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Singing Bowls, a Guide to Healing through Sound

7. Harmonious Sound Waves

All music, based upon melody and rhythm is the earthly representative of heavenly music.

When I first heard the singing bowls, I perceived a harmonious blend of sounds. After a few weeks I began to distinguish the different tones in each bowl. These are the partials, and a trained musician is able to appreciate their richness immediately. Singing bowls are actually far more complex and rich in partials than anyone can distinguish completely. A computer with a microphone and a sound program helps us look a little deeper into the intriguing world of sound waves, showing how beautiful and complex the sound waves of the singing bowls really are.
To fully comprehend the sound analysis of singing bowls, an understanding of the acoustics of bells is needed. The ring of a bell, or singing bowl, is always focused near the rim, where the maximum elastic movement and resonance occur. To get the full potential of the sound it is necessary to strike a bowl on its rim. Each bowl has unique patterns of vibrations that produce a great number of sound frequencies. Visualize a tuning fork as a thin vertical slice of a bowl, then imagine the bowl as a set of tuning forks bound together. Although a singing bowl has no stem, like a tuning fork there is a node (point of no movement) near the base, with the wall of the bowl resonating like the prongs of a tuning fork. Both tuning forks and singing bowls have other intermediate nodal regions associated with higher frequencies, or partials. In a tuning fork the intensity of the partials is purposely subdued, while in a bowl or bell the series of frequencies along nodal regions between the rim and the node at the base is enhanced. Irregularities in the cutaway profile of a singing bowl are responsible for the character of the vibrations, determining not only the frequencies the partials but also their intensities. This explains the wide range in sound quality, timbre, and partials among the hand-crafted singing bowls. No two are alike.
One of the most obvious differences when comparing singing bowls to each other is the duration of the sound. Some have a long sustained ring, while others stop within a few seconds. The duration of the ring is dependent upon material, size and profile. The metal must be sufficiently hard. Hardness depends upon the proportion of tin in relation to copper. More than one percent of lead will weaken the alloy and have an adverse effect on the duration of the tone. Larger masses of metal are capable of more acoustical energy and will vibrate longer. Among my collection of singing bowls, the ones that vibrate the longest with the strongest sound have considerably more mass in relation to their size. The thick walls and rims produce strong, clear sounds with harmonious partials.
When the rim of a singing bowl is hit by a striker, the bowl’s wall is momentarily forced out of the round into a more elliptic shape. The distortion sets the small tuning-fork-like cross-sectional areas into vibration. In the region of impact, they are forced outward, while one-quarter around the bowl’s periphery from the point of strike the motion responds inwardly. The alternating inward and outward movement produces the fundamental pitch and the hum tone. The result is four nodal points on the circumference which produce four equidistant vertical nodal lines, or meridians, from the rim to the nodal point at the base of the bowl. Further nodal meridians are formed because the bowl’s initial deformation is not a pure ellipse. When a bowl is partially filled with water and struck, the sound waves originating from these four meridians can clearly be seen rippling over the surface. When you strike the bowl (with a padded striker, you don’t want to break the bowl!) a little harder, the elastic movement of the bowl’s wall will cause a fountain of water drops to shoot up and out of the bowl.
What tones are generated when a bowl is struck? Upon impact, the strike tone dominates momentarily. It wavers and cannot be measured as a specific frequency. Below the fundamental is the hum tone, vibrating the longest as it fades into infinity. The partials of higher frequencies are of a shorter duration. In contrast to singing bowls, bells are tuned to specific partials, mainly a minor third, fifth, and octave. Because the acoustics of bells is complex, tuning is essential to the harmonious blending of tones when two or more bells are heard simultaneously. Because singing bowls are not tuned, combinations of two bowls will not always sound right. However, some will produce interesting and pleasing effects when played together.
With standard bells the frequencies of the first six or seven partials adhere to a strict harmonious relationship to the fundamental. The upper partials can be inharmonious because the vibrating segments of the bell, where these upper partials are produced, are shorter. In sound analysis, this is even more so with singing bowls. Only the first and second partials have a near harmonious relationship to the fundamental. The rest of the upper partials are inharmonious. Irregularities in the shape of a bowl also contributes to this considerably.
After a singing bowl has been struck tonal decay sets in as the energy is dissipated. The partials fade at different rates. This shows up clearly in spectral analysis. Higher frequencies dampen more rapidly than lower ones. Tonal decay undergoes a subtle and continuing transformation which varies according to the size and profile of the individual bowl. Spectral analysis also shows the cyclic pulsation (periodic rising and falling in intensity) of the partials. The intensity of some partials pulsates strongly (a couple of times a second), while the others are almost constant. Where does this flux come from? In a good bell perfect symmetry produces an even ring. Occasionally the ring of a bell is uneven, creating beats or pulsation. The causes of this are in the casting: lack of symmetry, irregularities in thickness, and the composition or homogeneity of the metal. This all occurs frequently with singing bowls. The hammering process gives an asymmetrical shape despite the general overall bowl shaped appearance. With hammer blows all over the surface, the thickness of the metal varies over the entire bowl. The homogeneity of the metal mixture is not ideal; some bowls even show areas of pure copper which turn green from oxidation. Some bowls have engraved inscriptions which distort the symmetry of the bell, changing the sound.
Are imperfections undesirable? If not too dominating, irregular pulsation contributes to the beneficial effects that singing bowls have on people. Pulsating sound waves cause hypnotic and trance states. Like listening to the waves of the ocean or to the heart beat of our mother they make us feel good. They have a soothing and calming effect. Studies have shown that vibrations from rhythmic sounds have a profound effect on brain activity. Drumming, rhythmic singing, and movement can transport a person into other realms of reality. Shamans use these methods for healing, but it is not exclusively their realm, people from all over the world use the same techniques to go into trance states.
I have done a sound analysis of some of my singing bowls. The computer generated diagrams show the patterns of sound waves when of all the tones are sounding at the same time. Each position on the diagram shows all the tones at that moment, each diagram shows a series of moments in time. As each tone has a different frequency and amplitude, tones will strengthen or cancel each other out as they go along. This produces many different patterns, which show up when we scale up the diagram. It is like a microscope, getting a bigger look at a smaller moment of time. Each magnification reveals more detail and more patterns. Each bowl is identified by a name I have given to it.

Diagram 1 is the wave patterns of the Deva bowl. The diagram length is 10 seconds. Groups of sound patterns rise and fall within other groups of sound patterns. All are harmoniously structured in sine waves, rising and falling as they travel across time.

Diagram 2 has a length of 0.81 seconds and shows a close-up of the sound patterns of the Sun bowl while it is being rubbed with a hardwood stick to accentuate the partials. Here the wave pattern is almost circular in form.

Diagram 3 shows the sound waves of the Diamond Ring bowl as it is rubbed with a hardwood stick. This produces a fluent sound pattern. The diagram length is 1.37 seconds.

Diagram 1, 2 and 3


These diagrams show how complex, well structured, and harmonious sound waves produced by good singing bowls can be. When a bowl is struck its sound waves emerge all together, creating an intriguing blend of composite waves and "wave packages".
This analysis of the singing bowls according to their frequencies yielded interesting results. Of prime importance is the fact that singing bowls do not emit well defined frequencies. The fundamental and its partials fluctuate around a central frequency which varies slightly each time the bowl is hit. A sample bowl produced a fundamental fluctuating between 269 and 271 Hz. The first partial fluctuated between 770 and 775 Hz. The third partial was between 1428 Hz and 1436 Hz. The first hit produced central frequencies of 270 Hz, 773 Hz, and 1432 Hz. The next hit will not produce exactly the same results. It might give central frequencies of 265 Hz, 766 Hz and 1402 Hz. The slightly different results depend on how the bowl is struck: the force, the angle, and the kind of striker. Nevertheless, each bowl has its own characteristic fundamental and partials stretching out over three to four octaves. Usually the first and second partial are close being to exact multiples of the fundamental, but the rest of the partials can deviate considerably from their respective places with the shape and quality of the bowls contributing to the distortion.
Some tones pulsate greatly in intensity, while others are very stable in their output. The fundamental tone is not always the strongest, usually the first or second partial dominates. Thin bowls and big bowls tend to have many partials (see diagram 4 and 5).

diagram 4 and 5

Small thick bowls, especially those with thick rims, have very few partials (see diagram 6).

diagram 6 and 7

The fundamentals of small thick bowls are two to three octaves higher than big bowls. When they are being rubbed, big and middle sized bowls produce three to five of the partials of the bowl. In general these larger bowls do not make a good sound when rubbed. Instead they give a scraping, metallic sound. Rubbing a small, thick bowl produces one clear tone. It can be the fundamental or the first or second partial, depending upon the bowl. These tones will be intense but pleasant to the ear. Like bells, singing bowls also have a weak hum tone one third of an octave lower than the fundamental.

Table 1 shows the Butterfly bowl, a middle sized thin bowl (9 ¼" wide, 4" high). It has a hum tone of 126 Hz. The fundamental is 328 Hz, close to E in our musical scale. All tones except one fluctuate widely in intensity. This is the first partial of 642 Hz; not only is it stable, it is the strongest in intensity. The bowl has a total of eight partials. (Partials over 3000 Hz are not strong enough to be heard or recorded).

Table 1


The Deva bowl is 8 ½" wide, 2 ½" high. It has a fundamental of 699 Hz (an F in the musical scale) with only two partials. This is one of my favorite bowls for private sessions. When rubbed it produces only the fundamental.

The Diamond Ring bowl (7 ¾" wide, 2 ½" high) is similar to the Deva bowl, but when rubbed the first partial of 1347 Hz is the only tone generated. The first partial of 1347 Hz is both the strongest tone when the bowl is hit, and the only tone generated when rubbed (see diagram 6 and 7).

The Sunset bowl, a middle sized thin bowl (9" wide, 3 ¾" high), does not allow a single tone to be generated when rubbed. Instead a series of four partials is produced.

Sometimes when two Singing Bowls, are played together, a "floating" sound effect is created. This happens when the tones of the two bowls are similar. The Sun bowl and Mercury bowl are a nice example. The frequencies of the Sun bowl are:
158, 427, 818, 1289, 1759, 2020, 2384 and 2560 Hz;
those of the Mercury bowl are:
129, 410, 780, 1249, 1782, 2030 and 2391 Hz.
The frequencies of my Fire bowl and Tree bowl are even closer together and thus create an especially dissonant sound effect.
The Fire bowl has 130, 379, 730, 1161, 1680 and 2262 Hz.
The Tree bowl has 128, 368, 721, 1162, 1678 and 2234 Hz.
The sound frequencies of singing bowls are more complex than they appear. They display interesting patterns of fundamental tones and partials resulting in unique musical instruments which, although not tuned, allow for an interesting sound display.

copyright 2001 by Dirk Gillabel