundamental frequency of the longitudinal mode of vibration of each string so tuned being separated from the frequency of its fundamental flexural mode by an integral number of semi-tone intervals.
- a method of tuning a substantial number of consecutive strings of a piano which comprises tuning both the fle'xural and the longitudinal modes of vibration of the said strings, the fundamental flexural modes of vibration of the said strings being tuned to the frequencies of notes' of the equally tempered scale, taking into-account deviations therefrom to compensate for the added sharpness of flexural overtones resulting from the flexural inharmonicity of the strings, the fundamental longitudinal modes being tuned to frequencies within the range 4000 to 5200 cents sharp at intervals of substantially cents relative to the fundamental flexural frequencies ofthe notes to which the strings are tuned, whereby the frequency of i the fundamental longitudinal mode of a given string so tuned which would sound flat to the musical ear, while tuning to the:
- each of' the strings of a plurality of adjacent consecutive notes has a longitudinal frequency tuned substantially the same number of cen-ts sharp with respeet to the frequency of the fundamental.
- a stringed instrument in which a substantial number of consecutive strings are tuned both for flexural modes of string vibration and for longitudinal modes of string vibration, the length and mass-to-stiffness ratios of the strings being so chosen that the fundamental longitudinal mode frequencies of said strings are tuned to the frequencies of partials or overtones of the fundamental flexural string modes.
- a stringed instrument in which a substantial number of consecutive strings are tuned for flexural modes of string vibration and for longitudinal modes of string vibration, the length and mass-to-stiffness ratios of the strings being so chosen that the fundamental longitudinal modes are tuned to frequencies within the range of about 40 to 52 semi-tones sharp at intervals of substantially one semi-tone relative to the fundamental flexural frequencies of the notes to which the said strings are tuned.
- each of the strings of a plurality of adjacent consecutive notes has a longitudinal fundamental frequency tuned substantially the same number of semi-tones sharp with respect to the frequency of the fundamental flexural mode of the string.
- a stringed instrument in which a substantial number of consecutive adjoining strings are tuned both for flexural modes of string vibration and for longitudinal modes of string vibration, the strings being tensioned to tune their fundamental flexural modes essentially to the frequencies of notes of the equally tempered scale, the lengths and mass-to-stiffness ratios of the strings being such that their fundamental longitudinal modes are also tuned essentially to the frequencies of the equally tempered scale, the fundamental frequencies of both flexural and longitudinal modes deviating from exact equal temperament by an amount just sufficient to compensate for the frequency shift of flexural overtones caused by the flexural inharmonicity of the strings.
- each of said strings has a tuned fundamental longitudinal frequency substantially no greater than 5000 Hz.
- a piano in which the adjoining strings in at least about the lower half of the scale of the instrument are tuned both for flexural modes of string vibration and for longitudinal modes of string vibration, said adjoining strings being tensioned to tune their fundamental flexural modes of string vibration to the frequencies of notes of the equally tempered scale, including such frequency deviations as are necessary to take into account the effects of the flexural inharmonicity of the strings, each of said adjoining strings having a length and mass-to-stiffness ratio such that the frequency of its fundamental longitudinal mode of string vibration is tuned approximately (u) cents sharp relative to its fundamental flexural frequency, (u) bemg any'integer between 40 and 52, the exact value ot the frequency of the longitudinal mode of each such tuned string being set to coincide substantially with the frequency of the fundamental flexural mode of vibration of other strings within the piano whose fundamental flexural modes are tuned (u) semi-tone intervals sharp with respect to the fundamental flexural frequency of each such tuned string.
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Description
Applications Claiming Priority (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
US68981567A | 1967-12-12 | 1967-12-12 |
Publications (1)
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US3523480A true US3523480A (en) | 1970-08-11 |
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US689815A Expired - Lifetime US3523480A (en) | 1967-12-12 | 1967-12-12 | Longitudinal mode tuning of stringed instruments |
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Cited By (4)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US5874685A (en) * | 1997-08-15 | 1999-02-23 | Ellis; James F. | Reduction of longitudinal modes in musical instruments strings |
US20110219933A1 (en) * | 2010-03-10 | 2011-09-15 | Thomastik-Infeld Gesellschaft M.B.H. | Musical string |
US9117421B2 (en) | 2012-02-09 | 2015-08-25 | Edward J. McMorrow | Fully tempered duplex scale |
WO2021016313A1 (en) | 2019-07-19 | 2021-01-28 | Taylor-Listug, Inc. D/B/A Taylor Guitars | Saddle and bridge for reducing longitudinal waves in a string instrument |
-
1967
- 1967-12-12 US US689815A patent/US3523480A/en not_active Expired - Lifetime
Cited By (4)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US5874685A (en) * | 1997-08-15 | 1999-02-23 | Ellis; James F. | Reduction of longitudinal modes in musical instruments strings |
US20110219933A1 (en) * | 2010-03-10 | 2011-09-15 | Thomastik-Infeld Gesellschaft M.B.H. | Musical string |
US9117421B2 (en) | 2012-02-09 | 2015-08-25 | Edward J. McMorrow | Fully tempered duplex scale |
WO2021016313A1 (en) | 2019-07-19 | 2021-01-28 | Taylor-Listug, Inc. D/B/A Taylor Guitars | Saddle and bridge for reducing longitudinal waves in a string instrument |
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Legal Events
Date | Code | Title | Description |
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AS | Assignment |
Owner name: GENERAL ELECTRIC CREDIT CORPORATION, A NY CORP., C Free format text: SECURITY INTEREST;ASSIGNOR:BPO ACQUISITION CORP., A DE CORP;REEL/FRAME:004297/0802 Effective date: 19840615 Owner name: SECURITY PACIFIC BUSINESS CREDIT INC., 10089 WILLO Free format text: SECURITY INTEREST;ASSIGNOR:BPO ACQUISITION CORP. A CORP OF DE;REEL/FRAME:004298/0001 Effective date: 19840615 |
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AS | Assignment |
Owner name: BPO ACQUISITION CORP., 180 GILBERT AVE., CINCINNAT Free format text: ASSIGNMENT OF ASSIGNORS INTEREST.;ASSIGNOR:D.H. BALDWIN COMPANY AN OH CORP.;REEL/FRAME:004385/0934 Effective date: 19840615 |
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AS | Assignment |
Owner name: BALDWIN PIANO & ORGAN COMPANY Free format text: CHANGE OF NAME;ASSIGNOR:BPO ACQUISTION CORP.;REEL/FRAME:004473/0501 Effective date: 19840612 |