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Topic: Pythagorean tuning

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Musical tuning

Just intonation

Meantone temperament

Well temperament

Wolf interval

Pythagorean comma

Equal temperament

Interval (music)

Lucy tuning

Mathematics of musical scales

Microtonal music

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	Pythagorean Tuning and Medieval Polyphony - Table of Contents
		Tuning data and the chromatic octave of C
		Rather, techniques of tuning and notation interact creatively with musical style in each period, and should all be taken into consideration in understanding and recreating the music of a given age.
		Readers interested in the practical details of Pythagorean tuning are encouraged to jump directly from Section 2 to Section 4.
	www.medieval.org /emfaq/harmony/pyth.html (766 words)

	Pythagorean tuning: Definition and Links by Encyclopedian.com
		Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on on the ratio 3:2.
		Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is the ratio of an octave.
		In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter.
	www.encyclopedian.com /py/Pythagorean-tuning.html (842 words)

	BAIN: A Pythagorean tuning of the diatonic scale
		The Pythagorean cult's preference for proportions involving whole numbers is evident in this scale's construction, as all of its tones may be derived from interval frequency ratios based on the first three counting numbers: 1, 2, and 3.
		For example, a Pythagorean tuning of the 12-note chromatic scale on C is shown in Fig.
		The Pythagorean semitone was also called the limma (left over), as it was calculated by the Greeks to be the difference (or amount left over) between a fourth and two whole tones.
	www.music.sc.edu /fs/bain/atmi02/pst (0 words)

	MTO 4.4: Scholtz, Algorithms for Tunings and Temperaments
		The tuning of D is equal to 204 cents, obtained by subtracting a an octave of 1200 cents from 702 + 702 = 1404.
		In Figure 5, the Pythagorean comma is indicated by the upper-case "P" and 1/12 of the Pythagorean comma by the lower-case "p." The direction of the arrow indicates the ascending harmonic interval.
		Compared to Pythagorean tuning, it is evident that the fifths to the right of C are all tuned a quarter-comma flat, while the fourths to the left of C become a quarter-comma sharp.
	www.societymusictheory.org /mto/issues/mto.98.4.4/mto.98.4.4.scholtz.html (8136 words)

	Tuning the Guitar
		Pythagorean tuning is named after the mathematician who created all those theorems that taunted us in high-school math classes.
		Tuning all of the strings to one string eliminates the potential for the passing small variations in tuning from string to string, by tuning 1 to 2, 2 to 3, 3 to 4, 4 to 5 and 5 to 6.
		If you tune by matching the harmonic at the 7th fret, you will be guaranteed to be putting the string slightly out of tune with your frets.
	timberens.com /essays/tuning.htm (699 words)

	Pythagorean Tuning
		Tune B3 to E4 and make a perfect 4th.
		Tune C#4 to F#3 and make a perfect 5th.
		Tune G#3 to C#4 and make a perfect 4th.
	www.music.indiana.edu /som/piano_repair/temperaments/pythagorean.html (238 words)

	Tuning Systems
		In order to feature these favored intervals, a tuning tradition may: use scales in which the notes are not equally spaced; avoid any notes or intervals which don't work with a particular tuning; change the tuning of some notes when the key or mode changes; or any combination of these techniques.
		A harpsichord that has been tuned using the Pythagorean system or just intonation may sound perfectly in tune in one key - C major, for example - and fairly well in tune in a related key - G major - but badly out of tune in a "distant" key like D flat major.
		In Western music, the tuning systems that have been invented and widely used that directly address this problem are the various temperaments, in which the tuning of notes is "tempered" slightly from pure intervals.
	cnx.org /content/m11639/latest (4745 words)

	pythagorean - 3-limit just intonation musical tuning system
		Thus, the early realization that a 53-tone Pythagorean chain is a nearly-closed system, which would lead to the further realization that 53edo is an excellent approximation of Pythagorean tuning.
		In 53-edo it is 1 degree, identical to the Pythagorean comma; in 306-edo it is 5 degrees, one less than the 6 degrees which compose the Pythagorean comma.
		Pythagorean tuning can be considered a particular instance of the schismic family of temperaments.
	tonalsoft.com /enc/p/pythagorean.aspx (1605 words)

	Pythagorean Tuning - More details
		Tuning ratios and measures in cents are shown relation to the lowest note of the octave range, the final in authentic modes but the fourth below it in plagal modes, with the final identified by the symbol "F" on the line above the tuning ratios.
		One quirk of Pythgorean tuning which we encountered in Section 4.2.2 is the "Wolf" fifth or fourth which results between the extreme notes of our tuning chain in fifths, g#-eb' or eb-g# in a standard scheme with Eb at one end of the chain and G# at the other.
		It is impossible to tune 12 pure fifths so as to arrive at an even octave; and it is impossible in any fixed 12-tone tuning to achieve pure fifths and also to obtain thirds (and sixths) in their simplest ratios.
	www.medieval.org /emfaq/harmony/pyth4.html (0 words)

	The Development of Musical Tuning Systems
		Intervals of Pythagorean Tuning - Ancient Greek Phrygian Mode (Descending)
		I have not ventured into modern tuning systems using the seventh and higher harmonics nor into division of the tonal spectrum other than into octaves of 12 semitones (though my software supports such features).
		There are many other excellent web sites on the subject of microtonal tuning systems and I have included links to just some of these on my links page.
	www.midicode.com /tunings (0 words)

	Pythagorean Tuning Forks from Carousel (Site not responding. Last check: )
		Each tuning fork is calibrated at a specific frequency to address different areas of healing and development..
		The most powerful tuning fork combination, the C and G tuners create a “perfect fifth” and open the gateway for healing and relaxation.
		The Otto Tuners are tuned to the frequencies of 32cps, 64cps and 128cps.
	www.carousel-music.com /nf/pythagorasnf.html (2610 words)

	Nydana Notation - Tuning
		It is a very practical tuning since we can modulate between different keys while using a set of only twelve notes per octave; each note having a fixed pitch.
		We can imitate Pythagorean tuning on a piano by playing sharp notes slightly harder, and flat notes slightly softer (a note will be perceived as being higher in pitch if you strike the key harder).
		Another tuning that should be mentioned here is the stretched equal-tempered tuning with justly tuned fifths and slightly wider octaves.
	home.swipnet.se /nydana/tuning.html (1535 words)

	The Mathematics of Tuning & Temperament, with audio examples.
		For the music to sound consonant (in tune) the different notes being sounded simultaneously need to be related in a way which is pleasing to the ear.
		The chromatic Pythagorean scale is formed by inserting semitones equal to 114 cents in such a way as to keep all perfect fifths true, except for the interval G#-Eb, which needs to be adjusted so that the intervals add up mathematically.
		The differences between the tuning systems are most evident in polyphonic music, particularly when chords and intervals incorporating the "fl notes" are used.
	www.pyxidium.u-net.com /Acoustics/MusicMaths/MusicMaths.html (1376 words)

	Pythagorean Tuning (Site not responding. Last check: )
		Pythagorean tuning is the oldest tuning system in European music heritage.
		According to legend, "Pythagoras discovered concords in the simple ratios among the divisions of a sounding string." Because the ancient Greeks believed numbers were the answer to all things, including the physical and the spiritual, so music was aligned with numbers.
		Pythagorean scales were built upon pure fourths and fifths.
	www.aug.edu /~cshotwel/pythagoreantuning.htm (175 words)

	The Pythagorean Sourcebook and Library
		The Pythagorean ethical and political tractates are especially interesting for they are based on the premise that the universal principles of Harmony, Proportion, and Justice govern the physical cosmos, and these writings show how individuals and societies alike attain their peak of excellence when informed by these same principles.
		The comprehensive introduction by David Fideler outlines the history of the Pythagorean school in the Classical and Hellenistic periods and carefully examines its philosophy and teachings, helping to situate the writings of this anthology in their proper historical and philosophical contexts.
		Contains over 20 engravings and illustrations, in addition to four appendices dealing with the formation of the musical scale in Pythagorean tuning, Pythagorean mathematical discoveries, etc. Many of the writings are prefaced by brief introductory notes; annotations explicate obscure passages and unfamiliar terms.
	phanes.com /pytsou.html (428 words)

	Tonalsoft Encyclopaedia of Tuning -- Septimal schisma as xenharmonic bridge?, (c)1998 by Margo Schulter
		As someone with a special interest in Pythagorean tuning, while I might not have any additional suggestions to contribute in the way of organizational questions, it occurs to me to share a few reflections on the relationship of this xenharmonic approach to others.
		Pythagorean tuning is not only a system of just intonation with a very long history, but also an amazingly successful system not only ideally suited to much Gothic music but inviting "Neo-Gothic extensions" approximating intervals in 5-based and 7-based systems.
		It is the difference between the Pythagorean comma (531441:524288, about 23.46 cents) and the syntonic comma (81:80, about 21.51 cents) which separates a number of basic 3-based intervals from their 5-based counterparts.
	sonic-arts.org /td/schulter/septimal.htm (1195 words)

	MTO 10.1: Wibberley, Syntonic Tuning: A Sixteenth-Century Composer's Soundscape
		In this example, the normal Pythagorean intervals are preserved including the ditone and semiditone (together with their compounds and inversions).
		For a composer using the Pythagorean sound world such dissonances are perfectly "in tune", and a normal property of the soundscape.
		The pitches on G consequently begin with their Pythagorean values so as to permit pure harmony with the raised B-flat; but as the B-flat changes to a B-natural the G is also raised by a Syntonic comma to permit pure harmonies with B, E and C).
	www.societymusictheory.org /mto/issues/mto.04.10.1/mto.04.10.1.wibberley2.html (5989 words)

	Pythagorean tuning - Cunnan
		Pythagorean tuning is a method of tuning the 12-note chromatic musical scale based on a series of perfect fifth intervals.
		Consequently, a problem of the Pythagorean tuning method is that no series of perfect fifths can be made equal to an octave.
		This resulted in higher octave notes being out of tune with respect to their lower octave counterparts.
	cunnan.sca.org.au /index.php?title=Pythagorean_tuning&redirect=no (162 words)

	Healing Sounds - Pythagorean Tuning Forks
		These tuning forks represent a powerful new way of using sound to resonate the body, brain and etheric fields.
		These tuning forks were developed by John Beaulieu, N.D., Ph.D., author of MUSIC AND SOUND IN THE HEALING ARTS.
		These specific tuning forks are of C (256 cycles per second) and G (384 cycles per second).
	www.healingsounds.com /articles/pythagorean-tuning-forks.asp (378 words)

	Tuning the Piano
		Lower notes are harder to tune because of their high harmonic content, and the higher notes are difficult because the amount of pin rotation needed to tune becomes smaller with higher pitch.
		Even in the inaudible treble, the duplex scale is “tuned” in the sense that the aliquot bar is placed at a location such that the length of the duplex part of the string is a harmonic length of the speaking section of the string in order to optimize the impedance (“aliquot” means fractional or harmonic).
		In harmonic tuning you tune, for example, the fundamental or a harmonic of the upper note to a higher harmonic of the lower note.
	members.aol.com /chang8828/tuning.htm (15440 words)

	Consonance and Dissonance - Issues in Tuning and Temperament
		There are four general ways of dealing with the Pythagorean comma: (1) ignore it, (2) minimize it, (3) adapt to it, or (4) hide it.
		A second way of minimizing the Pythagorean comma is to limit the music-making to a small number of notes.
		It is possible to tune a pentatonic scale so that most intervals are fairly close to their just values.
	www.music-cog.ohio-state.edu /Music829B/tuning.html (1078 words)

	SomaEnergetics Pythagorean History
		The Pythagoreans were known for their abilities to cure many physical, psychological and spiritual ailments through the use of this special music.
		BioSonic Repatterning with Pythagorean tuning forks, developed by John Beaulieu, N.D., Ph.D., is a natural method of healing using tuning forks based on the sonic ratios inherent in nature.
		When we tap the tuning forks, we produce pure musical intervals based on precise mathematical proportions known as the Pythagorean tunings.
	www.lightwithin.com /SomaEnergetics/3Products_PythagoreanHistory.htm (0 words)

	The Pythagorean Sourcebook and Library
		The Pythagorean ethical and political tractates are especially interesting for they are based on the premise that the universal principles of Harmony, Proportion, and Justice govern the physical cosmos, and these writings show how individuals and societies alike attain their peak of excellence when informed by these same principles.
		The comprehensive introduction by David Fideler outlines the history of the Pythagorean school in the Classical and Hellenistic periods and carefully examines its philosophy and teachings, helping to situate the writings of this anthology in their proper historical and philosophical contexts.
		Contains over 20 engravings and illustrations, in addition to four appendices dealing with the formation of the musical scale in Pythagorean tuning, Pythagorean mathematical discoveries, etc. Many of the writings are prefaced by brief introductory notes; annotations explicate obscure passages and unfamiliar terms.
	www.phanes.com /pytsou.html (0 words)

	Pythagorean Tone Generators
		The successive multiplication by 3/2, the musical fifth, is the method used by the Pythagoreans for generating the musical scale.
		Each instrument is tuned to the full esoteric octave, from Pythagorean middle C to high C, and can produce the Perfect Fifth tone (which contains all other tones in creation) and the true sound of Om.
		For the Pythagorean tuning to be experienced most effectively, play the bottom note “C” and then add one of the upper seven notes for harmony.
	www.crucible.org /pythagorean_tone_generators.htm (1181 words)

	Tonalsoft(TM) Tonescape(TM) tutorial: how to use "File \| New \| Tonespace...", part 1
		The identity-vector exists in most tunings around the world, and is the interval by which the pattern of intervals within the scale repeats itself, and so does the naming of the notes.
		Thus, the chromatic-semitone in pythagorean tuning -- i.e., the interval which separates a letter-name nominal from the same nominal with a sharp or flat, as that between "A" and "A#" -- is about 114 cents.
		The Lattice now shows us that if the pythagorean tuning is notated with the nearest degree of 12-edo, then 12 steps along the 3-axis takes us back to the same note from which we started.
	tonalsoft.com /support/tonescape/tut-tonespace.aspx (4782 words)

	PIANO PAGE - Piano Technicians Guild - Everything about Pianos, Tuning, Service, Repair, History, Find a Technician
		A clear, concise, up-to-date guide to buying, maintaining, and tuning this instrument that includes a short history and discussion of the different types of harpsichord and their use in different kinds of music.
		Polly's "most unusual doghouse in the world" is under the piano, where she is close up to her owner giving piano lessons and practicing Bach, Mozart, and Brahms in the hope of appearing in Carnegie Hall one day.
		Theory and practice of piano tuning covers all aspects of the theory, practice and "philosophy" of piano tuning from elementary to advanced level.
	www.ptg.org /resources-books.php (0 words)

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