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THE 22 SHRUTI CHAKRA The 22 note wheel The diagram below is a very simplified version of a drawing by Jacques Dudon which demonstrates how the modern equally tempered musical scale is derived from selected absolute harmonics. The original drawing includes more layers of arcs inside the circle. While the intervals of the fifth and fourth are very close in both tunings, other intervals in an equally tempered scale can be seen to deviate to some extent from their ‘ideal’ ratios. This diagram shows the closest simple ratios with factors of 2 and 3 in red, and ratios with factors of 2, 3 and 5 in blue. The equally tempered intervals are shown in green. The construction is based on taking the logarithm to base 2 of the ratios, and using this to find the angle at which each ratio plots. An octave (double the frequency) thus makes one turn. So, for example, the fifth in a justly intoned tuning makes an angle of 360 × log 2 (3/2) = 210.5865º (clockwise from 12 o’clock), while the fifth in an equally tempered tuning makes an angle of 360 × 7/12 = 210.0º. Jacques Dudon is one of a group of French musicians called DEDALUS who have an online magazine, microMégas. How to tune your fixed pitch instrument depends largely on how many keys it will have to play in. A wind-chime would be tuned best to just intervals. A set of steel ‘pans’ would probably be tuned using equal temperament, but I'd be interested to hear a justly intoned steel band. Ulf Kronmen would know if this is possible. These extracts are from the introduction to the Just Intonation Primer, by David B. Doty available from The Just Intonation Network. Throughout the middle ages, Western music was theoretically based on what is called Pythagorean intonation, a subset of Just Intonation based on ratios composed only of multiples of two and three. Pythagorean tuning is characterized by consonant octaves, perfect fourths and perfect fifths. Eventually, theorists were forced to partially abandon the Pythagorean framework of the middle ages in order to explain the existence of consonant thirds and sixths, because the most consonant possible thirds and sixths are based on ratios involving five, first mentioned by the English monk Walter Odington (c.1300). In general, music theorists of the Italian Renaissance came to agree with the proposition of the Venetian Gioseffe Zarlino (1517-1590) that consonace was the product of ratios of the integers one through six (the so-called senario). Fixed pitch fretted and keyboard instruments. A fixed pitch instrument intended to play in perfect Just Intonation in more than a few closely related keys requires far more than twelve notes per octave, an arrangement that had already become a standard by the fifteenth century. Some instruments with more than twelve notes per octave were built, but the mainstream adopted a different solution. Many different schemes of temperament were proposed, they eventually coalesced into a type of tuning known as meantone temperament. In England, meantone continued to be the predominant tuning, at least for organs, until the middle of the nineteenth century. The practical adoption of equal temperament, like its invention, was a gradual process, occuring at different rates in different countries. It apparently did not strike most of the theorists of the seventeenth and eighteenth centuries as problematic that, although they formed the basis for the whole of contemporary harmonic practice, simple-ratio intervals were gradually being purged from musical practice in favour of tempered aproximations. By substituting 12 equally-spaced tones for a universe of subtle intervallic relationships [they] effectively painted western music into a corner from which it has not yet succeded in extricating itself. The availability of electronic instruments with programmable tuning means a conventional keyboard can now be used to play a virtually unlimited number of different pitches. Changing the pitches available on a digital synthesizer simply means changing the data values in a tuning table, or switching to a different table. I just like the blues . . . Clyde plays electric bass J.J. Cale           ...and on that note, have you tried playing blues harmonica? It’s not an easy skill to learn, you might like to try the ocarina instead! These two instruments are played together in Frank Capra’s film ‘Meet John Doe’ (1941) starring Gary Cooper. OCARINA – an ancient vessel or globular flute now becoming better known especially since the recent Zelda series of games from Nintendo. ( I thought Zelda was the Martian witch in Gerry Anderson’s Terrahawks! ) Ocarinas are closed vessels unlike the open tube of a flageolet (tin whistle) or most other wind instruments. This means that they have a limited range, usually just over an octave, because they cannot be played in higher registers. Also, since the pitch produced depends mainly on the area of the holes that are covered or uncovered, rather than the distance of the holes from the mouthpiece, they can be made in many different designs, hole patterns and fingerings. The pitch is also more dependent on the breath strength, particularly the lower notes, so practice your scales from the top. They have a beautiful tone, like the bird-calls some people can make with their hands. You may remember the theme from ‘The Good, the Bad and the Ugly’. The name ‘Ocarina’ is derived from the Italian ’Oca‘, little goose; the French call them ‘cou-cou’ and the Chinese ‘hsan’, ‘Xun’ or ‘tschuan’. Stone-age examples have been found in Africa, and they are traditional instruments in Central and South America. Brought to Europe by the Spanish in the early 16th century, they became popular toys until in the mid 19th century Guiseppe Donati in Budrio started making ocarinas which could accurately play more than an octave. The ocarina then gradually started to be accepted as a true musical instrument. In the 1960s the 4-hole ‘English style’ or cross-finger ocarina was developed by John Taylor. The 6-Hole English style ocarina has another two thumb-holes underneath. This gives a better tuning to the top Do, an accurate bottom Do# and Re#, and an extra top Do# and Re. See the typical fingering below, the lower two holes are the thumb-holes underneath, and the top four holes are played with the first and second fingers. Low Ti is achieved, with practice, by ‘shading’ the whistle hole with the nose. Ti Do Re Me Fa So La Ti Do Re         Do# Re#   Fa# So# La#   Do#   6-hole ocarinas are made by John Langley of Langley Ocarinas (among others) in a range of three D sizes and three G sizes. Suitable for even quite small children, these are available from The Ocarina Workshop who also publish a set of four books, ‘Play Your Ocarina’. There is an online product list, but the ordering system at this site is in Japanese. Your local music store may stock or be able to order them, possibly cheaper. Hands Music Centre of Kingston-on-Thames, Tel. +44 (0) 20 8546 9156, have more information. Ocarina sites – info and sales. www.songbirdocarina.com     Zelda songs & an ocarina for the games. www.ocarina.demon.co.uk     Lots of history. www.potterymaking.org     Other links. Ocarina sites – sales. Anita Feng www.hollowtop.com     This is a large site dedicated to outdoor living. www.villagecraftsmen.com   also has other traditional instruments and craftwork.

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