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Creating The Pythagorean Scale We are told that Pytharoras experimented with an device called the Monochord (which literally means “one string”) by his student Philolaus. This was a single stringed instrument with a moveable bridge and by positioning the bridge in different positions it was possible to play different notes on the string. His reported aim in his analysis of the vibrations of the strings was to define the “music of the spheres” and was thus an attempt to understand the heavens. He, along with other Greeks, were of the opinion that the heavenly bodies moved in a form of musical precision and that by analysing music one gained an insight into the movement of the heavens. As early as Ancient Babylon, mathemeticians believed that the heavens were governed by ratios of integers and it is perhaps from here that Pythagoras found the inspiration to experiment with the instrument. Whatever the inspiration, there is little evidence that any theory was used in the tuning of musical scales prior to the life of Pythagoras. Conversely, the start of music theory being important to the tuning of instruments almost certainly begins with the words of Philolaus. It is likely that prior to the advent of the Pythagorean ratios that musical scales were very varied and perhaps even unique to the individual. By using at least two monochords, Pythagoras was trying to measure the lengths at which the notes from the strings would ring together in what could be considered “perfect” harmony. It is highly likely that he believed that these notes would be found at whole number ratios along the length of a string. It is also quite likely that he was happy to find that this appeared to be the case. Through his experimentation, he discovered that strings at a length ratio of 2:1 provided a consonant sound, an interval the Greeks called a diapason (dia means across, between or through in Greek). The second consonance was found at a length ratio of 3:2, which became named the diapente. The final consonance was found at a length ratio of 4:3 and was called a diatessaron. These ratios, when stated together, formed the ratio 1:2:3:4 which will no doubt have reinforced Pythagoras’ beliefs. 2 Music theory in Ancient Greece was based around the Tetrachord (in Greek, tetra means four). The four notes were tuned to notes in a descending order. The first and fourth notes were separated by the interval of a diatessaron. The two other strings were tuned to one of a number of intervals, the size of which depended on the musical scale. It was soon noticed that the two intervals 3:2 and 4:3 could be multipled together to become the 2:1 ratio. For example: 3 2 x x 4 3 = = 12 6 = = 2 1 It was also noted that the interval between the notes could be found by dividing one by the other. This interval was known as the Whole Tone and is defined as the ratio 9:8. The calculation would be: 3 2 x x 3 4 = = 9 8 This definition of a tone persists for centuries and does not fully disappear from music theory until as late as the eighteenth century. When two tetrachords are placed within an octave and separated by a tone the resulting scale is known as a Diatonic scale. The two tetrachords in a diatonic scale are often called diatonic tetrachords. It is widely believed that Pythagoras constructed a tuning system which he based on the interval ratios that he discoved using the monchord. The exact method he used for achieving this is a matter for debate between music historians, but the belief most commonly held is that the intervals are calculated using the 3:2 ratio. As the note found at two thirds of the string length is consonant with the note found at the full length, then the note found by a further shortening of the string is consonant with that found at the two thirds point. By extending the string by a third, we find the consonant note suggested by the 4:3 ratio. If this longer string is then extended by a third of its new length the resulting note is consonant. Chapter 2: A Condensed History We can continue the process of both shortening and lengthening the string by the 2:3 ratio to find a series of steps which denote musically related notes. Mathematics at the time saw no reason to go further than a cubed number as only three dimensions were visible to the naked eye. Due to this opinion, Pythagoras probably achieved his seven note scale by multiplying the ratios by themselves to create a squared and a cubed ratio - thus making 3 powers of the ratio 3:2 in each direction. For the 2:3 , the squared ratio is: 2 3 x x 2 3 = = 4 9 2 3 = = 8 27 And the cubed ratio is: 4 9 x x For the 3:2, the squared ratio is: 3 2 x x 3 2 = = 9 4 3 2 = = 27 8 And the cubed ratio is: 9 4 x x These fractions describe a set of notes which span from under half the length of the original string to almost four times its length. Using the 2:1 ratio to shorten or lengthen the string, we can place all of the calculated notes in the same range. The range we chose is between the original length of the string and a point described by the 2:1 ratio itself. i.e. half the original string length. So, the calculation for the first power of the 3:2 ratio would be: 3 2 x x 1 2 = = 3 4 This gives us the third of the “perfect” intervals that found by the 3:4 ratio - and this is within the range we require. When the result is still outside the range the 2:1 ratio is applied a second time Metal In Theory Ratio (3 : 2) ^ (3 : 2) ^ (3 : 2) ^ 1:1 (2 : 3) ^ (2 : 3) ^ (2 : 3) ^ 3 2 1 1 2 3 Calculation 27 9 3 1 2 4 8 / / / / / / / 8 4 2 1 3 9 27 Adjustment 1:4 1:4 1:2 1:1 1:1 2:1 2:1 Adjusted Ratio 27 : 32 9 : 16 3:4 1:1 2:3 8:9 16 : 27 Fig 2.01 : Pythagorean Series using the 3:2 ratio Fig 2.01 shows the calculations, the adjustment ratios and the adjusted lengths of a string. Seven was an especially important number to the Greeks as it denoted the number of heavenly bodies, not including stars, that they were aware of. These were the Sun, the Moon, the Earth, Mercury, Venus, Mars and Jupiter. To Pythgoras, the idea that his newly defined scale had seven notes was very appealing and this fact was perhaps the reason that the scale became an accepted part of music theory. Because there are seven notes, the scale is referred to as a heptatonic scale - hepta means seven and tonos means tone. Additionally, the eighth note when ascending through the scale is defined by the 2:1 ratio and is called the octave (octa means eight). Now, despite our extensive use of a twelve note scale, the interval between two notes at a 2:1 ratio is known by a name which has its origin in a seven note scale which is more than 2,000 years old. When the seven calculated string lengths are reordered from longest to shortest, they define what is referred to as the Pythagorean Heptatonic scale (see Fig 2.02). It stands as the first scale built on a basis in mathematics and as such begins a process of mathematical analysis in music that lasts to this day. Ratio (2 / 3) (2 / 3) (3 / 2) (3 / 2) (2 / 3) (2 / 3) (3 / 2) (1 / 2) ^ ^ ^ ^ ^ ^ ^ ^ 0 2 3 1 1 3 2 1 Calculated Ratio 1 8 27 3 2 16 9 1 / / / / / / / / 1 9 32 4 3 27 16 2 Decimal 1.0000 0.8889 0.8438 0.7500 0.6667 0.5926 0.5625 0.5000 Ratio to Next 8/9 243 / 256 8/9 8/9 8/9 243 / 256 8/9 8/9 Roman Letter D C B A G F E D Fig 2.02 : Pythagorean Heptatonic Scale values in order 3 It can be seen in Fig 2.02 that many of the intervals between notes are described as a ratio of 8:9, which we have already seen was defined as a Whole Tone. In this table is another ratio of a special significance - the ratio 243:256 - known as the Semitone. We can see how this figure is arived at by divining the ratio between the second and third degrees of the scale. 27 32 / / 8 9 = = 27 32 x x 9 8 = = 243 256 The letters assigned to the notes in Fig 2.02 are at first counter-intuitive to the modern musician - the letters seem to be the mirror image of what would be expected in a modern scale. This is because the Ancient Greeks listed their scales in descending order as opposed to the ascending order which is commonly used today. The letters themselves are actually derived from Roman music theory, although the Romans used fifteen letters to describe their scale to the modern system’s seven. Whilst the mathematical theory is of importance to the modern musician, a full scale analysis of the Greek theory system lies beyond the scope of this book. That said, a short detour into the outlines of the system is worthwhile as the concepts which drive it also drive the theory behind the modern musical system defined fifteen hundred years later. Tetrachord Hyperbolaiôn Diezeugmenôn Mesôn Hypatôn Roman Note P O N M L K I H G F E D C B A Greek Note Nêtê Paranêtê Tritê Nêtê Paranêtê Tritê Paramesê Mesê Likhanos Parhypatê Hypatê Likhanos Parhypatê Hypatê Proslambanomenos Planet Saturn Jupiter Mars Sun Venus Mercury Moon - Saturn Jupiter Mars Sun Venus Mercury Moon Fig 2.03: The Greater Perfect System of Ancient Greece 4 The Greater Perfect System (Systêma Teleion Meizon) was a Greek scale that was built on a set of four stacked tetrachords called the Hypatôn, Mesôn, Diezeugmenôn and Hyperbolaiôn tetrachords. Each of these tetrachords contains the two fixed notes that bound it. Fig 2.03 shows these notes in the darker shade of grey, whilst the tetrachords are highlighted in a lighter shade. The cousin of the Greater Perfect, the Lesser Perfect System, was built on three stacked tetrachords - the Hypatôn, Mesôn and Synêmenôn. The first two of these are the same as the first two tetrachords of the Greater Perfect, whilst the third tetrachord is placed above the Mesôn. When viewed together, with the Synêmenôn tetrachord placed between the Mesôn and Diezeugmenôn tatreachords, they make up the Immutable System (Systêma Ametabolon) which is also referred to as the Unmodulating System. Returning to the Pythagorean mathematics behind the system, we can further analyse the intervals between any two notes in the scale. The results of which are found in ratio form in Fig 2.04. The titles of the columns indicate the note from which the interval should be measured whilst the titles of the rows indicate the target notes. The ratios are stated in the form: Target Length : Initial Length We can see from the table that all but one of the five note intervals are defined by the ratio 2:3 (a true harmonic fifth). The remaining interval between the notes F and B can be found by analysis to be one semitone smaller than a harmonic fifth. Fig 2.04 also shows us that all but one four note interval is defined by the ratio 3:4 (a true harmonic fourth). The exception to the rule is that found between B and F which is found to be a semitone larger than the harmonic fourth. These two exceptions when considered together should, like the harmonic fourth and fifth, define the ratio of an octave (1:2). If we analyse this we find: 729 1024 * * 512 729 = = 1 2 Chapter 2: A Condensed History D 1:1 8:9 27 : 32 3:4 2:3 16 : 27 9 : 16 1:2 D C B A G F E D C 9 : 16 1:1 243 : 256 27 : 32 3:4 2:3 81 : 128 9 : 16 B 16 : 27 128 : 243 1:1 8:9 64 : 81 512 : 729 2:3 16 : 27 A 2:3 16 : 27 9 : 16 1:1 8:9 64 : 81 3:4 2:3 G 3:4 2:3 81 : 128 9 : 16 1:1 8:9 27 : 32 3:4 F 27 : 32 3:4 729 : 1024 81 : 128 9 : 16 1:1 243 : 256 27 : 32 E 8:9 64 : 81 3:4 2:3 16 : 27 128 : 243 1:1 8:9 D 1:2 8:9 27 : 32 3:4 2:3 16 : 27 9 : 16 1:1 Fig 2.04: Pythagorean Heptatonic Scale intervals When we consider the difference between the two we would expect to find them identical - a fifth minus a semitone should equal a fourth plus the same semitone. What we find is somewhat different: 729 / 1024 / 512 729 = 729 x = 1024 x 729 512 = = 531441 524288 This ratio describes an interval fractionally over an octave and demonstrates a fault in Pythagorean mathematics. Known as the Pythagorean Comma, it was the focus of much analysis over the next fifteen hundred years as mathematically inclined theorists attempted to solve the problem it defined. Intervals between two consecutive notes all measure either a tone (8:9) and a semitone (243:256). Three note intervals also have two possible ratios, of which the larger is two tones (64:81) and is referred to as a Ditone in Greek theory. The smaller of the two three note intervals in a tone and a semitone (27:32). The four note interval found between the B and the F (512:729) is three tones and as such is known as a Tritone, as is the interval between the notes F and B (729:1024). Because there are two possible tritones, the interval was avoided. Ratio (2 / 3) (2 / 3) (3 / 2) (2 / 3) (3 / 2) (1 / 2) ^ ^ ^ ^ ^ ^ 0 2 1 1 2 1 Calculated Ratio 1 9 4 3 16 2 / / / / / / 1 8 3 2 9 1 Decimal 1.0000 1.1250 1.3333 1.5000 1.7778 2.0000 Ratio to Next 9/8 32 / 27 9/8 32 / 27 9/8 9/8 Roman Letter Fig 2.05: Pythagorean Pentatonic Scale values in order Metal In Theory D E G A C D The six note intervals in the table can be found in two forms - four tones (81:128) or four tones plus a semitone (16:27). Finally, the seven note intervals also appear in two forms - five tones (9:16) or five tones plus a semitone (128:243). If the last step of both the ascending and descending series is ignored, the scale generated contains five notes. This is a Pentatonic scale and can be found in many styles of music, not least far eastern (Chinese, Japanese et al.) and European folk music. The Pentatonic version of the scale, shown in Fig 2.05, removes the two least correct members of the tonal series. This version of the table also inverts interval ratios to list the note letters in a modern context. The use of a Pentatonic scale removes the Tritone and hence removes the inaccuracy within the system - all fourths are now “perfect” fourths and all fifths are similarly “perfect”. Our understanding of Ancient Greek music itself is rather limited by the fact that no notational system existed with which to record the note sequences. However, a number of Ancient Greek scales have become part of the lexicon of modern music, mostly thanks to the work of Kathleen Schlesinger in her 1939 book “The Greek Aulos” (an aulos is a type of flute). From archaeological evidence she lists a set of scales which form the basis of Ancient Greek music. More recent research has since brought the accuracy of her work into question, but the scales in her book still persist. Whatever the truth of Schlesinger’s analysis, her work helps to cement the link between Ancient music theory and that of the modern era. As with much of music theory, her work will probably be augmented by the discoveries of future generations rather than being lost. 5