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Music Theory/Music and Mathematics: Frequently Asked Questions

Here I am working on a compilation of frequently asked questions in the field of music theory, mostly from postings in the newsgroup rec.music.theory. Since many (surprisingly many!) questions of music theory involve mathematics, I think this is not a bad place for it.
(My original intention was to make just a MuTh FAQ - questions directly related to topics of the mathematical music theory whose main representant is Guerino Mazzola. However, having left the university, I am now not so close to places where these questions are asked "frequently"; so there are just a few introductory questions in a - rather underdeveloped - separate section.)
If you have critics, suggestions or comments of any kind concerning this page, feel free to contact me. If you have a music theory question you would like to have answered, best write it into the Newsgroup.

Contributors to this page

Since large parts of this page are made up from newsgroup postings, I cannot take all the credits (or the blames) for me - except for the selection of the questions which, of course, reflects my personal taste (and also has a slight emphasis towards mathematics). In many cases I cite the original postings; there I mention the authors. Namely, essential contributions have come from Dr. Matt Fields and Margo Schulter <mschulter@value.net>.

Other websites answering music theory questions

If a question is not found here, you may try one of the websites below. Many questions are already answered elsewhere; and I thought I'd better offer links to the answers instead of writing the same things again. (I try to avoid redundancies, but I do not always succeed.)

Eric's Treasure Trove of Music
ThinkQuest Music Dictionary
Online dictionaries of music theory. Each of them answers much more questions than this page here...
Gary Ewer's Easy Music Theory
Greg's Music Theory Page
Music Theory For Songwriters
MIBAC (Music Instruction By A Computer)
Ars Nova Music Theory Q & A (new 2008-03-12)
Websites offering basic music theory knowledge - may serve as starting addresses for beginners.
Early Music FAQ.
A broad range of topics of the field of european medieval and renaissance music, including a Pythagorean Tuning FAQ.
A jazz improvisation primer
A good address for questions concerning jazz, including its own FAQ.

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Charter of rec.music.theory

This newsgroup shall be for the discussion of any topic related to music theory. Examples of appropriate topics are: the relationship between harmony and melody, the evolving role of harmony and melody in western music, how to derive the modes of a scale, contrasts between theoretical approaches to composition in jazz and classical music, theory of non-western music traditions, what a triad is, how to teach music theory, uses of rhythmic techniques in composition, how theory can guide improvisation, what counterpoint is, the origin of music, the history of music theory, and so on. The groups shall be an open and welcome place for both high-level discussions among those knowledgeable about music theory and those who are beginners and are interested in discussing the most basic elements of music theory for purposes of edification.

Posting which is of a commercial nature or which promotes business interests is strictly prohibited. No post which advertises for sale any goods or services, whether from a business or a private party may be posted to rec.music.theory. Extoling the virtues of a product or service by which one profits is limited to one post per month. It is permitted for somebody to answer questions or comments which are directed to them about a product or service in which they have a business interest.

It is permitted to post a simple pointer to a web site of any sort no more than once a month if the text of the pointer doesn't violate the rest of this charter. Long quotations or text, published or unpublished, or posting of text which is substantially similar to text previously posted is prohibited (FAQs and administrative posts excepted).

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Questions

Miscellaneous

Scales and modes

For scales see also the tuning section of this FAQ.
For definition of modes: see one of the beginner websites.
For use of modes in medieval music see Early Music FAQ.
For use of modes in jazz see A jazz improvisation primer.

"Local" musical structures

"Global" musical structures

Tuning, temperament, consonance, dissonance

MaMuTh-specific

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Answers

What good is mathematics for music anyway?!?

Mathematics is basically a kind of language, and it turns out to be quite well-suited to express many musical facts. I am aware of the fact that not all people share my enthusiasm about mathematics; I want to assure everybody that I am well aware it invariably has its limits - knowing the limits of a theory is something I consider inherent in an adequate use of any theory, not just mathematical music theory.

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What is music good for in mathematics?

Mathematical methods in music may not only be just useful in music but also the other way round: if you have a description of music in mathematical terms, you automatically have also a description of mathematics in musical terms - so mathematical facts and rules are made "audible": a new way to shape mathematical intuition (and maybe another idea to make mathematics less hated in schools...). A famous example for this is Douglas G. Hofstadter's book "Gödel, Escher, Bach".

Another, maybe even more important point is that every application of mathematics can stimulate research in mathematics. An extreme example for this is physics - in fact a very large part of mathematics (maybe even the majority) actually originated in physics questions.

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Recommendations for books of music theory

From: Ken Durling <kdurling@earthlink.net>
There are some standard - "classic" - books that you should know about if you don't, including Walter Piston's "Harmony"; George Wedge's two volumes (diatonic and chromatic) of "Applied Harmony"; Arnold Schoenberg's "Harmonielehre" and "Structural Functions of Harmony." Others will suggest more of them...
Lesser known, but very interesting as it attempts to unify traditional 4-part techniques and jazz nomenclature are Gordon Delamont's two volumes of "Modern Harmoninc Techniqie."
There are a few books on counterpoint (the single most important dispiline for a composer, IMO) - including ones by Piston, Kennan and others. But here I think the best teacher is the repertoire.
On a more elementary level (the above all start right in with 4-part writing), I like Owen Reed's "Basic Music" and Ernest Toch's "The Shaping Forces in Music."
Leo Kraft's two-volume integrated approach is excellent - what the hell is that called? :-) "Modus?"
And then there's one of my all-time favorites. for a creative, inclusive approach - Robert Cogan's "Sonic Design." Brilliant discussion of many aspects of music and how they interrelate.
Hindemith's "Elementary Training for Musicians" is indispensable for training solid musicianship, and Robert Starer has an excellent book called "Rhythmic Training."
There are a number of good books on structure and form, including Walter Berry's classic "Form in Music" (think I have that right) and Leon Stein's "Structure and Style" with its accompanying anthology. A number of books exist on Schenkerian approach, but others will have more educated suggestions than I. I believe Alan Forte's book is considered a classic. Charles Rosen's books on Classical Style and Sonata Forms belong on every bookshelf.
For twelve-tone writing, a good introduction is Reginald Smith-Brindle's "Serial Composition." John Rahn's "Basic Atonal Theory" will interest some, as well.

From: fields@login.itd.umich.edu (Dr.Matt)
Aldwell and Schachter, Harmony and Voice Leading is a classic

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What is a 12-tone series?

From: Ken Durling <kdurling@earthlink.net>
It is an ordering, chosen by the composer, of the 12 available pitch classes into a series of notes that is treated as a motivic and harmonic source. A chromatic scale through one octave is a "twelve tone series," albeit not a very interesting one.
Treatment varies across the spectrum from a purely "melodic" approach (how the set "sounds" - I believe this was Schonberg's original intention) to treating it more purely "numerically" - i.e. as a matrix of intervals that can be used in any way without necessarily trying to keep any large part of it intact melodically.
There are theoretically 48 versions, all equally usable, of any series you invent:
the original and its transposition to the 11 other pitches the retrograde (same thing backwards) and it 11 transpositions
the inversion (same intervals going the other way) and its 11 . . . the retrograde inversion ( you're catching on) and its 11 . . .
so you have quite a large menu of available pitch sets to draw from.
One of the original ideas, in Schonberg's terms, was that the 12 notes are related "only to each other" - in other words there is no tonal center, no heirarchy of Tonic, Dominant, etc. The focus was to be purely on the motivic fabric.
In practice it played out (is playing out?) quite differently, and there turned out to be a very broad spectrum of approaches to the technique. Stravinsky "dabbled" in it - I think he felt obligated - his 12-tone pieces are not his best. However, part of the technique was natural to him - that of motivic cell thinking, and he used the technique freely - creating 9, 10 and 11-note "series", rotating series, and more.

More information can be found on a very recommendable introduction into the serial composition technique , written by Dr. Matt.

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Where do the italian pitch names do, re, mi, fa, sol come from?

Essentially contributed by Sybrand Bakker <postbus@sybrandb.demon.nl> and Martha De Francisco.
The do stems from the scale do re mi fa sol la. Originally this do was called 'ut'. This note names derive from the hymn Ut queant laxis (11th century), where the text starts as follows:
UT queant laxis
REsonare fibris
MIra gestorum
FAmuli tuorum,
SOLve polluti
LAbii reatum,
Sancte Ioannes.
The ut was located on a c, the re on a d, the mi on an e, and so on. This ut was perfectly movable.
When a new name for the seventh, or leading, note of our octave was desired, Erich Van der Putten suggested, in 1599, the syllabic BI of "labii", but a vast majority of musical theorists supported the happier thought of the syllable SI, formed by the initial letters of the two words of the last line. (taken from http://www.newadvent.org/cathen/15244a.htm).
In the 16th century Hubert Waelrant replaced the ut by a do as he judged the ut syllable difficult to pronounce (This is a Latin u, which was pronounced differently by French. Flemish, Germans, English and so on). In some countries (France, Belgium) the do (and the other syllables) became fixed in the course of the centuries, replacing the orginal note names.

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What is the golden mean, and what importance does it have in music?

From: Dave Webber <dave@musical.demon.co.uk>.
Consider a line split into two parts of lengths 1 and x (so that the total length is 1+x) and define x to be the shorter section (ie x<1 ).
The ratio of the lengths of the two parts is x/1 = x.
The ratio of the length of the longer part to the whole is 1/(1+x)
The golden mean is defined by demanding that these are equal:

x = 1/(1+x)

The solution of this equation (there are two but the other one is negative) is

x = ( sqrt(5)-1 ) / 2

or approximately

0.618033989....

In pictorial art, possibly because of the above derivation, you will often find the horizon drawn at a height predicted by the golden mean - so that the area of the land to the area of the painting is the same ratio as the area of the sky to the area of land (or the other way around). I don't think the great artists did this by calculation and measurement - it just looked about right :-)

In music, the golden mean has been used in various ways. Some prominent uses are in the field of global structures and tuning.

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Why is the pitch B called H in Germany?

From: "M. Schulter" <mschulter@value.net>
Date: 2000/08/04
The modern "sharp" sign, and also the "natural" sign, derive from the medieval sign of "B-quadratum" or square-B, which show the hexachord syllable B-mi, a semitone below C-fa. This is also known as B-durum or "hard B," and appears similar or identical to a modern "natural" sign.
In the German tradition, the sign "h" also derived from this "square" or hard B. During the 14th and 15th centuries, the "sharp" sign evolved as a variant of the "square-B" sign, and the two signs were often used synonymously, both showing that a note should be sung as "mi" -- often inviting motion to the semitone above.
In contrast, the modern "flat" sign derives from the medieval sign of "B-molle" or a "soft" or "rounded" B, showing the hexachord syllable B-fa, a semitone above A-mi.

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Who is Albert Silverman?

(I do not really like this question - but a rec.music.theory FAQ would not be complete without it...)
Essentially contributed by Jo Totland <jtotland1@chello.no>
Albert is a frequent poster to the forum rec.music.theory, with his own unique theory of music, that does not fit into the traditional framework. The merits of his music theory can probably be discussed to no end. Many regard him as a usenet cook, or crank, although some find his postings occasionally useful or funny. Too often, flamewars result as a result of his postings. These - as any flamewars - are usually non-productive, and should be dealt with as flamewars usually are dealt with (best by not feeding the flame).

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What is "spanish phrygian"?

Pieces in spanish flamenco music (and also in spanish classical music) often have a melodic and harmonic structure that spanish theorists refer to as "Dorico Flamenco", and which is defined as follows:

For "classical" ears, this sounds much like harmonic minor ending on fifth position. Since the melodies are phrygian, phrygian is sometimes referred to as the "spanish mode"; there exist also other designations like "spanish phrygian" (emphasizing the raised third as a difference to "plain" phrygian) or even "spanish major" (since the root chord is major).

This mode or variations of it are also used in jewish music (Klezmer) and in some traditional oriental and arabian styles.

From: Pepsinogen <pepsinogen@aol.com>
A good example is the scale derived by tonicizing the 5th of a harmonic minor scale (i.e., E-F-G#-A-B-C-D-E). This is known as Hijaz in the Arab world and Ahavah Rabah to Jewish musicians. Makes for some wonderfully haunting melodies.

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Examples for use of modes in existing music?

From: Jon Riley <jon@jonriley.freeserve.co.uk>
Miles Davis "Kind Of Blue" album is the best known (and probably earliest) use of modes in a jazz context. The track "So What" is in D dorian mode, with a middle eight in Eb dorian. "All Blues" is mostly G mixolydian, switching to G dorian and G minor blues. "Flamenco Sketches" is a kind of loose improvisation on various modes (phrygian, mixolydian and dorian IIRC). Coltrane's "Impressions" is the same tonality as "So What" - only the tune is different.
Van Morrison's "Moondance" is in A dorian on the verse (A minor key on the chorus). And "Gloria" (his first single with Them) is in E mixolydian.
But if we're getting into rock/pop... the Beatles "Tomorrow Never Knows" is C mixolydian, a fine example of a mode focussed on its tonic chord, with no changes.
Mixolydian is very common in rock, though it's usually mixed with major (Ionian) or minor (aeolian/dorian) or blues tonalities.
Mixolydian and dorian are common in folk too - which is how they came to rock and jazz. (And both have an affinity with blues.)
Scarborough Fair is dorian.
She Moved Through The Fair and Blackwaterside are mixolydian.

To the above I have only a slight correction: "Kind Of Blue" is the album that made modal jazz popular, but the first modal jazz album is "Milestones", one year earlier, by the same band.
Other examples for the use of modes in pop/rock are Queen's "Innuendo", Rainbow's "Gates of Babylon" and Deep Purple's "Strangeways" which are partly in spanish phrygian.

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Why is melodic minor downward different from upward?

From: Matthew Laney <mslaney@hotmail.com>
I think the idea in the melodic minor is to be able to have a leading tone in the melody by raising the 7th, but not an Augmented 2nd in the scale and therefore you raise the 6th as well. When you are going down it is not necessary and many (if not most) times the leading tone isn't wanted because of it's strong tendency to go up. That is why you have the different forms of minor scales. Natural minor is the base form, harmonic show the most common use of harmony since the V (and vii) chords are usually altered to have a leading tone, and melody uses both. They aren't really as independent as they are made out to be, but are usually mixed together.

From: Laurence Payne <lp@laurenceDELETETHISpayne.freeserve.co.uk>
Because people decided that way sounded good and used it in their music.
When a pattern is used frequently in real music, it is useful to pin it down as 'theory' and practice it as an exercise. That's what happened.

From: Thomas A. Korth <tkorth1@home.com>
The theoretical justification has to do with the approach from scale degree 7 up to 1. Years of musical practice established this minor second "leading tone" relationship so it was included in all scales, major & minor. The resulting scale is harmonic minor, which contains an interval only rarely used melodically - the augmented 2nd between scale degrees 6 & 7. 6 is raised to "correct" this, and both 6 & & 7 revert to the "lowered" form descending to provide a downward leading tone to the dominant.

From: ian@beathoven.com (paramucho)
Date: 2001-08-06
I think of it as the natural minor modified to fit the expectations of a public fed on the diatonic major, and hungry for their final close.
The same kind of urge that leads to a major tonic chord at the close of a minor piece, although I think that goes even further back.

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What is a cadence?

From: Dr.Matt <fields@zip.eecs.umich.edu>.
In general it's a standard formula. Western classical music has a tradition of saving the standard formulas for endings, so when you hear one, it signals that something is ending.
In early chant, cadences were standard bits of melody that marked the end of a clause or sentence. The reader would sustain a high note, then finally come down from it (Latin: cadere) by way of the standard bit of melody.
With the invention of polyphony (music with simultaneous sounds planned out), the notion of cadences marking endings was retained, but cadences began incorporating particular intervals between voices--they became standard harmonic changes. Cadences as harmony patterns evolved over 1200 years. For examples from near the beginning of the cycle, look in Early Music FAQ. The near-end of the cycle embraces such phenomena as the turnaround in some kinds of jazz.
18th-century classical music was built on hierarchical patterns of cadences, the most prominent being the Perfect Authentic Cadence, the Half-Cadence, the Inauthentic Cadence, and the Deceptive Cadence. When you sense that a writer is using 18th-century music as a main reference, chances are that by "cadence" they mean that repertoire of 4 harmonic patterns, plus the more rare Plagal Cadence.

From: Joe Mizzi <joe@whoarecords.com>.
To summarize cadences:

Connection to MaMuTh:
A cadence is often at the end of a modulation. There, one of its purposes is to establish the new key. This latter property led to Mazzola's definition of cadencial sets.

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What is a picardy third?

From: Gary Ewer <easy@SPAMNOTmusictheory.halifax.ns.ca>
Date: Wed, 14 Nov 2001 13:19:16 GMT
The exact etymology of the word is not specifically known, but it refers to the major third interval used at the end of works that are in a minor key. There are lots of examples of this, particularly in the Baroque era - for example, the end of Bach's C#-minor Fugue in the Well-Tempered Clavier, Book I.

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What is special in the chord progression IV-V?

From: Inotmark <inotmark@aol.comnospam>.
It is strong for a number of reasons:
1) it has no tones in common
2) it has a tritone between root of IV and third of V
3) it is sufficient to define a key without the tonic chord appearing.
Schönberg calling it superstrong may be due to 3 above.

Connection to MaMuTh:
Property (3) means, in Mazzola's terminology, that the chord set {IV, V} is cadencial.

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How do you modulate?

A modulation (change of key) can be seen as a tripartite process (the following scheme is from Schönberg's "Harmonielehre"):

  1. A neutral phase where the old key is weakened, e.g. by using ambiguous chords (chords that are common in both keys), or by entering quickly into the new key and coming back at once.
  2. A fundamental step to mark the turning point where the old key is left. This is typically done by using chords from the new key that do not belong to the old key.
  3. A cadence to establish the new key.

With this pattern at hand, it is easy to determine which chord is to use in which phase. E.g. if you want to modulate from C major to G major, ambiguous triads are C, Em, G and Am while all triads containing the f# are candidates to mark the fundamental step.

The most "usual" keys to modulate into are those that are close in the circle of fifth. One reason for this is the availability of neutral chords - note that for major keys more than 2 steps away along the circle of fifths there are no common triads at all.

In practice, this pattern is not followed slavishly. The neutral phase may be lengthy or left out alltogether, the cadence may be omitted and a fundamental step to yet another new key may appear instead (there might be different opinions whether these are several modulations or just one) - and/or the fundamental step might be followed by another neutral phase where there is no key established at all etc. Sometimes culturally generated characteristics even "override" the mathematical laws; e.g. the sequence D-G might be recognized as the establishment of G major because of the long historical persistence of the V-I progression - although both chords are in fact ambiguous (they are also part of D major - an undoubted establishment of G major would be D7-G or C-D-G). OTOH, long use of certain progressions may "wear out" their effect. E.g. an isolated D-G or A-Dm in a context of C major (quite common in the time of Mozart) may pass almost unnoticed.

From: Dr. Matt <fields@qix.rs.itd.umich.ed>.
That sounds quite a bit like the rhetorical model Ratner quotes out of 18th-century treatises, where the three phases are indication, confirmation, and establishment, and are marked by the use of materials from the new key, materials that can only be parsed in the new key e.g. the tritone resolution of the new key, and a perfect authentic cadence in the new key, respectively.

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What are vertical and horizonal melodies?

I have seen these two words in the context of modal jazz (see also the scales and modes section of this FAQ). Modal jazz (developed since the end of the 50's by John Coltrane, Miles Davis et al.) is said to have a more "horizontal" structure whereas "traditional" jazz is structured more "vertically" - which means in this context that in traditional jazz improvisations are much based on (you might say: limited to) chord progressions; in modal jazz, on the other hand, improvisations typically go on for long periods without any chord change, and the interest is more turned towards the melodic line (see also the statement by Joey Goldstein below). It is a matter of whether a chord is seen as is ("vertical") or more as a scale/base for improvisation ("horizontal" or generally "linear") - the latter view being important in all kinds of jazz, but especially so since the 60's. (For more information about modal jazz, see A jazz improvisation primer.)

From: Joey Goldstein <nospam@nowhere.net>
In jazz circles we use the terms like this:
Horizontal melodies are based on a relationship with the overall key of the music or with the key of the moment during a modulation. Vertical melodies are bsed around the notes in the chord of the moment.
Examples:
A horizontal melody over this chord progression, in Bb major (the bridge of I Got Rhythm btw) would use notes mostly from the Bb major scale.:

D7 / / / | / / / / |G7 / / / | / / / / |
C7 / / / | / / / / |F7 / / / | / / / / |

A vertical melody would utilize more of the chromatic notes from outside the Bb major scale that are found within the chord tones.
Vertical melodies tend to run up and down the chord tones and the chord scales in an up and down vertical manner with many notes used from outside the key. Horizontal melodies are often less angular and have less accidentals.

From: Mark D. Lew <markdlew@earthlink.net>
In the choral world I occasionally hear "vertical" and "horizontal" used to describe aspects of arranging a tune for SATB. The arranger who pays attention to what chord is set for each note of the melody is said to be focusing on the "vertical", whereas one who takes care that each individual part has good movement is said to be focusing on the "horizontal".
In this context (which is not exactly the same as the "vertical" and "horizontal" discussed elsewhere), an arranger who is primarily pianist will tend to think vertically, whereas an arranger who is primarily a singer will tend to think horizontally. I've seen plenty of choral arrangements in which the focus is exclusively vertical with the horizontal completely neglected, but I can't recall any that are equally unbalanced in the other direction.
I've found that many jazzy choral arrangements (probably those written by jazz pianists) tend to be extremely vertical. Thus, when thoroughly practiced they sound terrific, but because of the bad horizontal lines they're very difficult to sing in tune, with the result that for a less practiced group they come out a bit muddy. This leads us directly to the question of which a composer/arranger should focus on: the ideal piece, assuming it is performed exactly as written; or the practical piece, taking into account the likelihood of how the real performance will sound.

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What are diagonal melodies?

From: Samuel Vriezen <sqv@xs4all.nl.getridofthisone>
I used that word to mean a melody that is a resultant of notes that follow each other, but not in the same voice. For instance, you have a 4-voice texture and the voices accentuate certain of their notes, but never two voices accentuating some note at the same time. It doesn't need to be a dynamic accent, the pitch could also be articulated by duration or register or whatever. In that way you get sort of a 'trompe l'oreille'-extra contrapuntal line which passes through the other voices, or emerging from them. I believe an understanding of this type of diagonal process is vital mostly for understanding how texture functions in counterpoint.

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What is a secondary dominant?

From: "David Webber" <dave@musical.demon.co.uk>
Date: Wed, 21 Nov 2001 23:25:50 -0000
The dominant is the fifth note of a scale or a chord built on it.

In the key of C, a passage ending G7-C goes from the dominant 7th (G7) to the tonic (C).

If you end D7-G7-C then you're effectively using D7 as a dominant 7th of G - or a "secondary dominant" of C.

(You might have thought that extending this from further up the cycle of 5ths to A7-D7-G7-C would mean that A7 is a "tertiary dominant", but in fact they gave up counting at "secondary" and the all the dominants of everything except the tonic (the original dominant) are called "secondary dominants".)

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What is a suite?

From: Steve Chandler <schandlr@dmreg.infi.net>
A Suite can be either of two things (perhaps more, but I'll describe two). It's either a collection of dances as in a baroque suite, e.g. Bach's English or French Suites or it is a condensation of a larger (longer) piece for concert performance. A good example of the latter would be Appalachian Spring, it's original form is a ballet for 13 (?) instruments which was shortened and arranged for orchestra as a suite.

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What is a fugue?

For this question there is a collection of newsgroup postings, put together by Dr. Matt.

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Applications of the golden mean for global musical structures?

(See Definition of the golden mean).

From: Inotmark <inotmark@aol.comnospam>.
In his book "Debussy in proportion" Howat demonstrates that the golden mean is consciously applied and he defines it to within fractions of a beat in extended pieces. The characteristic first appeared in Debussy's compositions after exposure to the concept at the world exposition in paris. Bartok marks his scores with specific timings based on his desire to maintain the proportions between various sections of his pieces. The concept of the golden section was quite consciously applied by these two composers. Others have followed.in their footsteps.

From: Dan Seriff <microtonal@sericap.com>
Bartok used it in his compositions by making things related to measure numbers adhere to the golden mean and the Fibonacci sequence.
Two examples:
String Quartet #4/III: Several important formal junctions in the slow movement fall in measures whose numbers are members of the Fibonacci sequence, 34 & 55.
Music for Strings Percussion & Celesta: I believe it is the first movement whose climax falls in the measure that exactly divides the movement into the two halves of the golden mean. I can't remember the measure number off the top of my head, and I haven't got a score. The ratio of the number of measures of the first and second halves (split over the climax) come out pretty close to the golden mean.

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Why is the octave divided into 12 semitones, and are there other possible divisions?

Essentially contributed by M.Schulter <mschulter@value.net>, with a contribution from Dr.Matt <fields@zip.eecs.umich.edu>.
The most widely used tuning system is 12-tet (12-tone equal temperament), where one octave is divided into 12 equidistant intervals. This leads to a scale that contains good approximations for a number of just intervals, namely the third and the fifth which are the building blocks for minor and major triads.
The division into 12 is, of course, not the only possible one. In the 16th century, for example, the first proposed equal (or virtually equal) temperaments for keyboard instruments involved not the division of the octave into 12 parts, but rather into 31 almost-equal parts (Vicentino), or likewise into 19 equal parts (Costeley, Salinas). If pure major thirds are important, then 31-tet is much the better choice than 12-tet. Also, in medieval and Renaissance tuning systems, G#/Ab (for example) are two separate notes - a property that is lost in 12-tet, but not in 19-tet or 31-tet.
The next higher important subdivision is 53-tet which has the speciality that it offers good approximations for both pure and pythagorean intervals.
There is, of course, a practical problem of playing a keyboard with 31 keys per octave (instruments constructed for this - involved several layers of keyboards) - which may be another reason that 31-tet (not to speak of 53-tet!) did not come to stay.
As it happens, a series of 12 pure fifths will produce almost a pure octave (actually a slightly larger interval) so the 12-tone chromatic scheme might have resulted from this fact.
Also, the fact that 12 is a composite number (it is the only one of the discussed above that is not prime!) provides some interesting ideas to toy with, like the notion of a symmetrical fully-diminished 7th chord.
Nowadays, the use of electronic media has made the access to arbitrary tunings quite easy, and there are some contemporary composers (such as Easley Blackwood or Ivor Darreg) who compose explicitly for 19-tet - and some, e.g. in the field of serial music, even like to use tunings such as 9-tet or 11-tet which do not offer anything corresponding closely to a 3:2 fifth or a 4:3 forth - a pleasing property if one wants to avoid usual harmonic associations.
Examples and a lot of informations about other tuning systems can be found on xenharmonic.wikispaces.com.

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What is a cent?

Essentially contributed by M.Schulter <mschulter@value.net>.
To express the difference in intervals between two tunings, one convenient unit is the cent, being equal to 1/100 of a 12-tet (equal temperament) semitone, so that there are 1200 cents in an octave.
For any frequency ratio r, the formula for its value x in cents is

x = 1200*ln(r)/ln(2)

where ln stands for the natural logarithm. For example, a pure 3:2 is about 701.955 cents; a 12-tet fifth is precisely 7/12 octave or 700 cents. This is a difference of not quite 2 cents, or about 1/100 of a whole-tone, not so dramatic. However, a pure 5:4 is around 386.31 cents, as opposed to 400 cents (4/12 octave) in 12-tet, and about 407.82 cents (81:64) in Pythagorean. Here 12-tet is a kind of compromise.

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Math formula for consonance?

It is an old idea, dating back (at least) to the ancient Greeks, that an interval is the more consonant the simpler the ratio between the frequencies is. There have been several approaches in history to formalize this idea, two prominent ones by Helmholtz and Leonhard Euler.
Searching for a consonance formula, we face one problem: if the pitch difference of 2 tones is below a certain boundary, the pitches will be perceived as the same - so musicians and musical instruments do not produce (and listeners do not hear) pitches with perfect accuracy, and, in consequence, there is no unique consonance value. Euler's formula is an example that illustrates these problems.
Another point is that consonance also depends on musical timbre (thus on the instrument). Helmholtz's approach can take this into account to a certain extent.
However, a simple, practical formula coinciding perfectly with music has not been found yet (and, probably, cannot be).

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Euler's formula for consonance?

Euler's approach is based on the prime factorization of frequency ratios. As is known, every integer a can be written in the form

a = (p1^k1) *(p2^k2)*...*(pn^kn)

where p1, p2,... pn are prime numbers. Then Euler defines the "gradus suavitatis" (degree of sweetness) function as:

G(a) = 1 + k1*(p1-1) + k2*(p2-1) +.... + kn * (pn-1).

If the frequency ratio of an interval is a fraction a/b (with gcd(a.b) = 1), then Euler defines G(a/b) = G(a*b). That way you can compute a numerical value for every rational interval. (Since this value is not high but low for "simple" ratios, Mazzola comments that it should better be called "gradus dissuavitatis"...)
In the table below some of the most important intervals are listed in increasing gradus suavitatis order.

Ratio Gradus suavitatis
1/2 (octave) 2
3/2 (fifth) 4
4/3 (forth) 5
5/4 (major third), 5/3 (major sixth) 7
6/5 (minor third), 9/8 (major whole tone), 8/5 (minor sixth) 8
9/5 (minor seventh) 9
10/9 (minor whole tone), 15/8 (major seventh) 10
16/15 (diatonic semitone) 11
45/32 (tritone) 14
81/64 (pythagorean major third) 15

As you see, there is a rough coincidence with music, but not a perfect one. E.g. the minor third is placed on the same level as the major whole tone, and the pythagorean major third is valued fairly dissonant: even worse than the tritone! The relevance of this formula is merely a historical one. Anyway, you may ask with right what the prime factorization of an integer has to do with consonance?!
There is just one point: the gradus suavitatis clearly measures the simplicity of an interval ratio. So one benefit of this formuila is that it forces us, if not to question, but in any case to clarify the concept of "simple interval ratio".

And irrational ratios?

In equal temperament, all interval ratios are irrational numbers, except for the octave and its multiples. For those, there is indeed NO gradus suavitatis defined. However, every irrational number can be approached arbitrarily close by a rational number (with arbitrarily complex ratios, of course); hence, since below a certain border pitch differences cannot be perceived as different anyway, this restriction is not really important.
This latter argument, however, shows a central weak point of the theory: the same mathematical property implies that arbitrarily close to any "simple" interval ratio there is another, arbitrarily "complex" one; the human perception not being able to tell them apart, which one should we take?. As I already remarked, this problem arises in any theory based on interval ratios - but Euler's formula, being highly incontinuous, looks especially bad under this aspect.

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Who was Pythagoras and what is his contribution to music theory?

Pythagoras was a greek philosopher and religious leader who lived in the 6th century B.C. His teaching was that the origin of all things lies in numbers (imagine that: a religion actually based on mathematics...). In the field of music, we owe him and/or his followers what I am tempted to call the first MaMuTh in history.

From: M. Schulter <mschulter@value.net>.
One complication in answering this question is that we have little if any _direct_ writings from Pythagoras himself, a statement true of various other pre-Socratic philosophers in ancient Greek. However, the "Pythagoreans" or followers of this teacher have indeed attributed to him some very important concepts transmitted by early medieval theorists such as Boethius (c. 480-524) and playing an influential role in medieval and later European music theory.

Most notably, Pythagoras and his followers associated the consonances - intervals which seem to "concord" or blend smoothly - with simple integer ratios. A legendary story tells us that he heard some blacksmiths harmoniously swinging their hammers, and then weighed the hammers, finding that they described the ratios 12:9:8:6. While the physics of hammers would not actually fit such a model so neatly, the lengths of strings (all things being equal) would.

Thus Pythagorean theory defines these four key intervals as the octave (12:6 or 2:1), the fifth (either 12:8 or 9:6, 3:2), the fourth (12:9 or 8:6, 4:3), and the major second or whole-tone (9:8).

It will be noted that all these numbers have ratios which are either _multiplex_ or n:1 (the 2:1 octave), or else _superparticular_ or n+1:n (the 3:2 fifth, 4:3 fourth, and 9:8 whole-tone, the last being equal to the difference of the fifth and fourth).

The Pythagoreans also recognized as concords the double octave (4:1) and twelfth or fifth-plus-octave (3:1), both multiplex ratios.
Generally, a Pythagorean approach to music attempts to explain sense experience in "rational" terms, a kind of double entendre: one should call upon the reason to be the ultimate arbitrator of musical questions, although taking into account the evidence of the senses; and musical intervals should be based on rational quantities in a mathematical sense, that is, upon ratios of integers.

The Pythagoreans -- as happened in other musical traditions, such as the Chinese -- described a method of tuning based on chain of pure fifths or fourths, a technique associated some 1500 years later with some of the greatest composed music of Western Europe, the polyphonic of the Gothic era (c. 1200-1420). A Pythagorean tuning generally yields pure fifths and fourths, also pure major seconds (9:8) and minor sevenths (16:9), not traditional Pythagorean "concords" but deemed to have some "compatibility" by certain medieval theorists, and rather active major thirds (81:64) and minor thirds (32:27). Also, the contrast between the generously wide major second at 9:8 and the narrow diatonic semitone (mi-fa, e.g. E-F, or B-C) at 256:243, can lend a pleasing quality to medieval chant as well as polyphonic cadences.

Around the middle of the 15th century, however, as Western European musicians favored more smooth thirds, this tuning was modified in various ways; and 16th-century theorists such as Fogliano, Vicentino, and Zarlino turned to other Classic Greek precedents such as Ptolemy and Aristoxenes to justify their new approaches to harmony, tuning, and temperament.

For a discussion of Pythagorean tuning and its aesthetic qualities in a medieval European setting, as well as a survey of some historical European tuning systems, visit the Pythagorean Tuning FAQ.

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Applications of the golden mean for tuning?

(See Definition of the golden mean).

From: M. Schulter <mschulter@value.net>
One famous application by Thorwald Kornerup uses the Golden Section -- the ratio toward which successive terms in the Fibonacci series tend

1 1 2 3 5 8 13 21 34 55 ...

of about 1.618... as the ratio between the diatonic and chromatic semitones in a regular tuning. This tuning, called Golden Meantone, is like a 16th-century meantone with the fifths at around 696.21 cents, or somewhere between 1/4-comma meantone (pure major thirds at 5:4) and Zarlino's 2/7-comma meantone (major and minor thirds equally impure by 1/7 syntonic comma, or ~3.07 cents). Here 1 cent is 1/1200 octave, or 1/100 of an equal semitone in 12-tone equal temperament (12-tET).
More recently, theorists Ervin Wilson and Keenan Pepper have described a tuning which I like very much where the same ratio applies, but in the other direction, between a large chromatic semitone and a small diatonic semitone, with fifths _wider_ than a pure 3:2 by around 2.14 cents. This is a very attractive tuning for 21st-century music in a "neo-Gothic" style derived from 13th-14th century Western European music, with major thirds and sixths a bit larger than Pythagorean, and minor thirds and sixths a bit smaller.
In this Fibonacci-like tuning, we have major and minor thirds close to the ratios of 14:11 and 13:11, and also diminished fourths and augmented seconds very close to the favored neo-Gothic ratios of 21:17 and 17:14. These are complex ratios, fitting a style where thirds and sixths are unstable intervals with considerable tension as well as some degree of "concord," rather in the medieval fashion.
Another use of a "Fibonacci-like" series was by theorists such as Joseph Yasser, who proposed an historical progression of tunings featuring divisions of the octave into 5, 7, 12, 19, 31, 50... parts: Yasser thus advocated 19-tone equal temperament as the next logical step after 12. Note that this isn't Fibonacci's series, but one constructed along analogous lines where each new term is equal to the sum of the previous two terms.
Another use of Fibonacci's series is to define integer ratios for intervals, specifically the "Fibonacci sixth" at or around 21:13, or 34:21, or 55:34, etc. -- approaching the ratio of Phi or the Golden Section itself, in this application representing an interval of around 833.09 cents. Note that these ratios take two successive terms of the series, often specifically from the portion 13, 21, 34, 55.
Musically, this kind of interval could be described as a "supraminor sixth" -- it's larger than a Pythagorean, meantone, or 12-tET minor sixth, and at the same time somewhat smaller than a usual major sixth or a "neutral sixth" closer to midway in between (say around 850 cents). In neo-Gothic music, it typically appears either as a regular augmented fifth (specifically in tunings with fifths at around 704 cents), or an interval resulting from two or more chains of fifths in a larger tuning set.
I use one variation on Pythagorean tuning with 24 notes per octave including some pure 34:21 supraminor sixths; they have a fascinating "nebulous" quality, and can seem at once complex and relatively concordant.
While I'm not sure about Stockhausen's specific connections with Fibonacci or Phi, I do know that he used various mathematically generated scales, so such a connection wouldn't surprise me; maybe someone more familiar with his works can give you a better answer on this point.

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What tuning system do orchestras and choirs use when not forced to equal temperament?

From: fields@arkanoid.gpcc.itd.umich.edu (Dr.Matt)
Date: Wed, 17 Apr 2002 13:53:12 GMT
Now that's an FAQ if I ever heard one. Orchestras and choirs use highly variable temperment, tending to adjust the final tonic triad of a work towards just intonation. String players tend towards local bits of pythagorian temperment. But in general, there's nothing on voices, winds, or strings that would ever force them to sing the same note at the same pitch every time, and in fact, part of the lore of learning to play is learning contexts in which it's conventional to do such things as "lower the third until it melds with the other voices", i.e. detune your instrument in the direction of a momentarily just intonation.
Woodwinds are built around idiosyncratic compromise scales, and would not be able to play much music convincingly in-tune were the players not able to constantly modify their pitches. Trumpets have two tuning slides handled with the small finger and thumb, and like any brass instrument, their pitch can be modified at the mouth. Horns have a hand in the bell dynamically modifying the escape speed of air at the end. Of course strings, trombones, and voices are a bit more obvious in their ability to dynamically tune...
If you listen to a lot of orchestras from around the world, you may begin to sense that orchestras have subtly different dialects for what counts as "in tune".

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What is higher, A# or Bb?

On a piano - and generally in 12-tone equal temperament - the same, of course! But this does not hold for, e.g., strings or voices where the players can - and often do unconsciously - play in just intonation (see also the answer to the previous question). Now, what is higher in just intonation?
For this, recall how the # and b come to existence at all - when music is to be played in different keys. F#, e.g., is used when playing in g major to make the dominant a major chord, and it is defined as a major third above the D. Similarly, a Bb (part of the subdominant of F major) is defined as a fourth below the F.
Now it gets a little mathematical (but only a little, don't be afraid): Recalling the interval ratios (e.g. from Euler's formula for consonance), we get: C to F is 4/3 (fourth), C to G is 3/2 (fifth), C to D is 9/8 (major whole tone), and D to F# is 5/4 (major third), which makes C to F# (9/8)*(5/4) = 45/32 and F to F# (45/32)/(4/3) = 135/128, about 1.0566875. When, on the other hand, we go from F# to G, we get an interval ratio of (3/2)/(45/32) = 16/15, about 1.066667 - a tiny little bit more: F# is a tiny little bit closer to F than to G. For the other sharp accidentals, the values are sometimes slightly different, but the trend is the same - and an analogous formula holds for the b's, which means that Bb is a tiny little bit closer to B than to A. The conclusion of this would be that Bb is higher than A#!

Unfortunately, this is not exactly the end of the story. For the tuning system sketched above is not the only one that has been relevant in the historical development that lead to the current 12-tone equal temperament. There is another approach, the pythagorean one, where notes are derived exclusively by chaining fifths - also a tuning that musicians somethimes unconsciously tend to play, especially in the violin family. In this tuning, the F# is defined as one fifh above the B, which was derived in turn as one fifth above the E and so on. And for this tuning, the analogous mathematical calculations as above yield that sharps are higher than flats - the exact opposite result...

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What is a cadencial set?

A cadencial set, or cadence-set, is Mazzola's terminology for a set of vertical sonorities that determine a key completely. E.G., the chords IV and V uniquely fix a certain major key, as well as the chords V7 and I - which may be one of the reasons that these became popular as parts of cadences.
Note that the property is based on sets, not on progressions - it is "purely vertical" and does not include the time component. Hence it is not sufficient for musical styles where beginning notes or chords are distinctive marks (such as the system of modes in medieval and renaissance music).
If a certain set is cadencial, then every set containing it is cadencial, too. So the really interesting things are minimal cadencial sets.
From this viewpoint, there is an interesting polarity between major and harmonic minor. From the 7 triads of the diatonic major scale we can build the following 5 minimal cadencial sets: {II, III}, {III, IV}, {IV, V}, {II, V} and {VII} - whereas in harmonic minor, every pair of triads forms a minimal cadence-set. Hence we have 21 unique minimal ways to fix a certain minor key - which is, BTW, the absolute maximum obtainable with 7 3-element sonorities.

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What is a harmonic braid?

"Harmonic braid", after an idea by Schönberg, is the name for the following mathematical construction: Take the 7 triads of a diatonic scale, numbered from I to VII. Draw a point for every chord. For every pair of chords having at least one tone in common, draw a connection between the points. For every triple of chords having at least one tone in common, draw a triangle between the points.
The result of this construction is, mathematically, a simplicial complex. It looks as follows:

It has the form of a moebius strip. On the border the triads are ordered in forth steps (or fifth, depending on the direction) - except for IV - VII which is a tritone.
Every triad has connections to 2 "parallel triads" on the opposite border of the braid. (The term "parallel triad" is used in the music theory of Hugo Riemann; the harmonic braid provides a geometric visualization.)

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What is a nerve?

The harmonic braid is an example for a mathematical construction called "nerve" that originated in topology where it is used for coverings (collections of subsets whose union is a certain set). In the case of the harmonic braid the triads form a covering of the diatonic scale.
Nerves play a prominent role in mathematical music theory. They are used to visualize modulations - as geometric structures connecting 2 harmonic braids - as well as in musical analyses of various kinds - which, BTW, nearly always involve coverings of some sort. (There is just the problem that nerves quickly become 4- or 5-dimensional...)

Sorry, no more answers!

© Hans Straub

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