# Pythagorean tuning

The syntonic tuning continuum, showing Pythagorean tuning at 702 cents.[1] Diatonic scale on C Play 12-tone equal tempered and Play just intonation. Pythagorean (tonic) major chord on C Play (compare Play equal tempered and Play just). Comparison of equal-tempered (black) and Pythagorean (green) intervals showing the relationship between frequency ratio and the intervals' values, in cents. The system had been mainly attributed to Pythagoras (sixth century BC) by modern authors of music theory, while Ptolemy, and later Boethius, ascribed the division of the tetrachord by only two intervals, called "semitonium", "tonus", "tonus" in Latin (256:243 x 9:8 x 9:8), to Eratosthenes. Method Pythagorean tuning is based on a stack of intervals called perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1. This succession of eleven 3:2 intervals spans across a wide range of frequency (on a piano keyboard, it encompasses 77 keys). Size of intervals (e.g. between E♭ and E)

Why the circle of fourths is so important when learning major scales | Hear and Play Music Learning Center Playing your major scales should be a part of your daily practice regimen. However, practicing them in a “circle of fourths” or “circle of fifths” pattern is even better. Let’s focus more on circle of fourths. If you type “circle of fourths” or “circle of fifths” in google, you can actually find a host of other examples. Notice that the keys go from: C >>> F >>> Bb >>> Eb and so forth. If this were a clock, C would be at 12 o’ clock. This is the optimal way to play your scales. Then play your F major scale all the way through (F G A Bb C D E F). Why the circle? Because music also happens to move in this same pattern (way beyond the scope of this article but I’ll touch on it a little bit). But here’s another reason to use the circle. Because it lets you know how related the major keys are to each other. If one just looked at a piano, they’d assume that C and Db, for example, were related because of how close they appear to each other on the piano. The reality is that C and F are more related.

Egyptian Fractions 1 Egyptian Fractions The ancient Egyptians only used fractions of the form 1/n so any other fraction had to be represented as a sum of such unit fractions and, furthermore, all the unit fractions were different! Why? Is this a better system than our present day one? In fact, it is for some tasks. This page explores some of the history and methods with puzzles and and gives you a summary of computer searches for such representations. This page has an auto-generated Content section which may take a second or two to appear. The calculators on this page also require JavaScript but you appear to have switched JavaScript off (it is disabled). 2 An Introduction to Egyptian Mathematics Some of the oldest writing in the world is on a form of paper made from papyrus reeds that grew all along the Nile river in Egypt. [The image is a link to David Joyce's site on the History of Maths at Clarke University.] So what was on them do you think? 2.1 Henry Rhind and his Papyrus scroll So what did it say? So

Circle of fifths Circle of fifths showing major and minor keys Nikolay Diletsky's circle of fifths in Idea grammatiki musikiyskoy (Moscow, 1679) In music theory, the circle of fifths (or circle of fourths) is a visual representation of the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys. More specifically, it is a geometrical representation of relationships among the 12 pitch classes of the chromatic scale in pitch class space. Definition Structure and use Pitches within the chromatic scale are related not only by the number of semitones between them within the chromatic scale, but also related harmonically within the circle of fifths. Octaves (7 × 1200 = 8400) versus fifths (12 × 700 = 8400), depicted as with Cuisenaire rods (red (2) is used for 1200, black (7) is used for 700). Diatonic key signatures The circle is commonly used to represent the relationship between diatonic scales. Play . History .

Harmony and Proportion: Pythagoras: Music and Space Pythagoras (6th century BC) observed that when the blacksmith struck his anvil, different notes were produced according to the weight of the hammer. Number (in this case amount of weight) seemed to govern musical tone... See if you can hear the sound in your imagination before it comes, by judging from the proportions of the string lengths. Again, number (in this case amount of space) seemed to govern musical tone. He also discovered that if the length of the two strings are in relation to each other 2:3, the difference in pitch is called a fifth: ...and if the length of the strings are in relation to each other 3:4, then the difference is called a fourth. Thus the musical notation of the Greeks, which we have inherited can be expressed mathematically as 1:2:3:4 All this above can be summarised in the following. (Another consonance which the Greeks recognised was the octave plus a fifth, where 9:18 = 1:2, an octave, and 18:27 = 2:3, a fifth;)

Harmonic series (music) Harmonic series of a string with terms written as reciprocals (2/1 written as 1/2). A harmonic series is the sequence of all multiples of a base frequency. Any complex tone "can be described as a combination of many simple periodic waves (i.e., sine waves) or partials, each with its own frequency of vibration, amplitude, and phase."[1] (Fourier analysis) A partial is any of the sine waves by which a complex tone is described. A harmonic (or a harmonic partial) is any of a set of partials that are whole number multiples of a common fundamental frequency.[2] This set includes the fundamental, which is a whole number multiple of itself (1 times itself). Typical pitched instruments are designed to have partials that are close to being harmonics, with very low inharmonicity; therefore, in music theory, and in instrument tuning, it is convenient to speak of the partials in those instruments' sounds as harmonics, even if they have some inharmonicity. An overtone is any partial except the lowest.

Microtonality, Tunings and Modes There is still a lot of good music waiting to be written in C major. — Arnold Schönberg One of the most liberating aspects about using a computer to compose music is that non-traditional “frequency space” can be explored in different ways without having to address the limitations of physical instruments or human performers. This sort of experimentation may involve working with raw, untempered Hertz values, just harmonic ratios, microtonal inflections, or “alternate” tuning systems, which we define very broadly to mean any tuning system other than the standard 12-tone equal tempered scale. But working with an expanded frequency space using MIDI synthesizers will quickly expose a strong bias of MIDI towards the popular music tradition and the Western tonal system in general. MIDI and Microtonal Tuning There are two basic methods for producing microtonal sound on a MIDI synthesizer. Channel Tuning Note by note tuning Equal division tuning Channel Tuning in Common Music Interaction 15-1. Cents

Untitled Document Mathematics, astronomy, geometry, geography, art, music and religion were all integrated in ancient times. Numerous surviving examples suggest that ancient Egyptian art was based on a strict canon that was followed for over 3,000 years. According to Plato, this was also true of ancient Egyptian music. Tone is based on the frequency of sound waves. The next higher octave may be calculated by doubling the frequency of each note and the next lower octave may be calculated by halving the frequency of each note. A string tuned to a base frequency of 440 waves per second (440 Hz) will also produce higher tones known as partials or harmonics. When the same note is played by two different instruments at the same time, the frequency of both notes is the same, so every wave of both notes aligns, resulting in optimal consonance. What Pythagoras described as perfect fifths and perfect fourths are notes with low integer ratios in relation to the base note, also resulting in consonant sound.

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