The teK Project centres on the importance of musical temperament, something almost unknown today. One reason why it remains so obscure is because it is difficult to investigate without either the right equipment or a lengthy period of training. However, modern computers make it possible to investigate the basics very easily, although putting to use what is learned still takes considerable time and effort. As with all true knowledge, one only receives reward in proportion to the effort one puts forth.
This page contains links to sixteen sound files and two zip files. The sound files can be played across the Internet by any computer with a sound-enabled browser. The zip files can be downloaded and used offline with any computer able to play .WAV files, and each contains eight of the sound files. The purpose of each of the files is explained in Table 4 below.
The following explanation constitutes a brief introduction to sound waveforms, musical intervals, and beats. A more detailed investigation of these topics will be provided in the second book of the teK Project, but the material provided here demonstrates several important concepts.
|Figure 1 A pure sinewave representing sound.|
The first concept to be understood is that of a sinewave (Figure 1). This elementary waveform will be familiar to all who completed high-school mathematics, but few understand just why it is so elementary and important. When representing sound, the horizontal axis of the graph denotes time and the vertical axis air-pressure, since sound waves are regular variations in the pressure of the air surrounding us.
|Figure 2 A sinewave is the 2D representation of a spiral.|
Figure 2 shows the reason why sinewaves are so fundamental in Nature: a sinewave is the two-dimensional representation of a spiral, that is, of an object that is both rotating around an axis and moving along it. The Moon rotates around the Earth, but the Earth also orbits the Sun, and the Sun itself is in motion about the galactic centre. This type of motion is therefore truly universal, and a familiarity with sinewaves is essential to understanding the natural world.
|Figure 3 The simplest sound has a sine waveform.|
A sinewave is also the waveform generated by any object oscillating or vibrating in the simplest possible way (Figure 3). If a vibrating string has a very light pen attached to its mid-point, and a sheet of paper is drawn rapidly past it, the shape traced out by the pen will be a sinewave.
|Figure 4 The left and right waveforms of the first .WAV file.|
Soundfile #1: 256-256-s-5.wav
A sound having the waveform of a sinewave gives a pure, bland tone. If your browser is sound-enabled, click the link above to hear it, or unzip the sine.zip file above and play the file with the same name. This file plays two identical notes through the left and right stereo channels. Each note is a pure sinewave with a frequency of 256Hz, and the notes are in phase - that is, their oscillations increase and decrease simultaneously.
|Left channel frequency||Right channel frequency||Waveform||Duration|
|Filename:||256||256||s or r||5|
|Meaning:||256Hz||256Hz||Sine or ramp||5 seconds|
|Table 1 Naming of the soundfiles.|
|Figure 5 Two notes with an interval of an octave.|
Soundfile #2: 256-512-s-5.wav
Figure 5 introduces one of the most important concepts in all of music - the interval of an Octave. The word comes from the Italian for eight, since there are eight notes in the modern scale. What is most important, however, is that two notes with the interval of an octave have frequency ratios of exactly 2:1 - that is, one has a frequency twice or one half of the other. The graph shows this clearly, with two cycles of the lower note occupying the same time as one cycle of the upper note. Soundfile #2 plays five seconds of two notes with frequencies of 256Hz and 512Hz.
|Figure 6 Two notes with an interval of a Perfect Fifth.|
Soundfile #3: 256-384-s-5.wav
The next most important musical interval after the Octave is the Perfect Fifth, so-called because it can be played using the first and fifth notes of the modern scale. A Perfect Fifth is the interval between two notes with a frequency ratio of 3:2. In Figure 6 it can be seen that there are two "humps" above the line for the note above, and three "humps" for the note below. The soundfile has notes with frequencies of 256Hz and (256 x 3/2 =) 384Hz. This interval has a very consonant sound, that is, it sounds pleasing and harmonious to our ears, and is the basis of all early and traditional music.
|Perfect Fifth||3:2||Less consonant|
|Minor Tone||10:9||More dissonant|
|Minor Semitone||17:16||Most dissonant|
|Table 2 The most important musical intervals.|
An important aspect of the musical art is the creative use of consonance and dissonance. A piece of music may begin with consonant sounds, move into strong dissonances, and then resolve the dissonance into a new consonance. This reflects many aspects of our emotional lives, since we all pass through periods of peace and harmony alternating with distress and uncertainty. The intervals in the table are not the only ones used. Many traditional musical styles use non-integer intervals that are strongly dissonant, but these usually have meaning only for those familiar with the tradition, and can sound harsh and unpleasant for those who have never heard them before. Some of these traditional intervals are contained in the notes of scales, others are generated by certain instruments. The Indonesian gamelan, for example, generates an intense and complex array of dissonances very strange to those unaccustomed to them, but strongly evocative of excitement and tension. They are thus very effective in generating appropriate moods for story-telling and legends.
|Figure 7 Two notes with a small frequency difference.|
Soundfile #4: 256-252-s-5.wav
It may be thought that if two notes are sounded with frequencies very close together, the resulting sound will be almost the same as a single note. Surprisingly, this is certainly not true. The reason has to do with beats, one of the most important considerations in tempering. The soundfile contains two notes with frequencies of 256Hz and 252Hz, a difference of less than 2%, but as playing it shows, the resulting sound varies in amplitude or volume (256 - 252 =) 4 times per second. This is true of any two notes sounded together - the difference in their frequencies will produce beat tones which can alter the sound quite markedly. The graph shows this phenomenon very clearly, as also that the resulting sound can be graphed by summing the amplitudes (distance along the Y-axis) of each note at each instant of time. Beat notes can be very useful at some times, as when tuning a guitar or other instrument, and most disconcerting at others. All who have flown in a multi-engined aircraft will remember the sound during take-off, when each engine is running at a slightly different speed. Once altitude is gained, the pilot trims the throttles to set each engine to the same speed, and the beats disappear.
|Delta||< 3Hz||Delta tends to be the highest in amplitude and lowest in frequency. It is quite normal, is the dominant rhythm in infants up to one year old, and in stages 3 and 4 of sleep.|
|Theta||3.5Hz - 7.5Hz||Theta is classed as "slow" activity. It is abnormal in awake adults, but perfectly normal in sleep and in children up to 13 years.|
|Alpha||7Hz - 13Hz||Brought out by closing the eyes and by relaxation, and abolished by eye opening or alerting by any mechanism (e.g. thinking, calculating). It is the major rhythm seen in normal relaxed adults and is present during most of life, especially beyond the thirteenth year when it dominates the resting pattern.|
|Beta||> 14Hz||Beta activity is 'fast' activity, and is accentuated by sedative-hypnotic drugs. It is generally regarded as a normal rhythm and is dominant in people who are alert or anxious, or who have their eyes open.|
|Table 3 The principal types of brainwaves in humans.|
One reason why beats are important when using music for meditation and similar purposes is that the beat frequencies generated by modern scales are in the same range as electrical waves in the human brain. This can be both desirable and undesirable. For the past few decades it has been used to generate altered states of consciousness by using headphones to play different notes in each ear. The brain will attempt to synthesize the frequency difference, resulting in the excitation of brainwaves at the beat frequency. This so-called binaural beat technique is the basis of many "meditation" tapes, and of training methods in such things as Remote Viewing and other psi techniques. To quote from a relevant paper:
"In the late 1950's and early 1960's research into the EEG effects of meditation began to reveal that the alpha rhythm appears different during meditation and may undergo long-term changes in persistent meditators. Anand, Chhina, and Singh (1961) reported that the EEG of meditators showed a high amplitude slowed alpha rhythm which gradually spread from the occipital to the frontal areas. Banquet (1973) also found high amplitude alpha rhythms during meditation. Additionally, Banquet noted a second stage of meditation in which theta frequencies appeared and moved from frontal to posterior channels. A third stage, which Banquet observed in only the most experienced meditators, was characterized by high-frequency beta waves over the whole scalp. Banquet also noted that during meditation alpha blocking did not occur to low intensity light and sound stimulation. Empson (1986) summarizes the recent research on meditation and concludes that the experience of meditation 'requires the constant maintenance of a fairly low level of arousal which allows the sort of dissociated, free-associative thinking that meditation entails'. The low-frequency, high-amplitude alpha rhythms generally found during meditation thus seem to represent a voluntary lowering of arousal by the meditator.
"These findings concerning the EEG activity of meditators sparked increased interest in the meanings of these rhythms and how to control them. Stewart (1974) observes that the interest in alpha brain wave biofeedback training appears to have originated from EEG monitoring of Zen and Yoga practitioners. The perceived link between meditation and alpha production influenced many to assume that increased alpha production would result in the ability to reap the benefits of meditation. This assumption has been a driving force behind the interest in alpha biofeedback training. However, over two decades of research into alpha biofeedback training indicates that this assumption is at best simplistic.
"Alpha biofeedback training was first introduced by Kamiya in 1962 (Kamiya, 1969) when he demonstrated that subjects who were required to guess whether or not alpha was present in their EEGs and were subsequently informed of their accuracy, could, within a few hours, correctly identify when they were producing alpha with high accuracy. He also found that those subjects who were successful in discrimination training could also produce or suppress alpha activity at will. He later successfully utilized auditory alpha-biofeedback devices which informed subjects of their alpha production through the presentation or absence of a tone generated by their alpha rhythms (Nowlis & Kamiya, 1970). The mental states which Kamiya's subjects associated with increased alpha production were reported to be feelings of relaxation, 'letting go,' and pleasant affect."
Traditional states of deep meditation are obtained by inner discipline rather than outer stimulation, which is why the simple assumption that binaural beats can serve the same purpose are "simplistic", as reported above. If music is used as the focus of meditation, it is desirable that it contain as few beats as possible in order to generate a sense of timelessness and unending stillness. In many ways this is the reverse of the above-mentioned simplistic assumption, the music serving to establish a repetitive "ground" or baseline from which consciousness can extract its own interpretations, rather than acting as a direct stimulant of frequencies which may not be in accord with other factors. Because modern music uses the equal-tempered scale it contains strong beat frequencies that do not create the same feelings of inner peace as can traditional styles.
The first book of the teK Project, Earthsong, explains the historical and scientific basis of these topics, and a forthcoming book will provide greater technical detail and practical exercizes for those interested in experimenting with sound, music, meditation, and altered states of consciousness.
The remaining soundfiles can be used for exploring these ideas further. Two versions of each of eight files are available, one using sinewaves and the other ramp (or sawtooth) waves. All except the 256-256 files have different frequencies in left and right channels. Their uses are given in Table 4. See the page on Musical Temperament for more information.
|Demonstrates beats between two notes close in frequency.|
|Basic sine and ramp waveforms, identical left and right.|
|The interval of a Major Tone, in which beats can be heard distinctly.|
|A pure Major Third, a ratio of 5:4.|
|An approximate equal-tempered Major Third, with beats just discernible in the ramp-wave version.|
|A pure Perfect Fifth, a ratio of 3:2.|
|A sharpened Perfect Fifth, with beats quite evident in the ramp-wave version.|
|A pure Octave, a ratio of 2:1.|
|Table 4 Soundfiles available on this page.|