3278:
2817:
42:
1868:
1856:
65:. One of the forks is being hit with a rubberized mallet. Although the first tuning fork hasn't been hit, the other fork is visibly excited due to the oscillation caused by the periodic change in the pressure and density of the air by hitting the other fork, creating an acoustic resonance between the forks. However, if a piece of metal is placed on a prong, the effect dampens, and the excitations become less and less pronounced as resonance isn't achieved as effectively.
159:
3565:
2657:
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45:
1041:
1005:
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933:
44:
49:
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43:
50:
1034:
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962:
926:
47:
545:
The resonance properties of a cylinder may be understood by considering the behavior of a sound wave in air. Sound travels as a longitudinal compression wave, causing air molecules to move back and forth along the direction of travel. Within a tube, a standing wave is formed, whose wavelength depends
428:
Higher tension and shorter lengths increase the resonant frequencies. When the string is excited with an impulsive function (a finger pluck or a strike by a hammer), the string vibrates at all the frequencies present in the impulse (an impulsive function theoretically contains 'all' frequencies).
1541:
where "n" here is an odd number (1, 3, 5...). This type of tube produces only odd harmonics and has its fundamental frequency an octave lower than that of an open cylinder (that is, half the frequency). This equation comes from the boundary conditions for the pressure wave, which treats the closed
1479:
of the fundamental note. For example, if the fundamental note of a closed pipe is C1, then overblowing the pipe gives G2, which is one-twelfth above C1. Alternatively we can say that G2 is one-fifth above C2 — the octave above C1. Adjusting the taper of this cylinder for a decreasing cone can tune
1848:
In the two diagrams below are shown the first three resonances of the pressure wave in a cylindrical tube, with antinodes at the closed end of the pipe. In diagram 1, the tube is open at both ends. In diagram 2, it is closed at one end. The horizontal axis is pressure. Note that in this case, the
1054:
In cylinders with both ends open, air molecules near the end move freely in and out of the tube. This movement produces displacement antinodes in the standing wave. Nodes tend to form inside the cylinder, away from the ends. In the first harmonic, the open tube contains exactly half of a standing
101:
of the strongest resonance. It will easily vibrate at those frequencies, and vibrate less strongly at other frequencies. It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other
2824:
This is a classic demonstration of resonance. A glass has a natural resonance, a frequency at which the glass will vibrate easily. Therefore the glass needs to be moved by the sound wave at that frequency. If the force from the sound wave making the glass vibrate is big enough, the size of the
1488:
by pinching open the dorsal thumb hole. Moving this small hole upwards, closer to the voicing will make it an "Echo Hole" (Dolmetsch
Recorder Modification) that will give a precise half note above the fundamental when opened. Note: Slight size or diameter adjustment is needed to zero in on the
537:
Any cylinder resonates at multiple frequencies, producing multiple musical pitches. The lowest frequency is called the fundamental frequency or the first harmonic. Cylinders used as musical instruments are generally open, either at both ends, like a flute, or at one end, like some organ pipes.
1455:
a tube which is closed at one end is called a "stopped pipe". Such cylinders have a fundamental frequency but can be overblown to produce other higher frequencies or notes. These overblown registers can be tuned by using different degrees of conical taper. A closed tube resonates at the same
562:
The table below shows the displacement waves in a cylinder closed at both ends. Note that the air molecules near the closed ends cannot move, whereas the molecules near the center of the pipe move freely. In the first harmonic, the closed tube contains exactly half of a standing wave
2225:
2315:
are nonnegative integers that cannot all be zero. If the small loudspeaker box is airtight, the frequency low enough and the compression is high enough, the sound pressure (decibel level) inside the box will be the same anywhere inside the box, this is hydraulic pressure.
190:
directly related to the mass, length, and tension of the string. The wavelength that will create the first resonance on the string is equal to twice the length of the string. Higher resonances correspond to wavelengths that are integer divisions of the
46:
685:
2884:, a piece for percussion and pre-recorded sounds, the resonances from the acoustic instruments form sonic bridges to the pre-recorded electronic sounds, that, in turn, prolong the resonances, re-shaping them into new sonic gestures."
1218:
is the radius of the resonance tube. This equation compensates for the fact that the exact point at which a sound wave is reflecting at an open end is not perfectly at the end section of the tube, but a small distance outside the tube.
1069:
an open tube, a note can be obtained that is an octave above the fundamental frequency or note of the tube. For example, if the fundamental note of an open pipe is C1, then overblowing the pipe gives C2, which is an octave above C1.
404:
458:
frequencies when other strings are sounded. For example, an A string at 440 Hz will cause an E string at 330 Hz to resonate, because they share an overtone of 1320 Hz (3rd overtone of A and 4th overtone of E).
2069:. In words, a complete conical pipe behaves approximately like an open cylindrical pipe of the same length, and to first order the behavior does not change if the complete cone is replaced by a closed frustum of that cone.
1138:
in air (which is approximately 343 metres per second at 20 °C ). This equation comes from the boundary conditions for the pressure wave, which treats the open ends as pressure nodes where the change in pressure
2739:
2636:
2416:
2091:
1226:; rather, it has a finite value, called radiation impedance, which is dependent on the diameter of the tube, the wavelength, and the type of reflection board possibly present around the opening of the tube.
871:
1062:. Note that the diagrams in this reference show displacement waves, similar to the ones shown above. These stand in sharp contrast to the pressure waves shown near the end of the present article.
806:
2806:
429:
Those frequencies that are not one of the resonances are quickly filtered out—they are attenuated—and all that is left is the harmonic vibrations that we hear as a musical note.
137:
on the membrane to detect sound. (For mammals the membrane has tapering resonances across its length so that high frequencies are concentrated on one end and low frequencies on the other.)
304:
1640:
1209:
1948:
1700:
1287:
1835:
1422:
2056:
1756:
1343:
1536:
1117:
247:
2547:
2500:
1788:
1375:
2825:
vibration will become so large that the glass fractures. To do it reliably for a science demonstration requires practice and careful choice of the glass and loudspeaker.
467:
The resonance of a tube of air is related to the length of the tube, its shape, and whether it has closed or open ends. Many musical instruments resemble tubes that are
1998:
1840:
where v is the speed of sound, L is the length of the resonant tube, d is the diameter of the tube, f is the resonant sound frequency, and λ is the resonant wavelength.
589:
1484:
point, or shared "wave/node" position will cancel the fundamental frequency and force the tube to resonate at a 12th above the fundamental. This technique is used in a
30:
This article is about mechanical resonance of sound including musical instruments. For a general description of mechanical resonance in physics and engineering, see
2449:
2273:
2313:
2293:
315:
121:, and the shape of a drum membrane. Acoustic resonance is also important for hearing. For example, resonance of a stiff structural element, called the
1890:
The resonant frequencies of a stopped conical tube — a complete cone or frustum with one end closed — satisfy a more complicated condition:
523:). Like strings, vibrating air columns in ideal cylindrical or conical pipes also have resonances at harmonics, although there are some differences.
2837:
has used acoustic instruments and sine wave generators to explore the resonance of objects large and small in many of his compositions. The complex
1055:
wave (antinode-node-antinode). Thus the harmonics of the open cylinder are calculated in the same way as the harmonics of a closed/closed cylinder.
94:
is defined in general terms concerning vibrational waves in matter, acoustic resonance can occur at frequencies outside the range of human hearing.
1475:
a cylindrical closed tube, a note can be obtained that is approximately a twelfth above the fundamental note of the tube, or a fifth above the
2220:{\displaystyle f={v \over 2}{\sqrt {\left({\ell \over L_{x}}\right)^{2}+\left({m \over L_{y}}\right)^{2}+\left({n \over L_{z}}\right)^{2}}}}
2673:
2566:
1887:
of a cone with both ends open, will have resonant frequencies approximately equal to those of an open cylindrical pipe of the same length.
1456:
fundamental frequency as an open tube twice its length, with a wavelength equal to four times its length. In a closed tube, a displacement
2349:
264:
in an open end pipe (that is, both ends of the pipe are open)). The speed of a wave through a string or wire is related to its tension
546:
on the length of the tube. At the closed end of the tube, air molecules cannot move much, so this end of the tube is a displacement
140:
Like mechanical resonance, acoustic resonance can result in catastrophic failure of the vibrator. The classic example of this is
3487:
817:
3029:
2947:
3102:
737:
3214:
1460:, or point of no vibration, always appears at the closed end and if the tube is resonating, it will have a displacement
567:-node). Considering the pressure wave in this setup, the two closed ends are the antinodes for the change in pressure Δ
2766:
714:
is that the pressure of the closed ends will follow that of the point next to them. Applying the boundary condition
1480:
the second harmonic or overblown note close to the octave position or 8th. Opening a small "speaker hole" at the
274:
1554:
1485:
583:
1587:
1156:
1059:
550:
in the standing wave. At the open end of the tube, air molecules can move freely, producing a displacement
3594:
3187:
3142:
1896:
1652:
1239:
1794:
1381:
2841:
2017:
17:
3326:
1706:
1293:
3447:
2061:
leading to resonant frequencies approximately equal to those of an open cylinder whose length equals
1498:
1079:
538:
However, a cylinder closed at both ends can also be used to create or visualize sound waves, as in a
532:
209:
2877:
2507:
2460:
3502:
1762:
1349:
680:{\displaystyle \Delta p(x,t)=p_{\text{max}}\cos \left({2\pi x \over \lambda }\right)\cos(\omega t)}
438:
76:
amplifies sound waves whose frequency matches one of its own natural frequencies of vibration (its
3382:
3239:
3207:
3119:
2923:
1222:
The reflection ratio is slightly less than 1; the open end does not behave like an infinitesimal
476:
1963:
3321:
3231:
2942:
Kinsler L.E., Frey A.R., Coppens A.B., Sanders J.V., "Fundamentals of
Acoustics", 3rd Edition,
3019:
2872:
spaces such as the 2-million-US-gallon (7,600 m) cistern at Fort Worden, WA, which has a
3462:
3360:
3264:
2078:
451:
192:
97:
An acoustically resonant object usually has more than one resonance frequency, especially at
58:
2816:
2424:
2258:
105:
Acoustic resonance is an important consideration for instrument builders, as most acoustic
87:
31:
8:
3482:
3392:
2340:
414:
3589:
3568:
3404:
3365:
3200:
2918:
2298:
2278:
1223:
447:
399:{\displaystyle f={n{\sqrt {T \over \rho }} \over 2L}={n{\sqrt {T \over m/L}} \over 2L}}
106:
3277:
3538:
3370:
3098:
3025:
2943:
1849:
open end of the pipe is a pressure node while the closed end is a pressure antinode.
520:
1867:
3548:
3467:
3431:
3399:
3350:
3259:
3150:
3073:
2961:
2861:
916:
512:
443:
200:
162:
String resonance of a bass guitar A note with fundamental frequency of 110 Hz.
153:
122:
1855:
3543:
3497:
3338:
3296:
3046:
2990:
2865:
1574:
is that the pressure of the closed end will follow that of the point next to it.
3492:
3477:
3424:
2452:
1578:
1457:
1147:
1135:
547:
3583:
3457:
3419:
3387:
3342:
3304:
3254:
3249:
3006:
2913:
2908:
2873:
2869:
1452:
515:) are acoustically similar to closed conical pipes with some deviations (see
2833:
Several composers have begun to make resonance the subject of compositions.
309:
So the frequency is related to the properties of the string by the equation
3452:
2898:
2853:
2834:
3472:
3284:
3154:
2656:
1472:
1066:
539:
479:). A pipe that is closed at one end and open at the other is said to be
55:
158:
3533:
3523:
3309:
2838:
2082:
2007:
is the distance from the small end of the frustum to the vertex. When
1954:
516:
134:
3078:
3314:
3244:
3223:
2903:
2845:
2734:{\displaystyle f={\frac {vd}{\pi }}{\sqrt {\frac {3}{8(0.85)D^{3}}}}}
2631:{\displaystyle f={\frac {vd}{\pi }}{\sqrt {\frac {3}{8L_{eq}D^{3}}}}}
2553:
500:
187:
130:
110:
91:
78:
73:
62:
35:
2411:{\displaystyle f={\frac {v}{2\pi }}{\sqrt {\frac {A}{V_{0}L_{eq}}}}}
1126:
is a positive integer (1, 2, 3...) representing the resonance node,
195:
wavelength. The corresponding frequencies are related to the speed
3409:
3377:
1461:
1443:
is the resonant sound frequency, and λ is the resonant wavelength.
564:
551:
496:
455:
261:
98:
3068:
Greene, Chad A.; Argo IV, Theodore F.; Wilson, Preston S. (2009).
2081:
and buildings. Rectangular buildings have resonances described as
2011:
is small, that is, when the cone is nearly complete, this becomes
256:
is the length of the string (for a string fixed at both ends) and
3355:
2893:
1884:
511:
as closed conical pipes, while most modern lip-reed instruments (
508:
126:
2852:
or other percussion instrument interact with room resonances in
1073:
Open cylindrical tubes resonate at the approximate frequencies:
2849:
2560:
For a spherical cavity, the resonant frequency formula becomes
2085:. For a rectangular box, the resonant frequencies are given by
1476:
183:
175:
114:
3528:
3072:. Proceedings of Meetings on Acoustics. ASA. p. 025001.
2757:
492:
179:
118:
3192:
1040:
1004:
968:
932:
554:. Displacement nodes are pressure antinodes and vice versa.
141:
3518:
504:
422:
171:
167:
3070:
A Helmholtz resonator experiment for the Listen Up project
2811:
2324:
The resonant frequency of a rigid cavity of static volume
2077:
Sound waves in a rectangular box include such examples as
1033:
997:
961:
925:
86:
The term "acoustic resonance" is sometimes used to narrow
3117:
1481:
1465:
1058:
The physics of a pipe open at both ends are explained in
866:{\displaystyle f={\frac {v}{\lambda }}={\frac {nv}{2L}}.}
2319:
1542:
end as pressure antinodes where the change in pressure Δ
432:
2667:=0 and the surface of the sphere acts as a flange, so
2769:
2676:
2569:
2510:
2463:
2427:
2352:
2301:
2281:
2261:
2094:
2020:
1966:
1899:
1883:
An open conical tube, that is, one in the shape of a
1797:
1765:
1709:
1655:
1590:
1501:
1384:
1352:
1296:
1242:
1159:
1082:
820:
740:
592:
318:
277:
212:
166:
In musical instruments, strings under tension, as in
2880:
composition professor and composer Kent
Olofsson's "
801:{\displaystyle \lambda ={\frac {2L}{n}};n=1,2,3,...}
450:. Strings or parts of strings may resonate at their
3067:
1492:A closed tube will have approximate resonances of:
586:), which gives the equation for the pressure wave:
571:; Therefore, at both ends, the change in pressure Δ
90:to the frequency range of human hearing, but since
2800:
2733:
2630:
2541:
2494:
2443:
2410:
2307:
2287:
2267:
2219:
2050:
1992:
1942:
1829:
1782:
1750:
1694:
1634:
1530:
1416:
1369:
1337:
1281:
1203:
1111:
865:
800:
679:
398:
298:
241:
3581:
2858:Koan: Having Never Written A Note For Percussion
2801:{\displaystyle f=72.6{\frac {d}{\sqrt {D^{3}}}}}
144:at the precise resonant frequency of the glass.
27:Resonance phenomena in sound and musical devices
3173:Rossing, Thomas D., and Fletcher, Neville H.,
491:pipe is open at both ends. Modern orchestral
34:. For a general description of resonance, see
3208:
3118:Acoustics research centre (14 January 2019).
2993:in 1987, discusses "open" and "closed" tubes.
731:gives the wavelengths of the standing waves:
462:
2989:. J. J. Weber, Leipzig. 1931. Translated by
1557:. The intuition for this boundary condition
1546:must have the maximal amplitude, or satisfy
687:. The intuition for this boundary condition
575:must have the maximal amplitude (or satisfy
533:Standing wave § Standing wave in a pipe
3215:
3201:
3168:Acoustical aspects of woodwind instruments
3086:
2451:is the equivalent length of the neck with
1468:point (length × 0.618) near the open end.
3095:The Science and Applications of Acoustics
3077:
2828:
2820:Breaking glass with sound using resonance
2072:
299:{\displaystyle v={\sqrt {T \over \rho }}}
3140:
3017:
2815:
1577:A more accurate equation considering an
1464:, or point of greatest vibration at the
1146:A more accurate equation considering an
499:behave as closed cylindrical pipes; and
157:
40:
3149:(4). University of Haddersfield Press.
3092:
3011:
2812:Breaking glass with sound via resonance
14:
3582:
2981:
2979:
2962:"Saxophone acoustics: an introduction"
1635:{\displaystyle f={nv \over 4(L+0.3d)}}
1204:{\displaystyle f={nv \over 2(L+0.6r)}}
557:
3196:
3024:. Taylor & Francis. p. 170.
2663:For a sphere with just a sound hole,
2320:Resonance of a sphere of air (vented)
1943:{\displaystyle kL=n\pi -\tan ^{-1}kx}
1695:{\displaystyle f={v \over 4(L+0.3d)}}
1282:{\displaystyle f={v \over 2(L+0.6r)}}
433:String resonance in music instruments
417:, ρ is the mass per unit length, and
1830:{\displaystyle \lambda ={4(L+0.3d)}}
1446:
1435:is the length of the resonant tube,
1417:{\displaystyle \lambda ={2(L+0.6r)}}
1049:
113:, such as the strings and body of a
2976:
2051:{\displaystyle k(L+x)\approx n\pi }
147:
24:
3177:. New York, Springer-Verlag, 1995.
3141:Olofsson, Kent (4 February 2015).
2655:
2552:
593:
495:behave as open cylindrical pipes;
25:
3606:
3181:
3175:Principles of Vibration and Sound
3120:"How to break a glass with sound"
3044:
2959:
2331:with a necked sound hole of area
811:And the resonant frequencies are
3564:
3563:
3276:
2744:In dry air at 20 °C, with
1866:
1854:
1843:
1751:{\displaystyle {f(4(L+0.3d))}=v}
1338:{\displaystyle {f(2(L+0.6r))}=v}
1039:
1032:
1003:
996:
967:
960:
931:
924:
268:and the mass per unit length ρ:
142:breaking a wine glass with sound
3134:
3049:. University of New South Wales
2964:. University of New South Wales
2255:are the dimensions of the box.
1555:the Sturm–Liouville formulation
1531:{\displaystyle f={nv \over 4L}}
1112:{\displaystyle f={nv \over 2L}}
584:the Sturm–Liouville formulation
242:{\displaystyle f={nv \over 2L}}
3170:. Amsterdam, Frits Knuf, 1969.
3166:Nederveen, Cornelis Johannes,
3111:
3097:. Springer. pp. 145–149.
3061:
3038:
2996:
2953:
2936:
2714:
2708:
2542:{\displaystyle L_{eq}=L+0.85d}
2495:{\displaystyle L_{eq}=L+0.75d}
2036:
2024:
1823:
1808:
1738:
1735:
1720:
1714:
1686:
1671:
1626:
1611:
1410:
1395:
1325:
1322:
1307:
1301:
1273:
1258:
1195:
1180:
1130:is the length of the tube and
674:
665:
611:
599:
201:wave traveling down the string
13:
1:
3222:
2929:
1783:{\displaystyle {f\lambda }=v}
1489:precise half note frequency.
1370:{\displaystyle {f\lambda }=v}
2502: for an unflanged neck
526:
72:is a phenomenon in which an
7:
3093:Raichel, Daniel R. (2006).
3018:Kuttruff, Heinrich (2007).
2887:
2868:regularly perform in large
1439:is the radius of the tube,
10:
3611:
3143:"Resonances and Responses"
3021:Acoustics: An Introduction
3003:Horns, Strings and Harmony
2650:d = diameter of sound hole
1993:{\displaystyle k=2\pi f/v}
943:= 880 Hz
905:= 440 Hz
530:
463:Resonance of a tube of air
436:
151:
117:, the length of tube in a
29:
3559:
3511:
3440:
3337:
3295:
3271:
3230:
2549: for a flanged neck
897:Molecular representation
2950:, Wiley, New York, 1982.
2876:with a 45-second decay.
1878:
439:String resonance (music)
3240:Architectural acoustics
3122:. University of Salford
2924:Reflection phase change
2234:is the speed of sound,
1431:is the speed of sound,
3327:Fletcher–Munson curves
3322:Equal-loudness contour
3232:Acoustical engineering
2878:Malmö Academy of Music
2829:In musical composition
2821:
2802:
2735:
2660:
2647:D = diameter of sphere
2632:
2557:
2543:
2496:
2445:
2444:{\displaystyle L_{eq}}
2412:
2309:
2289:
2269:
2221:
2079:loudspeaker enclosures
2073:Closed rectangular box
2052:
1994:
1944:
1831:
1784:
1752:
1696:
1636:
1532:
1418:
1371:
1339:
1283:
1205:
1113:
867:
802:
681:
400:
300:
243:
163:
66:
3463:Hermann von Helmholtz
3361:Fundamental frequency
3265:Sympathetic resonance
3188:Standing Waves Applet
3047:"Helmholtz Resonance"
2848:and decrescendo on a
2819:
2803:
2736:
2659:
2633:
2556:
2544:
2497:
2446:
2413:
2310:
2290:
2270:
2268:{\displaystyle \ell }
2222:
2053:
1995:
1945:
1832:
1785:
1753:
1697:
1637:
1533:
1419:
1372:
1340:
1284:
1206:
1114:
868:
803:
682:
401:
301:
244:
161:
152:Further information:
54:Experiment using two
53:
3155:10.5920/divp.2015.48
2767:
2674:
2567:
2508:
2461:
2425:
2350:
2299:
2279:
2259:
2092:
2018:
1964:
1897:
1795:
1763:
1707:
1653:
1646:Again, when n is 1:
1588:
1499:
1382:
1350:
1294:
1240:
1157:
1080:
894:Wave representation
818:
738:
590:
316:
275:
210:
188:resonant frequencies
102:than its resonance.
88:mechanical resonance
32:Mechanical resonance
3595:Musical instruments
3483:Werner Meyer-Eppler
3393:Missing fundamental
2341:Helmholtz resonance
558:Closed at both ends
186:and so forth, have
3366:Frequency spectrum
2919:Sympathetic string
2844:of a swell shaped
2822:
2798:
2731:
2661:
2628:
2558:
2539:
2492:
2441:
2408:
2305:
2285:
2265:
2217:
2048:
1990:
1940:
1827:
1780:
1748:
1692:
1632:
1528:
1414:
1367:
1335:
1279:
1224:acoustic impedance
1201:
1109:
863:
798:
677:
448:string instruments
396:
296:
239:
164:
70:Acoustic resonance
67:
3577:
3576:
3539:Musical acoustics
3371:harmonic spectrum
3079:10.1121/1.3112687
3031:978-0-203-97089-8
2948:978-0-471-02933-5
2796:
2795:
2729:
2728:
2696:
2626:
2625:
2589:
2406:
2405:
2372:
2308:{\displaystyle n}
2288:{\displaystyle m}
2215:
2203:
2168:
2133:
2109:
1690:
1630:
1526:
1447:Closed at one end
1277:
1199:
1107:
1060:Physics Classroom
1050:Open at both ends
1047:
1046:
858:
835:
760:
653:
624:
513:brass instruments
394:
383:
382:
352:
341:
340:
294:
293:
237:
51:
16:(Redirected from
3602:
3567:
3566:
3468:Carleen Hutchins
3400:Combination tone
3287:
3280:
3260:String vibration
3217:
3210:
3203:
3194:
3193:
3159:
3158:
3147:Divergence Press
3138:
3132:
3131:
3129:
3127:
3115:
3109:
3108:
3104:978-0387-26062-4
3090:
3084:
3083:
3081:
3065:
3059:
3058:
3056:
3054:
3042:
3036:
3035:
3015:
3009:
3007:Arthur H. Benade
3000:
2994:
2983:
2974:
2973:
2971:
2969:
2957:
2951:
2940:
2862:Pauline Oliveros
2807:
2805:
2804:
2799:
2797:
2794:
2793:
2784:
2780:
2760:, this becomes
2740:
2738:
2737:
2732:
2730:
2727:
2726:
2725:
2700:
2699:
2697:
2692:
2684:
2637:
2635:
2634:
2629:
2627:
2624:
2623:
2622:
2613:
2612:
2593:
2592:
2590:
2585:
2577:
2548:
2546:
2545:
2540:
2523:
2522:
2501:
2499:
2498:
2493:
2476:
2475:
2450:
2448:
2447:
2442:
2440:
2439:
2417:
2415:
2414:
2409:
2407:
2404:
2403:
2402:
2390:
2389:
2376:
2375:
2373:
2371:
2360:
2339:is given by the
2314:
2312:
2311:
2306:
2294:
2292:
2291:
2286:
2274:
2272:
2271:
2266:
2226:
2224:
2223:
2218:
2216:
2214:
2213:
2208:
2204:
2202:
2201:
2189:
2179:
2178:
2173:
2169:
2167:
2166:
2154:
2144:
2143:
2138:
2134:
2132:
2131:
2119:
2112:
2110:
2102:
2057:
2055:
2054:
2049:
1999:
1997:
1996:
1991:
1986:
1949:
1947:
1946:
1941:
1930:
1929:
1870:
1858:
1836:
1834:
1833:
1828:
1826:
1789:
1787:
1786:
1781:
1773:
1757:
1755:
1754:
1749:
1741:
1701:
1699:
1698:
1693:
1691:
1689:
1663:
1641:
1639:
1638:
1633:
1631:
1629:
1606:
1598:
1581:is given below:
1573:
1563:
1552:
1537:
1535:
1534:
1529:
1527:
1525:
1517:
1509:
1451:When used in an
1423:
1421:
1420:
1415:
1413:
1376:
1374:
1373:
1368:
1360:
1344:
1342:
1341:
1336:
1328:
1288:
1286:
1285:
1280:
1278:
1276:
1250:
1210:
1208:
1207:
1202:
1200:
1198:
1175:
1167:
1150:is given below:
1118:
1116:
1115:
1110:
1108:
1106:
1098:
1090:
1043:
1036:
1007:
1000:
971:
964:
935:
928:
917:fundamental tone
876:
875:
872:
870:
869:
864:
859:
857:
849:
841:
836:
828:
807:
805:
804:
799:
761:
756:
748:
730:
720:
713:
703:
693:
686:
684:
683:
678:
658:
654:
649:
638:
626:
625:
622:
581:
444:String resonance
405:
403:
402:
397:
395:
393:
385:
384:
381:
377:
365:
364:
358:
353:
351:
343:
342:
333:
332:
326:
305:
303:
302:
297:
295:
286:
285:
248:
246:
245:
240:
238:
236:
228:
220:
203:by the equation
154:Vibrating string
148:Vibrating string
123:basilar membrane
52:
21:
3610:
3609:
3605:
3604:
3603:
3601:
3600:
3599:
3580:
3579:
3578:
3573:
3555:
3507:
3498:D. Van Holliday
3436:
3405:Mersenne's laws
3339:Audio frequency
3333:
3297:Psychoacoustics
3291:
3290:
3283:
3269:
3226:
3221:
3184:
3163:
3162:
3139:
3135:
3125:
3123:
3116:
3112:
3105:
3091:
3087:
3066:
3062:
3052:
3050:
3043:
3039:
3032:
3016:
3012:
3001:
2997:
2991:Lawrence Gwozdz
2984:
2977:
2967:
2965:
2958:
2954:
2941:
2937:
2932:
2890:
2866:Stuart Dempster
2831:
2814:
2789:
2785:
2779:
2768:
2765:
2764:
2721:
2717:
2704:
2698:
2685:
2683:
2675:
2672:
2671:
2618:
2614:
2605:
2601:
2597:
2591:
2578:
2576:
2568:
2565:
2564:
2515:
2511:
2509:
2506:
2505:
2468:
2464:
2462:
2459:
2458:
2432:
2428:
2426:
2423:
2422:
2395:
2391:
2385:
2381:
2380:
2374:
2364:
2359:
2351:
2348:
2347:
2329:
2322:
2300:
2297:
2296:
2280:
2277:
2276:
2260:
2257:
2256:
2253:
2246:
2239:
2209:
2197:
2193:
2188:
2184:
2183:
2174:
2162:
2158:
2153:
2149:
2148:
2139:
2127:
2123:
2118:
2114:
2113:
2111:
2101:
2093:
2090:
2089:
2075:
2019:
2016:
2015:
1982:
1965:
1962:
1961:
1922:
1918:
1898:
1895:
1894:
1881:
1874:
1871:
1862:
1859:
1846:
1804:
1796:
1793:
1792:
1766:
1764:
1761:
1760:
1710:
1708:
1705:
1704:
1667:
1662:
1654:
1651:
1650:
1607:
1599:
1597:
1589:
1586:
1585:
1565:
1558:
1553:in the form of
1547:
1518:
1510:
1508:
1500:
1497:
1496:
1449:
1391:
1383:
1380:
1379:
1353:
1351:
1348:
1347:
1297:
1295:
1292:
1291:
1254:
1249:
1241:
1238:
1237:
1176:
1168:
1166:
1158:
1155:
1154:
1099:
1091:
1089:
1081:
1078:
1077:
1052:
1015:= 1760 Hz
979:= 1320 Hz
850:
842:
840:
827:
819:
816:
815:
749:
747:
739:
736:
735:
722:
715:
705:
695:
688:
639:
637:
633:
621:
617:
591:
588:
587:
582:in the form of
576:
560:
535:
529:
465:
441:
435:
386:
373:
369:
363:
359:
357:
344:
331:
327:
325:
317:
314:
313:
284:
276:
273:
272:
229:
221:
219:
211:
208:
207:
156:
150:
74:acoustic system
41:
39:
28:
23:
22:
15:
12:
11:
5:
3608:
3598:
3597:
3592:
3575:
3574:
3572:
3571:
3560:
3557:
3556:
3554:
3553:
3552:
3551:
3546:
3536:
3531:
3526:
3521:
3515:
3513:
3512:Related topics
3509:
3508:
3506:
3505:
3500:
3495:
3493:Joseph Sauveur
3490:
3485:
3480:
3478:Marin Mersenne
3475:
3470:
3465:
3460:
3455:
3450:
3444:
3442:
3438:
3437:
3435:
3434:
3429:
3428:
3427:
3417:
3412:
3407:
3402:
3397:
3396:
3395:
3390:
3385:
3375:
3374:
3373:
3363:
3358:
3353:
3347:
3345:
3335:
3334:
3332:
3331:
3330:
3329:
3319:
3318:
3317:
3312:
3301:
3299:
3293:
3292:
3289:
3288:
3281:
3273:
3272:
3270:
3268:
3267:
3262:
3257:
3252:
3247:
3242:
3236:
3234:
3228:
3227:
3220:
3219:
3212:
3205:
3197:
3191:
3190:
3183:
3182:External links
3180:
3179:
3178:
3171:
3161:
3160:
3133:
3110:
3103:
3085:
3060:
3037:
3030:
3010:
2995:
2975:
2952:
2934:
2933:
2931:
2928:
2927:
2926:
2921:
2916:
2911:
2906:
2901:
2896:
2889:
2886:
2830:
2827:
2813:
2810:
2809:
2808:
2792:
2788:
2783:
2778:
2775:
2772:
2742:
2741:
2724:
2720:
2716:
2713:
2710:
2707:
2703:
2695:
2691:
2688:
2682:
2679:
2654:
2653:
2652:
2651:
2648:
2639:
2638:
2621:
2617:
2611:
2608:
2604:
2600:
2596:
2588:
2584:
2581:
2575:
2572:
2551:
2550:
2538:
2535:
2532:
2529:
2526:
2521:
2518:
2514:
2503:
2491:
2488:
2485:
2482:
2479:
2474:
2471:
2467:
2453:end correction
2438:
2435:
2431:
2419:
2418:
2401:
2398:
2394:
2388:
2384:
2379:
2370:
2367:
2363:
2358:
2355:
2327:
2321:
2318:
2304:
2284:
2264:
2251:
2244:
2237:
2228:
2227:
2212:
2207:
2200:
2196:
2192:
2187:
2182:
2177:
2172:
2165:
2161:
2157:
2152:
2147:
2142:
2137:
2130:
2126:
2122:
2117:
2108:
2105:
2100:
2097:
2074:
2071:
2059:
2058:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2001:
2000:
1989:
1985:
1981:
1978:
1975:
1972:
1969:
1951:
1950:
1939:
1936:
1933:
1928:
1925:
1921:
1917:
1914:
1911:
1908:
1905:
1902:
1880:
1877:
1876:
1875:
1872:
1865:
1863:
1860:
1853:
1845:
1842:
1838:
1837:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1803:
1800:
1790:
1779:
1776:
1772:
1769:
1758:
1747:
1744:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1702:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1666:
1661:
1658:
1644:
1643:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1605:
1602:
1596:
1593:
1579:end correction
1539:
1538:
1524:
1521:
1516:
1513:
1507:
1504:
1448:
1445:
1425:
1424:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1390:
1387:
1377:
1366:
1363:
1359:
1356:
1345:
1334:
1331:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1289:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1253:
1248:
1245:
1212:
1211:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1174:
1171:
1165:
1162:
1148:end correction
1143:must be zero.
1136:speed of sound
1120:
1119:
1105:
1102:
1097:
1094:
1088:
1085:
1051:
1048:
1045:
1044:
1037:
1030:
1027:
1024:
1021:
1016:
1009:
1008:
1001:
994:
991:
988:
985:
980:
973:
972:
965:
958:
955:
952:
949:
944:
937:
936:
929:
922:
919:
914:
911:
906:
899:
898:
895:
892:
889:
886:
883:
880:
874:
873:
862:
856:
853:
848:
845:
839:
834:
831:
826:
823:
809:
808:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
759:
755:
752:
746:
743:
676:
673:
670:
667:
664:
661:
657:
652:
648:
645:
642:
636:
632:
629:
620:
616:
613:
610:
607:
604:
601:
598:
595:
559:
556:
528:
525:
464:
461:
437:Main article:
434:
431:
407:
406:
392:
389:
380:
376:
372:
368:
362:
356:
350:
347:
339:
336:
330:
324:
321:
307:
306:
292:
289:
283:
280:
250:
249:
235:
232:
227:
224:
218:
215:
149:
146:
26:
9:
6:
4:
3:
2:
3607:
3596:
3593:
3591:
3588:
3587:
3585:
3570:
3562:
3561:
3558:
3550:
3547:
3545:
3542:
3541:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3520:
3517:
3516:
3514:
3510:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3488:Lord Rayleigh
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3458:Ernst Chladni
3456:
3454:
3451:
3449:
3446:
3445:
3443:
3439:
3433:
3430:
3426:
3423:
3422:
3421:
3420:Standing wave
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3394:
3391:
3389:
3388:Inharmonicity
3386:
3384:
3381:
3380:
3379:
3376:
3372:
3369:
3368:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3348:
3346:
3344:
3340:
3336:
3328:
3325:
3324:
3323:
3320:
3316:
3313:
3311:
3308:
3307:
3306:
3303:
3302:
3300:
3298:
3294:
3286:
3282:
3279:
3275:
3274:
3266:
3263:
3261:
3258:
3256:
3255:Soundproofing
3253:
3251:
3250:Reverberation
3248:
3246:
3243:
3241:
3238:
3237:
3235:
3233:
3229:
3225:
3218:
3213:
3211:
3206:
3204:
3199:
3198:
3195:
3189:
3186:
3185:
3176:
3172:
3169:
3165:
3164:
3156:
3152:
3148:
3144:
3137:
3121:
3114:
3106:
3100:
3096:
3089:
3080:
3075:
3071:
3064:
3048:
3041:
3033:
3027:
3023:
3022:
3014:
3008:
3004:
2999:
2992:
2988:
2982:
2980:
2963:
2956:
2949:
2945:
2939:
2935:
2925:
2922:
2920:
2917:
2915:
2914:Standing wave
2912:
2910:
2909:Reverberation
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2891:
2885:
2883:
2879:
2875:
2871:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2840:
2836:
2826:
2818:
2790:
2786:
2781:
2776:
2773:
2770:
2763:
2762:
2761:
2759:
2755:
2751:
2747:
2722:
2718:
2711:
2705:
2701:
2693:
2689:
2686:
2680:
2677:
2670:
2669:
2668:
2666:
2658:
2649:
2646:
2645:
2644:
2643:
2642:
2619:
2615:
2609:
2606:
2602:
2598:
2594:
2586:
2582:
2579:
2573:
2570:
2563:
2562:
2561:
2555:
2536:
2533:
2530:
2527:
2524:
2519:
2516:
2512:
2504:
2489:
2486:
2483:
2480:
2477:
2472:
2469:
2465:
2457:
2456:
2455:
2454:
2436:
2433:
2429:
2399:
2396:
2392:
2386:
2382:
2377:
2368:
2365:
2361:
2356:
2353:
2346:
2345:
2344:
2342:
2338:
2334:
2330:
2317:
2302:
2282:
2262:
2254:
2247:
2240:
2233:
2210:
2205:
2198:
2194:
2190:
2185:
2180:
2175:
2170:
2163:
2159:
2155:
2150:
2145:
2140:
2135:
2128:
2124:
2120:
2115:
2106:
2103:
2098:
2095:
2088:
2087:
2086:
2084:
2080:
2070:
2068:
2065: +
2064:
2045:
2042:
2039:
2033:
2030:
2027:
2021:
2014:
2013:
2012:
2010:
2006:
1987:
1983:
1979:
1976:
1973:
1970:
1967:
1960:
1959:
1958:
1956:
1937:
1934:
1931:
1926:
1923:
1919:
1915:
1912:
1909:
1906:
1903:
1900:
1893:
1892:
1891:
1888:
1886:
1869:
1864:
1857:
1852:
1851:
1850:
1844:Pressure wave
1841:
1820:
1817:
1814:
1811:
1805:
1801:
1798:
1791:
1777:
1774:
1770:
1767:
1759:
1745:
1742:
1732:
1729:
1726:
1723:
1717:
1711:
1703:
1683:
1680:
1677:
1674:
1668:
1664:
1659:
1656:
1649:
1648:
1647:
1623:
1620:
1617:
1614:
1608:
1603:
1600:
1594:
1591:
1584:
1583:
1582:
1580:
1575:
1572:
1568:
1561:
1556:
1550:
1545:
1522:
1519:
1514:
1511:
1505:
1502:
1495:
1494:
1493:
1490:
1487:
1483:
1478:
1474:
1469:
1467:
1463:
1459:
1454:
1444:
1442:
1438:
1434:
1430:
1407:
1404:
1401:
1398:
1392:
1388:
1385:
1378:
1364:
1361:
1357:
1354:
1346:
1332:
1329:
1319:
1316:
1313:
1310:
1304:
1298:
1290:
1270:
1267:
1264:
1261:
1255:
1251:
1246:
1243:
1236:
1235:
1234:
1232:
1227:
1225:
1220:
1217:
1192:
1189:
1186:
1183:
1177:
1172:
1169:
1163:
1160:
1153:
1152:
1151:
1149:
1144:
1142:
1137:
1133:
1129:
1125:
1103:
1100:
1095:
1092:
1086:
1083:
1076:
1075:
1074:
1071:
1068:
1063:
1061:
1056:
1042:
1038:
1035:
1031:
1029:4th harmonic
1028:
1026:3rd overtone
1025:
1022:
1020:
1017:
1014:
1011:
1010:
1006:
1002:
999:
995:
993:3rd harmonic
992:
990:2nd overtone
989:
986:
984:
981:
978:
975:
974:
970:
966:
963:
959:
957:2nd harmonic
956:
954:1st overtone
953:
950:
948:
945:
942:
939:
938:
934:
930:
927:
923:
921:1st harmonic
920:
918:
915:
912:
910:
907:
904:
901:
900:
896:
893:
890:
887:
884:
881:
878:
877:
860:
854:
851:
846:
843:
837:
832:
829:
824:
821:
814:
813:
812:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
757:
753:
750:
744:
741:
734:
733:
732:
729:
725:
718:
712:
708:
702:
698:
691:
671:
668:
662:
659:
655:
650:
646:
643:
640:
634:
630:
627:
618:
614:
608:
605:
602:
596:
585:
579:
574:
570:
566:
555:
553:
549:
543:
541:
534:
524:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
482:
478:
474:
470:
460:
457:
453:
449:
445:
440:
430:
426:
424:
421:is the total
420:
416:
412:
390:
387:
378:
374:
370:
366:
360:
354:
348:
345:
337:
334:
328:
322:
319:
312:
311:
310:
290:
287:
281:
278:
271:
270:
269:
267:
263:
260:= 1, 2, 3...(
259:
255:
233:
230:
225:
222:
216:
213:
206:
205:
204:
202:
198:
194:
189:
185:
181:
177:
173:
169:
160:
155:
145:
143:
138:
136:
132:
128:
124:
120:
116:
112:
108:
103:
100:
95:
93:
89:
84:
82:
80:
75:
71:
64:
60:
57:
37:
33:
19:
3503:Thomas Young
3453:Jens Blauert
3441:Acousticians
3414:
3174:
3167:
3146:
3136:
3124:. Retrieved
3113:
3094:
3088:
3069:
3063:
3051:. Retrieved
3045:Wolfe, Joe.
3040:
3020:
3013:
3002:
2998:
2987:Das Saxophon
2986:
2985:Kool, Jaap.
2966:. Retrieved
2960:Wolfe, Joe.
2955:
2938:
2899:Music theory
2881:
2857:
2854:James Tenney
2835:Alvin Lucier
2832:
2823:
2753:
2749:
2745:
2743:
2664:
2662:
2640:
2559:
2420:
2336:
2332:
2325:
2323:
2249:
2242:
2235:
2231:
2229:
2076:
2066:
2062:
2060:
2008:
2004:
2002:
1952:
1889:
1882:
1847:
1839:
1645:
1576:
1570:
1566:
1559:
1548:
1543:
1540:
1491:
1470:
1450:
1440:
1436:
1432:
1428:
1426:
1230:
1228:
1221:
1215:
1213:
1145:
1140:
1131:
1127:
1123:
1121:
1072:
1064:
1057:
1053:
1023:4th partial
1018:
1012:
987:3rd partial
982:
976:
951:2nd partial
946:
940:
913:1st partial
908:
902:
810:
727:
723:
716:
710:
706:
700:
696:
689:
577:
572:
568:
561:
544:
536:
488:
484:
480:
472:
468:
466:
442:
427:
418:
410:
408:
308:
265:
257:
253:
251:
196:
165:
139:
104:
96:
85:
77:
69:
68:
61:at the same
56:tuning forks
3473:Franz Melde
3448:John Backus
3432:Subharmonic
3285:Spectrogram
2882:Terpsichord
2870:reverberant
2752:in metres,
2335:and length
1473:overblowing
1067:overblowing
540:Rubens Tube
521:false tones
517:pedal tones
473:cylindrical
452:fundamental
193:fundamental
125:within the
107:instruments
81:frequencies
59:oscillating
18:Closed tube
3584:Categories
3534:Ultrasound
3524:Infrasound
3310:Bark scale
3126:17 January
2930:References
2839:inharmonic
2083:room modes
1955:wavenumber
1953:where the
879:Frequency
531:See also:
501:saxophones
446:occurs on
135:hair cells
111:resonators
3590:Acoustics
3415:Resonance
3315:Mel scale
3245:Monochord
3224:Acoustics
3053:1 January
2968:1 January
2904:Resonance
2846:crescendo
2694:π
2587:π
2369:π
2263:ℓ
2121:ℓ
2046:π
2040:≈
1977:π
1932:
1924:−
1916:−
1913:π
1799:λ
1771:λ
1386:λ
1358:λ
833:λ
742:λ
669:ω
663:
651:λ
644:π
631:
594:Δ
527:Cylinders
497:clarinets
487:while an
338:ρ
291:ρ
131:inner ear
99:harmonics
92:acoustics
79:resonance
63:frequency
36:Resonance
3569:Category
3410:Overtone
3378:Harmonic
2888:See also
2842:partials
2343:formula
1560:∂(Δp)/∂x
1549:∂(Δp)/∂x
1486:recorder
1462:antinode
1229:So when
717:∂(Δp)/∂x
690:∂(Δp)/∂x
578:∂(Δp)/∂x
565:antinode
552:antinode
509:bassoons
456:overtone
262:Harmonic
3356:Formant
2894:Harmony
1885:frustum
1134:is the
891:Name 3
888:Name 2
885:Name 1
481:stopped
469:conical
415:tension
413:is the
184:violins
176:guitars
133:allows
129:of the
127:cochlea
3549:Violin
3383:Series
3101:
3028:
2946:
2874:reverb
2850:tamtam
2641:where
2421:where
2295:, and
2230:where
1477:octave
1427:where
1233:is 1:
1214:where
1122:where
1013:4 · f
977:3 · f
941:2 · f
903:1 · f
882:Order
563:(node-
507:, and
493:flutes
485:closed
409:where
252:where
180:pianos
115:violin
3544:Piano
3529:Sound
3343:pitch
3305:Pitch
3005:, by
2758:hertz
1957:k is
1879:Cones
1453:organ
1019:n = 4
983:n = 3
947:n = 2
909:n = 1
505:oboes
475:(see
199:of a
172:harps
168:lutes
119:flute
3519:Echo
3425:Node
3351:Beat
3341:and
3128:2019
3099:ISBN
3055:2015
3026:ISBN
2970:2015
2944:ISBN
2864:and
2777:72.6
2748:and
2712:0.85
2534:0.85
2487:0.75
2248:and
2241:and
2003:and
1458:node
704:and
548:node
519:and
489:open
477:bore
423:mass
109:use
3151:doi
3074:doi
2856:'s
2756:in
1920:tan
1818:0.3
1730:0.3
1681:0.3
1621:0.3
1564:at
1562:= 0
1551:= 0
1482:Phi
1471:By
1466:Phi
1405:0.6
1317:0.6
1268:0.6
1190:0.6
1065:By
721:at
719:= 0
694:at
692:= 0
660:cos
628:cos
623:max
580:= 0
483:or
471:or
454:or
83:).
3586::
3145:.
2978:^
2860:.
2275:,
1569:=
726:=
709:=
699:=
542:.
503:,
425:.
182:,
178:,
174:,
170:,
3216:e
3209:t
3202:v
3157:.
3153::
3130:.
3107:.
3082:.
3076::
3057:.
3034:.
2972:.
2791:3
2787:D
2782:d
2774:=
2771:f
2754:f
2750:D
2746:d
2723:3
2719:D
2715:)
2709:(
2706:8
2702:3
2690:d
2687:v
2681:=
2678:f
2665:L
2620:3
2616:D
2610:q
2607:e
2603:L
2599:8
2595:3
2583:d
2580:v
2574:=
2571:f
2537:d
2531:+
2528:L
2525:=
2520:q
2517:e
2513:L
2490:d
2484:+
2481:L
2478:=
2473:q
2470:e
2466:L
2437:q
2434:e
2430:L
2400:q
2397:e
2393:L
2387:0
2383:V
2378:A
2366:2
2362:v
2357:=
2354:f
2337:L
2333:A
2328:0
2326:V
2303:n
2283:m
2252:z
2250:L
2245:y
2243:L
2238:x
2236:L
2232:v
2211:2
2206:)
2199:z
2195:L
2191:n
2186:(
2181:+
2176:2
2171:)
2164:y
2160:L
2156:m
2151:(
2146:+
2141:2
2136:)
2129:x
2125:L
2116:(
2107:2
2104:v
2099:=
2096:f
2067:x
2063:L
2043:n
2037:)
2034:x
2031:+
2028:L
2025:(
2022:k
2009:x
2005:x
1988:v
1984:/
1980:f
1974:2
1971:=
1968:k
1938:x
1935:k
1927:1
1910:n
1907:=
1904:L
1901:k
1873:2
1861:1
1824:)
1821:d
1815:+
1812:L
1809:(
1806:4
1802:=
1778:v
1775:=
1768:f
1746:v
1743:=
1739:)
1736:)
1733:d
1727:+
1724:L
1721:(
1718:4
1715:(
1712:f
1687:)
1684:d
1678:+
1675:L
1672:(
1669:4
1665:v
1660:=
1657:f
1642:.
1627:)
1624:d
1618:+
1615:L
1612:(
1609:4
1604:v
1601:n
1595:=
1592:f
1571:L
1567:x
1544:p
1523:L
1520:4
1515:v
1512:n
1506:=
1503:f
1441:f
1437:r
1433:L
1429:v
1411:)
1408:r
1402:+
1399:L
1396:(
1393:2
1389:=
1365:v
1362:=
1355:f
1333:v
1330:=
1326:)
1323:)
1320:r
1314:+
1311:L
1308:(
1305:2
1302:(
1299:f
1274:)
1271:r
1265:+
1262:L
1259:(
1256:2
1252:v
1247:=
1244:f
1231:n
1216:r
1196:)
1193:r
1187:+
1184:L
1181:(
1178:2
1173:v
1170:n
1164:=
1161:f
1141:p
1139:Δ
1132:v
1128:L
1124:n
1104:L
1101:2
1096:v
1093:n
1087:=
1084:f
861:.
855:L
852:2
847:v
844:n
838:=
830:v
825:=
822:f
796:.
793:.
790:.
787:,
784:3
781:,
778:2
775:,
772:1
769:=
766:n
763:;
758:n
754:L
751:2
745:=
728:L
724:x
711:L
707:x
701:0
697:x
675:)
672:t
666:(
656:)
647:x
641:2
635:(
619:p
615:=
612:)
609:t
606:,
603:x
600:(
597:p
573:p
569:p
419:m
411:T
391:L
388:2
379:L
375:/
371:m
367:T
361:n
355:=
349:L
346:2
335:T
329:n
323:=
320:f
288:T
282:=
279:v
266:T
258:n
254:L
234:L
231:2
226:v
223:n
217:=
214:f
197:v
38:.
20:)
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