TW201608559A - Method for design a board with harmonics sound and percussion instruments with boards - Google Patents
Method for design a board with harmonics sound and percussion instruments with boards Download PDFInfo
- Publication number
- TW201608559A TW201608559A TW103128869A TW103128869A TW201608559A TW 201608559 A TW201608559 A TW 201608559A TW 103128869 A TW103128869 A TW 103128869A TW 103128869 A TW103128869 A TW 103128869A TW 201608559 A TW201608559 A TW 201608559A
- Authority
- TW
- Taiwan
- Prior art keywords
- soundboard
- harmonic
- simple harmonic
- model
- design
- Prior art date
Links
Landscapes
- Auxiliary Devices For Music (AREA)
- Electrophonic Musical Instruments (AREA)
Abstract
Description
本發明係關於一種音板的設計方法及具有音板之擊樂器；特別是關於一種具簡諧倍頻音之音板的設計方法及具有該音板之擊樂器。 The present invention relates to a method for designing a soundboard and a percussion instrument having a soundboard; and more particularly to a method for designing a soundboard having a simple harmonic frequency and a percussion instrument having the same.
在交響樂團中，通常具有弦樂器(stringed instruments)及擊樂器(percussion instrument)，弦樂器係如：鋼琴(piano)或吉他(guitar)等，以吉他弦為例，其振動模態之自然頻率包含基音(fundamental frequency，即第1個自然頻率)及泛音(overtones，即第2、3、4、…個自然頻率)，當該泛音與基音具有整數比例關係時，即稱為「具有簡諧倍頻音(harmonics sound)」，可使音色較為諧和柔美(consonance)(請詳參「王栢村，蘇集銘，2006，吉他弦之振動與聲音特性探討，中華民國音響學會第十九屆學術研討會論文集，台南，論文編號：A7」論文)。另一方面，擊樂器可略分為膜鳴樂器及體鳴樂器，體鳴樂器略如：木琴(xylophone)、顫音琴(vibraphone)或鐵琴(metallophone)等。 In the symphony orchestra, there are usually stringed instruments and percussion instruments, such as pianos or guitars. For example, guitar strings, the natural frequency of the vibration mode includes the pitch. (fundamental frequency, the first natural frequency) and overtones (the 2nd, 3rd, 4th, ... natural frequencies). When the overtone has an integer proportional relationship with the pitch, it is called "having a simple harmonic frequency." Harmonic sound can make the sound more harmonious and consonance (please refer to "Wang Baicun, Su Jiming, 2006, vibration and sound characteristics of guitar strings, the 19th Symposium of the Republic of China Sound Society" Set, Tainan, paper number: A7" paper). On the other hand, the percussion instrument can be slightly divided into a film-sounding instrument and a body-sounding instrument, and the body-sounding instrument is slightly like: a xylophone, a vibraphone, or a metallophone.
以市面上之鐵琴為例，略可分為桌上鐵琴、箱型鐵琴、立奏鐵琴及行進時使用的鐘琴等，通常由數個長度不一的長條金屬板作為音板(鐵琴片)，音板的長度可決定主音階(基音)，惟該音板之基音與泛音間未具備簡諧倍頻特性，使得習用音板經過敲擊後發出的泛音與基音呈現不和諧狀態(dissonance)，仍需加以改良以符合演奏所需。 Take the iron hammer on the market as an example. It can be divided into a table harpsichord, a box-type xylophone, a stand-up harpsichord, and a carillon used for traveling. It is usually made up of several long metal plates of different lengths. Plate (iron hammer), the length of the soundboard can determine the main scale (pitch), but the pitch and the overtone of the soundboard do not have a simple harmonic frequency multiplier, which makes the overtone and pitch of the conventional soundboard after the tapping Dissonance still needs to be improved to meet the needs of the performance.
為改善上述情況，已有習知技術利用貝茲曲線(Bezier curve)特性設計鐵琴片，如：中華民國公開第201025287號「具和弦片之鐵琴構造」專利申請案。惟，鐵琴片外型與發音音階有關，當欲設計具有特定音階之鐵琴片時，因貝茲曲線之貝茲點需以試誤法(try and error)重複嘗試，方能找出可發出特定音階的鐵琴片外型，導致鐵琴片外型之調整與更改不易，且設計時間長致使製造成本不易降低，故而難以針對特殊演奏需求設計鐵琴片。 In order to improve the above situation, the prior art has used the Bezier curve characteristic to design a metal piano piece, such as the Patent Application No. 201025287 of the Republic of China, "The Iron Peg Structure with a Chord Piece". However, the shape of the hammer is related to the pronunciation scale. When you want to design a hammer with a specific scale, the Bezier point of the Bezier curve needs to be tried repeatedly by try and error. The shape of the iron sheet that emits a specific scale makes the adjustment and modification of the shape of the iron sheet difficult, and the design time is long, making the manufacturing cost difficult to reduce, so it is difficult to design the iron sheet for special performance requirements.
有鑑於此，有必要改善上述先前技術的缺點，研發出具簡諧倍頻音之音板的設計方法，以符合實際需求，提升其實用性。 In view of this, it is necessary to improve the shortcomings of the above prior art, and develop a design method of a soundboard with a simple harmonic frequency to meet practical needs and improve its practicability.
本發明係提供一種具簡諧倍頻音之音板的設計方法，可利用多簡諧波曲線設計該音板之外型，有效地簡化音板的設計過程。 The invention provides a design method of a soundboard with a simple harmonic frequency sound, which can be designed by using a multi-harmonic harmonic curve to effectively simplify the design process of the soundboard.
本發明係提供一種具有音板之擊樂器，可利用多簡諧波曲線外型決定音板經敲擊所產生的基音及簡諧倍頻音。 The invention provides a percussion instrument with a soundboard, which can determine the pitch and harmonic octave produced by the knocking of the soundboard by using the shape of the multi-harmonic harmonic curve.
本發明揭示一種具簡諧倍頻音之音板的設計方法，係由一電腦系統執行，包含下列步驟：產生一音板模型，該音板模型設有二曲緣及二直邊，各直邊分別連接該二曲緣，各曲緣分別沿二相互垂直之軸線呈鏡像對稱，各曲緣之對稱中心由一多簡諧波函數之一波峰向外延伸形成平滑曲線，該多簡諧波函數包含一偏位量、一第一簡諧波及一第二簡諧波；將該偏位量、該第一簡諧波之振幅、週期、該第二簡諧波之振幅、週期、相位及該二直邊之間距設為一設計變數組；設定該音板模型的敲擊中心點符合一振動限制條件；計算三目標頻率與該音板模型之三自然頻率的誤差平方和，利用一數值逼近法修正該三自然頻率，直到取得該誤差平方和之最小值，依據該修正後的三自然頻率修正該音板模型之設計變數組，依據該修正後的設計變數組修正該音板模型的形狀；及於該音板模型設置二組裝 孔，各組裝孔位於該二曲緣之間的軸線上。 The invention discloses a design method of a soundboard with a simple harmonic frequency sound, which is executed by a computer system and comprises the following steps: generating a soundboard model having two curved edges and two straight sides, each straight The two curved edges are respectively connected to each other, and each curved edge is mirror-symmetrical along two mutually perpendicular axes, and the symmetry center of each curved edge extends outward from a peak of a multi-harmonic harmonic function to form a smooth curve, and the multi-harmonic harmonic The function includes a bias amount, a first simple harmonic, and a second simple harmonic; the offset amount, the amplitude of the first simple harmonic, the period, the amplitude of the second simple harmonic, the period, and the phase And the distance between the two straight edges is set as a design variable array; setting a tap center point of the soundboard model conforms to a vibration limit condition; calculating a square sum error of the three target frequencies and the natural frequency of the soundboard model, using one The numerical approximation method corrects the three natural frequencies until the minimum value of the sum of the squares of the errors is obtained, and the design variable array of the soundboard model is corrected according to the corrected three natural frequencies, and the soundboard model is corrected according to the modified design variable array. shape And to the model provided two installation soundboard A hole, each of which is located on an axis between the two curved edges.
所述設計方法之多簡諧波函數為一雙簡諧波函數，係如下列方程式所示：
所述設計方法之振動限制條件為該音板模型的敲擊中心點之模態振型的絕對值大於一位移閥值，該位移閥值大於零。 The vibration limiting condition of the design method is that the absolute value of the mode shape of the tap center point of the soundboard model is greater than a displacement threshold, and the displacement threshold is greater than zero.
所述設計方法之誤差平方和的計算方式係如下列方程式所示：
所述設計方法之數值逼近法為牛頓法。 The numerical approximation method of the design method is the Newton method.
所述設計方法另包含由該電腦系統將該音板模型的形狀轉成一輸出檔案，用以加工一音板，使該音板具有簡諧倍頻音。 The design method further includes converting, by the computer system, the shape of the soundboard model into an output file for processing a soundboard such that the soundboard has a simple harmonic frequency.
本發明另揭示一種具有音板之擊樂器，包含：一架體；及數個音板，結合於該架體，各音板設有二曲緣及二直邊，各音板之直邊分別連接同一音板之二曲緣，各音板之曲緣分別沿二相互垂直之軸線呈鏡像對稱，各曲緣之對稱中心由一多簡諧波函數之一波峰向外延伸形成平滑曲線，該多簡諧波函數包含一偏位量、一第一簡諧波及一第二簡諧波，該第一、二簡諧波分別具有一振幅、一週期及一相位。 The invention further discloses a percussion instrument with a soundboard, comprising: a frame body; and a plurality of sound plates, combined with the frame body, each sound plate is provided with two curved edges and two straight edges, and the straight edges of the sound plates are respectively respectively The two curved edges of the same soundboard are connected, and the curved edges of the soundboards are mirror-symmetrical along two mutually perpendicular axes, and the symmetry center of each curved edge extends outward from a peak of a multi-harmonic harmonic function to form a smooth curve. The multi-mode harmonic function includes a bias amount, a first simple harmonic and a second simple harmonic, and the first and second simple harmonics respectively have an amplitude, a period and a phase.
所述擊樂器之多簡諧波函數為一雙簡諧波函數，係如下列方程式所示：
所述擊樂器之各音板之弧線段的平滑曲線不同。 The smooth curves of the arc segments of the soundboards of the percussion instrument are different.
所述擊樂器之音板之材質為金屬、合金、木質、竹材或塑膠。 The soundboard of the percussion instrument is made of metal, alloy, wood, bamboo or plastic.
上揭具簡諧倍頻音之音板的設計方法，主要由該電腦系統使該音板模型之弧線段形成上述簡諧波函數之曲線，另以數值逼近法自動修正該音板模型，據以加工一音板之外型，使該音板產生簡諧倍頻音；另可配合該音板模型之不同外型參數，使該音板發出不同音階，可用以製作具有音板之擊樂器。藉此，本案可以達到「自動修正外型」及「減少設計時間」等功效，可確實避免該習知利用貝茲曲線特性設計鐵琴片技術所致的「鐵琴片外型調整與更改不易」及「設計時間長致使製造成本不易降低」等問題。 The design method of the soundboard with a simple harmonic frequency is mainly obtained by the computer system, the arc segment of the soundboard model is formed into a curve of the simple harmonic function, and the soundboard model is automatically corrected by a numerical approximation method. To process a soundboard outside shape, the soundboard produces a simple harmonic frequency sound; and the different shape parameters of the soundboard model can be used to make the soundboard emit different scales, and can be used to make a soundboard with a soundboard. . In this way, the case can achieve the effects of "automatically correcting the appearance" and "reducing the design time". It can be avoided that the design of the iron sheet is not easy to use due to the design of the characteristics of the sheet metal. And "the problem of long design time, making manufacturing costs difficult to reduce".
〔本發明〕 〔this invention〕
1‧‧‧曲緣 1‧‧‧曲缘
11a,11b‧‧‧弧線段 11a, 11b‧‧‧ arc segments
2‧‧‧直邊 2‧‧‧ Straight edge
3‧‧‧組裝孔 3‧‧‧Assembled holes
Lx‧‧‧直邊間距 L x ‧‧‧ Straight edge spacing
Ly1‧‧‧第一波峰間距 L y1 ‧‧‧first peak spacing
Ly2‧‧‧第二波峰間距 L y2 ‧‧‧second peak spacing
M‧‧‧音板模型 M‧‧‧ soundboard model
MR‧‧‧有限元素模型 M R ‧‧‧ finite element model
P1‧‧‧第一波峰 P1‧‧‧ first peak
P2‧‧‧第二波峰 P2‧‧‧ second peak
X,Y‧‧‧軸線 X, Y‧‧ axis
Xc‧‧‧孔距 X c ‧‧‧ hole spacing
S1‧‧‧建模步驟 S1‧‧‧Modeling steps
S2‧‧‧修模步驟 S2‧‧‧Modification steps
S3‧‧‧加工步驟 S3‧‧‧Processing steps
y(x)‧‧‧雙簡諧波函數 y(x)‧‧‧double simple harmonic function
y0‧‧‧偏位量 y 0 ‧‧‧ partial amount
y1(x)‧‧‧第一簡諧波 y 1 (x)‧‧‧first simple harmonic
y2(x)‧‧‧第二簡諧波 y 2 (x)‧‧‧second simple harmonic
A1,λ1,φ1‧‧‧第一簡諧波之振幅、週期、相位 A 1 , λ 1 , φ 1 ‧‧‧ amplitude, period, phase of the first simple harmonic
A2,λ2,φ2‧‧‧第二簡諧波之振幅、週期、相位 A 2 , λ 2 , φ 2 ‧‧‧ amplitude, period, phase of the second simple harmonic
第1圖係本發明之具簡諧倍頻音之音板的設計方法實施例的流程示意圖。 Fig. 1 is a flow chart showing an embodiment of a method for designing a soundboard with a simple harmonic frequency sound according to the present invention.
第2圖係本發明之具簡諧倍頻音之音板的設計方法實施例之音板模型俯視圖。 Fig. 2 is a plan view showing a soundboard model of a method for designing a soundboard with a simple harmonic frequency sound according to the present invention.
第3圖係本發明之具簡諧倍頻音之音板的設計方法實施例之音板模型 的右上部分之有限元素模型示意圖。 Figure 3 is a soundboard model of a design method of a soundboard with a simple harmonic frequency sound according to the present invention. Schematic diagram of the finite element model in the upper right part.
第4圖係本發明之具簡諧倍頻音之音板的設計方法實施例之雙簡諧波函數示意圖。 Fig. 4 is a schematic diagram showing the double simple harmonic function of the embodiment of the design method of the soundboard with simple harmonic frequency sound of the present invention.
第5圖係雙簡諧波曲線與貝茲曲線的比對示意圖。 Figure 5 is a schematic diagram of the comparison between the double simple harmonic curve and the Bezier curve.
第6圖係利用貝茲曲線作為音板模型的弧線段之音板的頻率響應圖。 Fig. 6 is a frequency response diagram of a soundboard using a Bezier curve as an arc segment of a soundboard model.
第7圖係利用雙簡諧波曲線作為音板模型的弧線段之音板的頻率響應圖。 Figure 7 is a frequency response diagram of a soundboard using a double simple harmonic curve as an arc segment of a soundboard model.
為讓本發明之上述及其他目的、特徵及優點能更明顯易懂，下文特舉本發明之較佳實施例，並配合所附圖式，作詳細說明如下：本發明全文所述之「鏡像對稱」(mirror symmetry)，係指一軸線通過一物體，該物體位於該軸線兩側之形狀呈對稱(symmetry)，使該軸線一側之物體形狀如同另一側之物體形狀的鏡像(mirror image)，係本發明所屬技術領域中具有通常知識者可以理解。 The above and other objects, features and advantages of the present invention will become more <RTIgt; "mirror symmetry" means that an axis passes through an object, and the shape of the object on both sides of the axis is symmetry, so that the object on one side of the axis is shaped like a mirror image of the object shape on the other side (mirror image) It is understood by those of ordinary skill in the art to which the present invention pertains.
請參閱第1圖所示，其係揭示本發明之具簡諧倍頻音之音板的設計方法實施例的流程示意圖。其中，該設計方法實施例可利用一電腦系統連接一資料庫作為執行架構，並由該電腦系統執行一設計作業軟體，如：專用應用程式或以有限元素分析軟體(如：ANSYS等)配合其內部函式庫等。在此實施例中，該設計作業軟體係以ANSYS有限元素分析軟體配合其內部函式庫作為實施態樣說明，惟不以此為限。該設計方法實施例包含一建模步驟S1及一修模步驟S2。 Please refer to FIG. 1 , which is a flow chart showing an embodiment of a method for designing a soundboard with a simple harmonic frequency sound according to the present invention. The design method embodiment may use a computer system to connect to a database as an execution architecture, and execute a design operation software by the computer system, such as a dedicated application or a finite element analysis software (such as ANSYS). Internal libraries, etc. In this embodiment, the design work soft system uses the ANSYS finite element analysis software with its internal function library as an implementation example, but is not limited thereto. The design method embodiment includes a modeling step S1 and a modification step S2.
該建模步驟S1，係由一電腦系統產生一音板模型，該音板模型設有二曲緣及二直邊，各直邊分別連接該二曲緣，各曲緣分別沿二相互垂直之軸線呈鏡像對稱，各曲緣之對稱中心由一多簡諧波函數之一波峰向外延伸形成平滑曲線，該多簡諧波函數包含一偏位量、一第一簡諧波及 一第二簡諧波，該第一、二簡諧波分別具有一振幅、一週期及一相位。另，由該電腦系統將該偏位量、該第一簡諧波之振幅、週期、該第二簡諧波之振幅、週期、相位及該二直邊之間距設為一設計變數組。另，由該電腦系統設定該音板模型的敲擊中心點符合一振動限制條件。 The modeling step S1 is to generate a soundboard model by a computer system. The soundboard model is provided with two curved edges and two straight edges, and the straight edges are respectively connected to the two curved edges, and the curved edges are respectively perpendicular to each other. The axis is mirror-symmetrical, and the center of symmetry of each curved edge extends outward from a peak of a multi-harmonic harmonic function to form a smooth curve, and the multi-harmonic harmonic function includes a bias amount, a first harmonic and A second simple harmonic, the first and second simple harmonics respectively have an amplitude, a period and a phase. In addition, the computer system sets the offset amount, the amplitude of the first simple harmonic, the period, the amplitude of the second simple harmonic, the period, the phase, and the distance between the two straight sides as a design variable array. In addition, the tapping center point of the soundboard model set by the computer system conforms to a vibration limiting condition.
在此實施例中，該多簡諧波函數係以雙簡諧波函數作為實施態樣說明，惟不以此為限；請一併參閱第2圖所示，該電腦系統可先產生該音板模型M，該音板模型M係如具有一定厚度之平板，該音板模型M具有二曲緣1及二直邊2，各直邊2分別連接該二曲緣1，各曲緣1分別沿二相互垂直之軸線X、Y呈鏡像對稱，各曲緣1具有呈鏡像對稱的二弧線段11a、11b，各弧線段11a、11b可由該多簡諧波函數之波形自該第一波峰P1(即該曲緣1之對稱中心)延伸形成平滑曲線，如：自該第一波峰P1延伸至第二波峰P2所形成之平滑曲線，惟不以此為限；該多簡諧波函數可選為一雙簡諧波函數或具有至少二簡諧波之函數，該雙簡諧波函數係如下式(1)所示：
舉例而言，以ANSYS有限元素模型分析時，可假設材料圍等向且為均值，並符合虎克定律之假設。若一材料之楊氏係數(Young’s Modulus)E=199.3GPa、蒲松比(Poisson’s ratio)ν=0.34，密度(density)ρ=7862.94(kg/m3)，則可建構一具簡諧倍頻音之音板模型的四等分之右上部分的有限元素模型MR(如第3圖所示)，元素採用線性四邊形殼元素(Shell 63)建構，分割採取Sweep mesh方式，共有1393個元素與4394個節點，邊界採用自由邊界(free-free)，不設定邊界條件，負荷條件於模態分析不須設定負荷，惟不以此為限。 For example, when analyzed with the ANSYS finite element model, the material can be assumed to be equi-directional and mean, and in accordance with Hooke's law. If a material's Young's Modulus E=199.3GPa, Poisson's ratio ν=0.34, density ρ=7862.94(kg/m 3 ), a simple harmonic octave can be constructed. The finite element model M R of the upper right part of the quarter of the soundboard model (as shown in Fig. 3), the element is constructed with a linear quadrilateral shell element (Shell 63), and the split adopts the Sweep mesh method, with a total of 1393 elements and 4394 For nodes, the boundary is free-free. The boundary conditions are not set. The load condition does not need to set the load for modal analysis, but not limited to this.
另，該電腦系統可將該偏位量y0、該第一簡諧波y1(x)之振幅A1、週期λ1、該第二簡諧波y2(x)之振幅A2、週期λ2、相位φ2及該二直邊之間距Lx設為一設計變數組D，如下式(2)所示：D=F(λ1,λ2,A 1,A 2,y 0,φ2,L x ) (2)又，該電腦系統可使該音板模型的敲擊中心點之模態振型(mode shape)的絕對值大於一位移閥值(threshold)，作為該振動限制條件，例如：若設該音板模型之中心點作為一敲擊點，則以ANSYS進行有限元素模型分析時，對應的模態振型中心點(敲擊點)必須沒有節線(即有位移，節線所在之處表示位移為零)，例如：該音板模型之中心點振動而產生位移時，該模態振型會因相對二方向的振動而產生正、負值，故，可取該模態振型的絕對值大於該位移閥值，以令音板在中心點敲擊而發出聲響，其係熟知有限元素模型分析技術人員可以理解，在此容不贅述。因此，可設定該中心點之位移閥值大於零，如：0.1~0.3，令音板在敲擊後產生特定音階之簡諧倍頻音。在此實施例中，該位移閥值可設為0.2，惟不以此為限。 In addition, the computer system can set the offset amount y 0 , the amplitude A 1 of the first simple harmonic y 1 (x), the period λ 1 , and the amplitude A 2 of the second simple harmonic y 2 (x), The period λ 2 , the phase φ 2 and the distance L x between the two straight sides are set as a design variable array D, as shown in the following formula (2): D = F (λ 1 , λ 2 , A 1 , A 2 , y 0 , φ 2 , L x ) (2) Further, the computer system can make the absolute value of the mode shape of the knocking center point of the soundboard model larger than a displacement threshold (threshold) as the vibration Restrictive conditions, for example, if the center point of the soundboard model is used as a tapping point, when the finite element model analysis is performed with ANSYS, the corresponding mode shape center point (tapping point) must have no pitch line (ie, Displacement, where the pitch line indicates displacement is zero. For example, when the center point of the soundboard model vibrates and generates displacement, the mode shape will produce positive and negative values due to vibration in the opposite directions, so it is preferable. The absolute value of the mode shape is greater than the displacement threshold, so that the soundboard is struck at the center point to make a sound, which is well known to the finite element model analysis technicians can understand Do not repeat them. Therefore, the displacement threshold of the center point can be set to be greater than zero, such as: 0.1~0.3, so that the soundboard produces a simple harmonic octave of a specific scale after tapping. In this embodiment, the displacement threshold can be set to 0.2, but not limited thereto.
另，本發明所屬技術領域中具有通常知識者參酌材料力學、靜力學、動力學等教科書，可由該音板模型之設計變數組計算出該音板模型之三自然頻率(第1,2,3個自然頻率，或稱基準頻率、2倍頻率、3倍頻率)，該自然頻率之計算方式可由有限元素分析軟體(如：ANSYS等)分 析得出，亦可由該音板模型之三自然頻率計算出該音板模型之設計變數組，係所屬技術領域中具有通常知識者可以理解，在此容不贅述。接著，進行該修模步驟S2。 In addition, in the technical field to which the present invention pertains, a person having ordinary knowledge can refer to a textbook of material mechanics, statics, dynamics, etc., and the natural frequency of the soundboard model can be calculated from the design variable array of the soundboard model (1, 2, 3). Natural frequency, or reference frequency, 2 times frequency, 3 times frequency), the natural frequency can be calculated by finite element analysis software (such as: ANSYS, etc.) It can be understood that the design variable array of the soundboard model can also be calculated from the natural frequency of the soundboard model, which can be understood by those having ordinary knowledge in the technical field, and will not be described here. Next, the mold modification step S2 is performed.
該修模步驟S2，係由該電腦系統計算三目標頻率與該音板模型之三自然頻率的誤差平方和，利用一數值逼近法修正該三自然頻率，直到取得該誤差平方和之最小值，依據該修正後的三自然頻率修正該音板模型之設計變數組，依據該修正後的設計變數組修正該音板模型的形狀。另於該音板模型設置二組裝孔，各組裝孔位於該二曲緣之間的軸線上，使該音板模型可作為後續執行音板加工作業之憑據。在此實施例中，如第2圖所示，該電腦系統可先計算該音板模型M之三自然頻率與三目標頻率之誤差平方和，作為使該設計變數組最佳化的一目標函數(objective function)，如下式(3)所示：
接著，該電腦系統可利用習知數值逼近法(approximation method)修正該三目標頻率，如：牛頓法(Newton’s method，又稱Newton-Raphson’s method)或有限元素分析軟體之最佳化函式(如：ANSYS之RANDOM、SUBP、SWEEP)等，直到取得該誤差平方和之最小值，以確認該三自然頻率與該三目標頻率最接近，再依據該修正後的三自然頻率(下稱分析頻率)將該音板模型之設計變數組D修正為D’(即修正後的設計變數組)，依據該修正後的設計變數組D’修正該音板模型之形狀，使該 修正後之形狀可經敲擊而發出簡諧倍頻音，該形狀可由有限元素分析軟體(如：ANSYS等)分析得出，係所屬技術領域中具有通常知識者可以理解，在此容不贅述。 Then, the computer system can correct the three target frequencies by using a conventional numerical approximation method, such as Newton's method (also known as Newton-Raphson's method) or optimization function of finite element analysis software (such as : RANDOM, SUBP, SWEEP of ANSYS, etc. until the minimum of the sum of the squares of the errors is obtained to confirm that the three natural frequencies are closest to the three target frequencies, and then based on the corrected three natural frequencies (hereinafter referred to as the analysis frequency) Correcting the design variable array D of the soundboard model to D' (ie, the modified design variable array), and correcting the shape of the soundboard model according to the modified design variable array D' The modified shape can be tapped to emit a simple harmonic octave. The shape can be analyzed by a finite element analysis software (such as ANSYS, etc.), which can be understood by those having ordinary knowledge in the technical field, and will not be described herein. .
請再參閱第2圖所示，該電腦系統可於該音板模型M設置二組裝孔3，各組裝孔3可位於該二曲緣1之間的軸線X上，該組裝孔3至另一軸線Y之孔距可為Xc，該音板模型M於該軸線X兩側之第一波峰間距可表示為Ly1，該音板模型M於該軸線X兩側之第二波峰間距可表示為Ly2，該音板模型M之直邊間距(即沿軸線X方向延伸之長度)表示為Lx，以調整該音板模型M加工成音板後敲擊發出的音頻。在此實施例中，僅以該第一波峰間距Ly1係小於該第二波峰間距Ly2作為實施態樣說明，惟該第一波峰間距Ly1亦可大於或等於該第二波峰間距Ly2，不應以此為限。 Referring to FIG. 2 again, the computer system can provide two assembly holes 3 in the soundboard model M, and each assembly hole 3 can be located on the axis X between the two curved edges 1 and the assembly holes 3 to another The pitch of the axis Y may be X c , and the first peak spacing of the soundboard model M on both sides of the axis X may be represented as L y1 , and the second peak spacing of the soundboard model M on both sides of the axis X may represent For L y2 , the straight edge spacing of the soundboard model M (i.e., the length extending along the axis X direction) is expressed as L x to adjust the sound produced by the soundboard model M after being processed into a soundboard and tapped. In this embodiment, only the first peak spacing L y1 is smaller than the second peak spacing L y2 as an embodiment, but the first peak spacing L y1 may also be greater than or equal to the second peak spacing L y2 . It should not be limited to this.
請再參閱第1圖所示，該具簡諧倍頻音之音板的設計方法實施例還可以包含一加工步驟S3，係由該電腦系統將該音板模型修正後的形狀轉成一輸出檔案，用以加工一音板，使該音板具有簡諧倍頻音。在此實施例中，該電腦系統可將該音板模型修正後的形狀匯出而形成該輸出檔案，供加工廠商據以加工(如：採用雷射加工技術等)該音板，該音板之材質可依實際需求選為金屬、合金、木質、竹材或塑膠等，惟不以此為限，使該音板具有特定音頻之簡諧倍頻音，以便製作鐵琴等具有音板之擊樂器。 Referring to FIG. 1 again, the embodiment of the method for designing a soundboard with a simple harmonic frequency can further include a processing step S3, wherein the modified shape of the soundboard model is converted into an output by the computer system. A file for processing a soundboard so that the soundboard has a simple harmonic frequency. In this embodiment, the computer system can reproduce the corrected shape of the soundboard model to form the output file, and the processing manufacturer can process the soundboard according to the processing (eg, using laser processing technology, etc.). The material can be selected as metal, alloy, wood, bamboo or plastic according to actual needs, but not limited to this, so that the soundboard has a simple harmonic octave of specific audio, so as to make a soundboard attack such as a hammer Musical instrument.
舉例而言，該具有音板之擊樂器具有一架體及數個音板(圖未繪示)，該架體可為習知擊樂器之架體，用以平置該數個音板，該數個音板結合於該架體；各音板設有二曲緣及二直邊，各音板之直邊分別連接同一音板之二曲緣，各音板之曲緣分別沿二相互垂直之軸線呈鏡像對稱，各曲緣之對稱中心由一多簡諧波函數之一波峰向外延伸形成平滑曲線，該多簡諧波函數包含一偏位量、一第一簡諧波及一第二簡諧波，該第一、二簡諧波分別具有一振幅、一週期及一相位。該偏位量、第一簡諧波及第二簡 諧波已說明如前，在此容不贅述。又，各音板之弧線段的平滑曲線可為不同，用以產生具有不同基音之簡諧倍頻音。 For example, the percussion instrument with a soundboard has a body and a plurality of soundboards (not shown), and the frame body can be a frame of a conventional percussion instrument for arranging the plurality of soundboards. The soundboards are combined with the frame body; each soundboard is provided with two curved edges and two straight edges, and the straight edges of the soundboards are respectively connected to the two curved edges of the same soundboard, and the curved edges of the soundboards are respectively separated by two The vertical axis is mirror symmetrical, and the symmetry center of each curved edge extends outward from a peak of a multi-harmonic harmonic function to form a smooth curve. The multi-transformed harmonic function includes a bias amount, a first simple harmonic and a The second simple harmonic, the first and second simple harmonics respectively have an amplitude, a period and a phase. The amount of deviation, the first simple harmonic and the second simple The harmonics have been explained as before and will not be described here. Moreover, the smooth curves of the arc segments of the respective soundboards can be different to generate a simple harmonic octave with different pitches.
另，當欲測試上述音板之音頻時，可利用敲擊鎚敲擊受測之音板，待以收音裝置(如：麥克風)收集音頻後，由電腦配合聲音量測程式產生分析結果，其係所屬技術領域中具有通常知識者可以理解，在此容不贅述。 In addition, when the audio of the above soundboard is to be tested, the soundboard can be tapped with a hammer, and after the audio is collected by a sounding device (such as a microphone), the computer and the sound measurement program generate an analysis result. It will be understood by those of ordinary skill in the art, and will not be described herein.
由下列表一可知常見的音階頻率，以A4音階為例，利用下列音階頻率方程式：b=a×2n/1200，其中a為已知的音階頻率，b為欲計算的音階頻率，n為間隔的音分(cent)，n可設為6 cents(人類對單音音階可分辨的音分範圍)，即可求得一誤差值，再以(a+b)、(a-b)分別作為已知音階頻率的上、下限數值。故，因A4音階所對應的音階頻率為440赫茲(Hz)，可求得容許的頻率範圍上、下限數值分別為441.53Hz與438.48Hz，A4音階頻率容許範圍約為±1.53(Hz)。由此可知，音階頻率越高，音階頻率容許範圍越大，更進一步發現，各個音階頻率容許範圍雖不同，但卻都與各音階間具有一定的百分比關係，如：A4音階之百分比為1.53Hz/440Hz×100%0.347%。因此，後續定義各音階頻率之誤差百分比在±0.347%之範圍內，即屬符合音準之範圍，後續僅以C6音階為例作為實施態樣說明，惟不以此為限。 The common scale frequency can be seen from the following list. Taking the A4 scale as an example, the following scale frequency equation is used: b=a×2 n/1200 , where a is the known scale frequency, b is the scale frequency to be calculated, and n is The interval of the cents (cent), n can be set to 6 cents (human to the resolution range of the monophonic scale), you can get an error value, and then (a + b), (ab) as The upper and lower limits of the known scale frequency. Therefore, since the scale frequency corresponding to the A4 scale is 440 Hz, the upper and lower limits of the allowable frequency range are 441.53 Hz and 438.48 Hz, respectively, and the allowable range of the A4 scale frequency is about ± 1.53 (Hz). It can be seen that the higher the scale frequency, the larger the allowable range of the scale frequency. It is further found that although the allowable range of each scale frequency is different, it has a certain percentage relationship with each scale, for example, the percentage of the A4 scale is 1.53 Hz. /440Hz×100% 0.347%. Therefore, the error percentage of the subsequent definition of each scale frequency is within the range of ±0.347%, which is the range of the pitch. The following is only the case of the C6 scale as an example, but not limited to this.
表一 音階頻率表
請參閱第4圖所示，其係本發明上述實施例之雙簡諧波曲線示意圖。其中，以雙簡諧波函數y(x)之波形自該第一波峰P1延伸至第二波峰P2之平滑曲線為例，該雙簡諧波函數y(x)之曲線可包含一偏位量y0、第一簡諧波y1(x)及第二簡諧波y2(x)，依據該雙簡諧波函數y(x)設計音板之曲緣外型(弧線段)，可令該音板發出C6音階(基音)，惟不以此為限。 Please refer to FIG. 4, which is a schematic diagram of a double simple harmonic curve of the above embodiment of the present invention. For example, the smooth curve of the waveform of the double simple harmonic function y(x) extending from the first peak P1 to the second peak P2 is taken as an example, and the curve of the double simple harmonic function y(x) may include a bias amount. y 0 , the first simple harmonic y 1 (x) and the second simple harmonic y 2 (x), according to the double simple harmonic function y(x), the curved outer shape (arc segment) of the soundboard is designed, Let the soundboard emit a C6 scale (pitch), but not limited to this.
請參閱第5圖所示，其係本發明上述實施例之雙簡諧波曲線與貝茲曲線的比對示意圖。其中，以MATALAB軟體描繪可發出C6音階之弧線段的雙簡諧波曲線、貝茲曲線分別表示為y(x)、Y，並可計算該雙簡諧波曲線、貝茲曲線之平均誤差(%)Erravg約為1.7057，最大誤差(%)Errmx約為6.921，最小誤差(%)Errmn約為-8.7031，惟不以此為限。 Please refer to FIG. 5, which is a schematic diagram of the comparison between the double simple harmonic curve and the Bezier curve of the above embodiment of the present invention. Among them, the double simple harmonic curve and the Bezier curve which can be used to describe the arc segment of C6 scale by MATALAB software are respectively represented as y(x), Y, and the average error of the double simple harmonic curve and the Bezier curve can be calculated ( %) Err avg is about 1.7057, maximum error (%) Err mx is about 6.921, and minimum error (%) Err mn is about -8.7031, but not limited to this.
又，請參閱第6及7圖所示，其分別為利用貝茲曲線、雙簡 諧波曲線作為弧線段之音板的頻率響應圖。其中，若利用ANSYS軟體進行簡諧分析，由第6圖可知，當以貝茲曲線作為弧線段之音板設定輸入力時，在1046.45、2093.05、3139.43赫茲(Hz)音階之振幅將同時產生峰值，可令該音板發出C6音階，惟不以此為限。由第7圖可知，當以雙簡諧波曲線作為弧線段之音板設定輸入力時，在1046.15、2092.37、3138.84(Hz)音階之振幅將同時產生峰值，可令該音板發出C6音階，惟不以此為限。 Also, please refer to Figures 6 and 7 for the use of Bezier curves and double simplicity. The harmonic curve is used as the frequency response diagram of the soundboard of the arc segment. Among them, if the ANSYS software is used for harmonic analysis, as shown in Fig. 6, when the input force is set by the soundboard with the Bezier curve as the arc segment, the amplitudes of the scales of 1046.45, 2093.05, and 3139.43 Hertz (Hz) will simultaneously generate peaks. This allows the soundboard to emit a C6 scale, but not limited to this. It can be seen from Fig. 7 that when the input force is set by the double simple harmonic curve as the soundboard of the arc segment, the amplitudes of the scales of 1046.15, 2092.37, and 3138.84 (Hz) will simultaneously generate peaks, which can cause the soundboard to emit C6 scales. However, it is not limited to this.
由此可知，雙簡諧波曲線y(x)與貝茲曲線Y皆可用於設計音板之外型。值得注意的是，音板之外型雖可利用貝茲曲線Y設計，惟貝茲曲線Y須以數個貝茲點yi共同決定其曲線變化，該曲線語音階之變化難以預測，且複雜度隨貝茲點的數量增加而提高，故，設計不同音階之音板外型時，須不斷的嘗試改變各貝茲點，直到貝茲曲線形成的音板外型符合需求為止，難以估計及縮短設計時間。相較之下，本發明上述利用雙簡諧波函數設計音板外型之實施例，可利用上述數值逼近法(如：牛頓法或有限元素分析軟體之最佳化函式)自動修正該音板之外型，而達到「自動修正外型」及「減少設計時間」等功效。 It can be seen that both the simple harmonic curve y(x) and the Bezier curve Y can be used to design the soundboard. It is worth noting that the shape of the soundboard can be designed by using the Bezier curve Y. However, the Bezier curve Y must determine the curve change with several Bezier points. The change of the phonetic level of the curve is difficult to predict and the complexity is complicated. As the number of Bezier points increases, so when designing the soundboard shape of different scales, it is necessary to constantly try to change the Bezier points until the shape of the soundboard formed by the Bezier curve meets the demand, which is difficult to estimate and shorten. Design time. In contrast, the above embodiment of the present invention for designing a soundboard appearance using a double simple harmonic function can automatically correct the sound by using the above numerical approximation method (for example, a Newton method or an optimization function of a finite element analysis software). The board is externally designed to achieve the functions of "automatically correcting the appearance" and "reducing the design time".
另，舉例說明本發明上述實施例針對不同音階之音板外型參數，請一併參閱第2圖所示。下列表二揭示音階F4~F7的直邊間距Lx、第一波峰間距Ly1、第二波峰間距Ly2、孔距Xc、雙簡諧波曲線之偏位量y0、第一簡諧波之振幅A1、週期λ1、第二簡諧波之振幅A2、週期λ2及相位φ2等參數，相位φ2之單位為弳(rad)，其餘參數之單位為公尺(m)。另，下列表三則為針對表二之音板外型參數作音階分析之結果，該音階分析可利用有限元素分析軟體進行，其係所屬技術領域中具有通常知識者可以理解，在此容不贅述。由表二及表三可知，本發明上述實施例確實可設計不同音階之音板，且該音板確實可發出簡諧倍頻音。 In addition, for example, the above embodiments of the present invention are directed to the soundboard appearance parameters of different scales, please refer to FIG. 2 together. Table 2 below reveals the straight edge spacing L x of the scales F4~F7, the first peak spacing L y1 , the second peak spacing L y2 , the hole spacing X c , the offset y 0 of the double simple harmonic curve, the first harmonic The amplitude of the wave A 1 , the period λ 1 , the amplitude A 2 of the second harmonic, the period λ 2 and the phase φ 2 , etc., the unit of the phase φ 2 is 弪 (rad), and the other parameters are in meters (m) ). In addition, the following three lists are the results of the scale analysis for the soundboard appearance parameters of Table 2. The scale analysis can be performed by the finite element analysis software, which can be understood by those having ordinary knowledge in the technical field. Narration. It can be seen from Table 2 and Table 3 that the above embodiments of the present invention can indeed design soundboards of different scales, and the soundboard can indeed emit simple harmonic octaves.
藉由前揭之技術手段，本發明之具簡諧倍頻音之音板的設計方法實施例的主要特點列舉如下：首先，由該電腦系統產生該音板模型，該音板模型設有二曲緣及二直邊，各直邊分別連接該二曲緣，各曲緣分別沿二相互垂直之軸線呈鏡像對稱，各曲緣具有呈鏡像對稱的二弧線段，各弧線段由一多簡諧波函數之平滑曲線所形成，該多簡諧波函數包含一偏位量、一第一簡諧波及一第二簡諧波，該第一、二簡諧波分別具有一振幅、一週期及一相位。另，由該電腦系統將該偏位量、該第一簡諧波之振幅、週期、該第二簡諧波之振幅、週期、相位及該二直邊之間距設為該設計變數組。另，由該電腦系統設定該音板模型的敲擊中心點符合該振動限制條件(如：該音板模型的敲擊中心點之模態振型的絕對值大於位移閥值)。之後，由該電腦系統計算三目標頻率與該音板模型之三自然頻率的誤差平方和，利用該數值逼近法修正該三自然頻率，直到取得該誤差平方和之最小值，依據該修正後的三自然頻率修正該音板模型之設計變數組，依據該修正後的設計變數組修正該音板模型的形狀。另於該音板模型設置二組裝孔，各組裝孔位於該二曲緣之間的軸線上。 The main features of the design method of the soundboard with simple harmonic frequency sound according to the technical means disclosed in the foregoing are listed as follows: First, the soundboard model is generated by the computer system, and the soundboard model is provided with two The curved edge and the two straight sides are respectively connected to the two curved edges, and each curved edge is mirror-symmetrical along two mutually perpendicular axes, each curved edge has a two-arc line segment with mirror symmetry, and each arc segment is composed of a plurality of curved lines. Forming a smooth curve of a harmonic function, the multi-mode harmonic function includes a bias amount, a first simple harmonic and a second simple harmonic, wherein the first and second simple harmonics respectively have an amplitude and a period And one phase. Further, the computer system sets the offset amount, the amplitude of the first simple harmonic, the period, the amplitude of the second simple harmonic, the period, the phase, and the distance between the two straight sides as the design variable array. In addition, the tapping center point of the soundboard model set by the computer system conforms to the vibration limiting condition (eg, the absolute value of the mode shape of the tapping center point of the soundboard model is greater than the displacement threshold). Thereafter, the computer system calculates a sum of squared errors of the three target frequencies and the three natural frequencies of the soundboard model, and corrects the three natural frequencies by using the numerical approximation method until the minimum value of the squared sum of the errors is obtained, according to the corrected The natural frequency corrects the design variable array of the soundboard model, and corrects the shape of the soundboard model according to the modified design variable array. In addition, the soundboard model is provided with two assembly holes, and each assembly hole is located on an axis between the two curved edges.
藉此，可由該音板模型據以加工一平板(如：金屬、合金、木質、竹材或塑膠等平板)成為一音板，使該音板之弧線段形成上述簡諧波函數之曲線，另可配合上述不同直邊間距、第一波峰間距、第二波峰間距、孔距，使該音板具有特定音階之簡諧倍頻音，更可進一步利用該音板組成各式擊樂器(如：上述具有音板之擊樂器或鐵琴等)，反觀習知鐵琴之音板無法產生簡諧倍頻音，本案具有功效上之增進。 Thereby, the soundboard model can be used to process a flat plate (such as a metal, alloy, wood, bamboo or plastic flat plate) into a soundboard, so that the arc segment of the soundboard forms the curve of the simple harmonic function, and The soundboard can be combined with the different peak spacing, the first peak spacing, the second peak spacing, and the pitch to make the soundboard have a simple harmonic octave of a specific scale, and the soundboard can be further utilized to form various types of musical instruments (eg: The above-mentioned percussion instrument or iron hammer with a soundboard, etc., in contrast, the soundboard of the conventional harpsichord cannot produce a simple harmonic octave, and the present invention has an effect enhancement.
承上，本發明之具簡諧倍頻音之音板的設計方法實施例，可利用電腦系統以數值逼近法自動修正該音板模型，據以加工一音板之外型，不同音板之部份外型規格可設計為一致，如：長度(即直邊間距)或寬度(即第一波峰間距與第二波峰間距中較大者)，以符合演奏者之使用需 求，本案上述實施例可以達到「自動修正外型」及「減少設計時間」等功效，可確實避免習知技術利用貝茲曲線特性設計鐵琴片所致「鐵琴片外型調整與更改不易」及「設計時間長致使製造成本不易降低」等問題。 According to the embodiment of the design method of the soundboard with simple harmonic frequency sound of the present invention, the soundboard model can be automatically corrected by a numerical approximation method by using a computer system, thereby processing a soundboard shape and different soundboards. Some exterior specifications can be designed to be consistent, such as: length (ie straight edge spacing) or width (ie the larger of the first peak spacing and the second peak spacing) to meet the player's needs In view of the above, the above embodiments can achieve the functions of "automatically correcting the appearance" and "reducing the design time", and can avoid the conventional technique of designing the iron sheet by using the characteristics of the Bezier curve. And "the problem of long design time, making manufacturing costs difficult to reduce".
雖然本發明已利用上述較佳實施例揭示，然其並非用以限定本發明，任何熟習此技藝者在不脫離本發明之精神和範圍之內，相對上述實施例進行各種更動與修改仍屬本發明所保護之技術範疇，因此本發明之保護範圍當視後附之申請專利範圍所界定者為準。 While the invention has been described in connection with the preferred embodiments described above, it is not intended to limit the scope of the invention. The technical scope of the invention is protected, and therefore the scope of the invention is defined by the scope of the appended claims.
S1‧‧‧建模步驟 S1‧‧‧Modeling steps
S2‧‧‧修模步驟 S2‧‧‧Modification steps
S3‧‧‧加工步驟 S3‧‧‧Processing steps
Claims (10)
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| TW103128869A TWI530939B (en) | 2014-08-21 | 2014-08-21 | Method for design a board with harmonics sound and percussion instruments with boards |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| TW103128869A TWI530939B (en) | 2014-08-21 | 2014-08-21 | Method for design a board with harmonics sound and percussion instruments with boards |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| TW201608559A true TW201608559A (en) | 2016-03-01 |
| TWI530939B TWI530939B (en) | 2016-04-21 |
Family
ID=56084817
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| TW103128869A TWI530939B (en) | 2014-08-21 | 2014-08-21 | Method for design a board with harmonics sound and percussion instruments with boards |
Country Status (1)
| Country | Link |
|---|---|
| TW (1) | TWI530939B (en) |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| TWI756780B (en) * | 2020-08-13 | 2022-03-01 | 國立屏東科技大學 | Frame of a vehicle and designing method thereof |
| TWI762856B (en) * | 2020-01-10 | 2022-05-01 | 國立屏東科技大學 | Bell plate with harmonics sound, percussion instrument with the bell plate, and design method of the bell plate |
Families Citing this family (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| TWI630601B (en) * | 2017-06-26 | 2018-07-21 | 國立屏東科技大學 | Semi-circular pipe with simple harmonic overtones, method for designing the semi-circular pipe, and percussion instrument with the semi-circular pipe |
| TWI620169B (en) * | 2017-12-11 | 2018-04-01 | 國立屏東科技大學 | Method for designing a bronze clock with harmonics sound |
-
2014
- 2014-08-21 TW TW103128869A patent/TWI530939B/en not_active IP Right Cessation
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| TWI762856B (en) * | 2020-01-10 | 2022-05-01 | 國立屏東科技大學 | Bell plate with harmonics sound, percussion instrument with the bell plate, and design method of the bell plate |
| TWI756780B (en) * | 2020-08-13 | 2022-03-01 | 國立屏東科技大學 | Frame of a vehicle and designing method thereof |
Also Published As
| Publication number | Publication date |
|---|---|
| TWI530939B (en) | 2016-04-21 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| TWI530939B (en) | Method for design a board with harmonics sound and percussion instruments with boards | |
| US20080000342A1 (en) | Soundboard for Acoustic Guitar | |
| Meurisse et al. | Experimental demonstration of the modification of the resonances of a simplified self-sustained wind instrument through modal active control | |
| Chauhan et al. | Vibro-acoustic modeling, numerical and experimental study of the resonator and its contribution to the timbre of Sarasvati veena, a South Indian stringed instrument | |
| TWI630601B (en) | Semi-circular pipe with simple harmonic overtones, method for designing the semi-circular pipe, and percussion instrument with the semi-circular pipe | |
| US20140033897A1 (en) | Musical keyboard in the form of a two-dimensional matrix | |
| TWI620169B (en) | Method for designing a bronze clock with harmonics sound | |
| TWI550597B (en) | Chimes with harmonics sound, a method for design the chimes, and percussion instruments with chimes | |
| Dalmont et al. | Piano strings with reduced inharmonicity | |
| TWI530938B (en) | Pipes with harmonics sound, a method for design the pipes and percussion instruments with pipes | |
| TWI762856B (en) | Bell plate with harmonics sound, percussion instrument with the bell plate, and design method of the bell plate | |
| TWI409801B (en) | Manufacturing method for a metalophone with a chord plate and a chord plate of metalophones | |
| TWI550596B (en) | Cymbals with harmonics sound, a method for design the cymbals, and percussion instruments with cymbals | |
| Laukkanen et al. | Acoustical effect of progressive undercutting of percussive aluminum bars | |
| TWI515718B (en) | Chord plate structure of metalophones | |
| McNeil et al. | Vibrational frequencies and tuning of the African mbira | |
| Beaton | Computational design and simulation of idiophone bars | |
| JP2018092009A (en) | Fret design method and fret string instrument | |
| Thomas et al. | Tonal optimization of bells utilizing evolutionary shape optimization | |
| Kirkland | Topographical optimization of structures for use in musical instruments and other applications | |
| Bissinger | The Hutchins–Schelleng Violin Octet After 50 Years | |
| WO2013172937A1 (en) | Systems and methods for designing and fabricating musical instruments | |
| Zhang | Acoustic Design of Elliptical Bells for Chime Sound Simulation | |
| Fan | Seminar Notes: The Mathematics of Music | |
| JP5641551B1 (en) | Keyboard instrument |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| MM4A | Annulment or lapse of patent due to non-payment of fees |