Definitions of tuning terms

My abbreviation for the usual and familiar 12-tone equal-tempered scale, the standard scale tuning used on keyboard, fretted string, and woodwind and brass instruments.

It is calculated by taking the twelfth root of each successive power of 2, from 0 to 11, with higher or lower "octaves" of these 12 notes assumed to be, and tuned as, equivalents. The 12th root of 2 can also be written as 2^(1/12), the next degree, the 12th root of 2², as 2^(2/12), etc., which is the way I normally write it on this website.

As any number to the 0th power equals 1, the starting note of the scale, or 2^(0/12), has the ratio 1:1.

As these proportions, with the exception of 2^(0/12), are irrational numbers, this scale was never tuned with absolute precision, with the exception of the "octaves" of the starting-note, until just a few decades ago, with the introduction of electronic instruments and tuners.

It is frequently abbreviated as 12-ET or 12tET by others.

An interval measurement invented by Alexander Ellis and appearing in his appendix to his translation of Helmhotz's On the Sensations of Tone.

A cent is calculated as the 1200th root of 2, or 2^(1/1200), with a ratio of approximately 1:1.0005777895. It is an irrational number.

The formula for calculating the cents-value of any ratio is:

cents = log₁₀(ratio) * [1200 / log₁₀(2)]

Cents are almost universally used as a small logarithmic measurement to compare interval sizes, however, I prefer Semitones.

one of several small intervals with a size of about an "eighth-tone" [= 0.25 Semitone = 25 cents].

The three commas which are most commonly encountered are the Syntonic, the Pythagorean, and the Septimal.

When used unqualified, it frequently refers specifically to the Syntonic Comma.

an Otonality (or Utonality) which contains, with possible octave displacements, every odd harmonic (or subharmonic, respectively), through the odd number n, of the fundamental tone, is said to be complete.

[from Paul Erlich]

An equal temperament with an integer number of notes per octave is consistent with JI through some odd limit if a complete chord of that limit is constructed in that equal temperament in the same way no matter which intervals are approximated.

If for all odd integers a, b, c such that 1 <= a <b <c <= n, the ET's best approximation of b/a plus the ET's best approximation of c/bequals the ET's best approximation of c/a, then the ET is consistent in the n-limit. For example, the smallest ETs consistent in the 11, 13, 15, and 17-limits are 22, 26, 29, and 58-tET, respectively.

a variety of small intervals were called diesis in ancient Greek theory. The term was revived in medieval Europe when the ancient Greek treatises became known and were translated into Latin. There was no one specific measurement.

a system of tuning based on a scale whose "steps" or degrees have logarithmically equal intervals between them, in contrast to the differently-spaced degrees of just intonation, meantone, well-temperament, or other tunings.

Usually, but not always, equal temperaments assume octave-equivalence, of which the usual 12-EQ is the most obvious example.

Examples of non-octave equal temperaments are Gary Morrison's 88-cET (88 cents between degrees), and Wendy Carlos's alpha, beta, and gamma scales [listen to them here].

the usual abbreviation for Equal Temperament.

A specific scale is expressed as, for example, 12-ET, or 12tET [= "12-tone equal temperament].

the smaller integers into which a larger integer may be divided.

A mathematical rule which states that any number can be described as the product of its prime-factors, with each prime in the series raised to various exponents, 0 or positive.

By extension, this method can also be used to factor rational proportions into the series of primes or odd-numbers, with 0, positive, or negative exponents, the negative exponents representing the denominator of the ratio.

This makes it easier to understand and visualize the mathematical relationships between ratios (especially small-number ratios), to make calculations between ratios by matrix addition rather than fraction multiplication, and in general, to avoid very large numbers in the proportional terms. It is an essential element in lattice diagrams.

one of the correlatives, "major" or "minor", in a tonality; one of the odd-number ingredients, one or several or all of which act as a pole of tonality (for example, 1-3-5-7-9-11 in Partch's theory)

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 75]

a number which has no decimal or fractional part.

A number which cannot be described as an exact ratio between two proportions. Irrational numbers have an infinite and non-repeating string of digits after the decimal point. Two of the most famous examples are pi and the 12-EQ Semitone.

Charles Lucy submitted the following:

From VAN NOSTRAND's Mathematics Dictionary:

IR-RA'TION-AL, adj. Irrational algebraic surface. The graph of an algebraic function in which the variable (or variables) irreducibly under a radical sign..

IRRATIONAL NUMBER.
A real number not expressible as an integer or quotient of integers; a nonrational number. The irrational numbers are those numbers defined by sets (A,B) of a Dedekind cut such that A has no greater member and B has no least member. Also, the irrational numbers are precisely those infinite decimals which are not repeating.

The irrational numbers are of two types, algebraic irrational numbers (irrational numbers are the roots of polynomial equations with rational coefficients:) transcendental numbers. e.g., e and pi.

the usual abbreviation for Just Intonation.

a system of tuning based on notes whose frequencies have small-integer rational relationships.

The scales produced are almost always unequally-spaced, and usually exhibit various types of symmetry.

When unqualified, "just intonation" generally means a 5-limit tuning, as described in my paper. Systems with a higher limit are frequently called extended just intonation.

A cycle of 3/2s ["5ths"], or of any other just ratio, will never return exactly to the frequency of the origin.

a visual representation of the mathematical relationships of musical ratios in 2-, 3-, or multi-dimensional space, consisting of points which represent the ratios as positions calculated according to the Fundamental Theorem of Arithmetic.

Lattices may be based upon two types of factoring: either odd or prime - similar to the two types of limit. In either case, a vector is drawn or imagined to represent each factor, with exponents represented as a series of points along that vector.

Angles between and lengths of the vectors are not standardized, with simple triangular or rectilinear lattices popular in ASCII text use. Probably the most complex diagrams have been designed by Erv Wilson, many of whose lattices form beautiful mandala-like designs. John Chalmers has made very complex diagrams of triangles representing tetrachords.

The precursors to musical lattice diagrams are the Lambdoma, Ellis's Duodenarium and Riemann's matrix charts. Harry Partch's Tonality Diamond is related but slightly different.

Adrian Fokker apparently designed the first 3-dimensional lattices with factors of 3, 5, and 7 represented. Theorists known for their lattice diagrams are Erv Wilson, Ben Johnston, John Chalmers, David Canright, Graham Breed, Paul Erlich, and myself.

[from Paul Erlich]

a system of tuning in which the two JI "whole tones" (with ratios of 9:8 and 10:9) are conflated into one "mean tone" which lies between the two, the objective being to produce JI "major 3rd"s with a ratio of 5:4.

The "5th" is either flatter or sharper (by a fraction of a Syntonic Comma) than the just 3:2, in order to provide a closed system where a "cycle of 5ths" eventually returns to the origin. Thus the names "Quarter-comma meantone", "2/9-comma meantone", etc.

Latin for "one-string": an instrument constructed of a sound-box and a single string, with several bridges used to delineate various rational or geometrical (irrational) intervals.

The instrument was not assumed to have any practical value in the playing of music, but was strictly for purposes of theoretical demonstration and experiment, and was widely used for such from the time of ancient Greece until about the baroque.

an organization of musical materials based upon the faculty of the human ear to perceive all intervals and to deduce all principles of musical relationship as an expansion from unity, as 1 is to 1, or -- as it is expressed in this work -- 1/1.

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 71]

the number common to all identities in the ratios of one tonality -- the common anchor; the characteristic of a series of ratios that determines them as a tonality. In the 8/5 Otonality the Numerary Nexus is 5, as seen in the sequence of the six Odentities:

11/10 [= 11/5 in Partch's theory]
9/5
7/5
6/5
1/1 (= 5/5)
8/5

In the 5/4 Utonality the Numerary Nexus is also 5, as seen in the sequence of the six Udentities:

20/11 [= 5/11]
10/9 [= 5/9]
10/7 [= 5/7]
5/3
1/1 (= 5/5)
5/4

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 75]

A tuning which provides neither the small-integer ratios of just-intonation, nor the equal "steps" or divisions of Equal Temperament.

The well-temperaments may be considered a subset of these, although in modern usage (such as Brian McLaren's pieces) these scale are also frequently non-"octave".

based on the unique property of the 2:1 ratio, commonly called the "octave", that although it is a different pitch from the origin 1:1, it seems to have the same aesthetic affect or properties as 1:1.

Traditional music theory assumes octave-equivalence, thus the letter-names of the notes repeat in the different "octaves".

Many tuning systems follow this approach, but not all.

Examples which do not are:

Modern acoustical research is showing evidence that most individual's perception of what is consonant is more complex than the long-held belief by many music-theorists and scientists that consonance is directly related to the size of the integer terms in the ratios and/or the size of the prime number factors.

[McLaren's website will have much information on and quotations from this research - one citation refers to an interval of 12.15 Semitones as that most commonly perceived as a consonant "octave".]

Johnny Reinhard wrote an interesting paper on a study he did of a song by two Sapmi (also known as Lapp) singers of northern Scandinavia. There were very minute but deliberate interval dissonances between them, and tiny changes in these intervals in each of the 9 repeating verses. One of the most prominent was a frequently-used mistuned harmonic "octave" which ranged from about 11.90 to 12.04 Semitones.

an integer which cannot be divided by 2; the series of every other integer beginning with 1.

The ancient Greeks are purported to have created the distinction between even and odd numbers because of their musical studies. Because the 2:1 frequency ratio, or "octave", gives a sound which so strongly resembles the 1:1 or starting pitch, the Greeks characterized this 2:1 relationship as female and even, with the 1:1 as male and odd.

one of the Otonality correlatives...; example: "the 9 Odentity of 8/5 unity is 9/5".

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 72]

a tonality expressed by the over numbers [numerators] of ratios having a Numerary Nexus -- in current musical theory, "major" tonality.

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 72]

a term coined by Johnny Reinhard to characterize the use of any variety of different tunings in the same piece of music.

A stunning example of this is Reinhard's piece Raven, where each phrase on each instrument and voice had its own unique character due to the use of completely individual tunings.

a number which has as its factors only itself and 1.

The prime series is infinite, and as yet no one has discovered a pattern in its progression. Primes are are important in number theory because of the Fundamental Theorem of Arithmetic.

The first fourteen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43.

Some theorists attach special importance to the prime numbers as they seem to each produce a unique affect in music, at least for the lower primes. Much is still unclear about this idea, but it has been explored by Harry Partch, Ben Johnston, Alain Daniélou, La Monte Young, Scott Makeig, and myself, among others.

My own feeling is that, although there are many factors that go into the feelings and perceptions we have of music, the primes seem to be a kind of cognitive archtype or template that helps us to grasp or understand the harmonic relationships we are hearing in music, and that the more sophisticated use the performer/ composer makes of subtle prime relationships, and the better the listener understands those relationships, whether intellectually, intuitively, or viscerally, the more rewarding the experience for both performer and listener.

There is much debate about this, in particular, where the boundaries of our perception of these effects lie within the primes and within the exponents of the primes, and to what extent ratios which have greatly differentiated prime-factors but which lie very close to each other in pitch resemble or differ from each other.

But it is undeniable that simultaneous notes bear harmonic proportional frequency relationships, and that the human cognitive apparatus will perceive those relationships in the simplest way it knows.

an adjective describing the construction of a scale or musical system by successions of 3/2's (just "perfect fifths")

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 75]

In JustMusic terminology, such a system would be described as a cycle of powers of 3. Pythagoras was given credit in ancient Greece for discovering the properties of musical ratio numbers, with an emphasis on what we would today call powers of 3.

The ratio 531441/ 524288, in JustMusic prime-factor notation designated as 3¹², with an interval size of approximately 0.23 Semitones [= 23.460+ cents].

It is the difference between the Pythagorean or 3-Limit "tritone" or "augmented 4th" of approximately 6.12 Semitones [= 729/512 = 3⁶ = 611.730+ cents] and the Pythagorean or 3-Limit "diminished 5th" of approximately 5.88 Semitones [= 1024/729 = 3^-6 = 588.269+ cents].

It was first described c. 300 BC by pseudo-Euclid in Divisions of the Canon.

a relationship, or interval, expressing the vibrations per second, or cycles, of the two tones concerned, generally in the lowest possible [integer] terms;...simultaneously a representative of a tone and an implicit relationship to a "keynote" -- or unity.

Euclid's definition: a mutual relation of two magnitudes of the same kind to one another in respect of quantity.

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 73]

a chord to which no note may be added without increasing the chord's odd limit is said to be saturated.

For example, the four saturated 9-limit chords are:

[from Paul Erlich, "Tuning, Tonality, and Twenty-Two Tone Temperament", Xenharmonikon 17, footnote 33]

See Graham Breed's Anomalous Saturated Suspensions for more info.

A term coined by Alexander Ellis in his translation of Helmhotz's On the Sensations of Tone, and originally spelled skhisma. It designates an extremely small interval, just barely discernible to human pitch-detection.

Unqualified, it is the difference between 5¹ [= 5/4 = 3.86 Semitone = 386.313+ cents] and
3^-8 [= 8192/6561 = 3.84 Semitone = 384.359+ cents] and has an interval size of approximately 0.02 Semitone [= 3⁸5¹ = 32805/32768 = 1.953+ cents].

As no combination of different prime numbers will ever produce ratios which have exactly the same interval size, if cycles of a particular ratio are calculated far enough, very small intervals like this eventually appear between ratios having different sets of prime factors. When these are under consideration, the term schisma is qualified with a latin word designating the higher prime, with the assumption that the other prime being compared is a more familiar one, almost always 3.

Thus, we get the septimal schisma, which is the difference between 7¹ [= 7/4 =
9.69 Semitone = 968.825+ cents] and 3^-14 [= 8388608/4782969 = 9.73 Semitone = 972.629+ cents] and has an interval size of approximately 0.04 Semitones [= 3^-147^-1 = 33554432/33480783 = 3.804+ cents].

Likewise, there is the nondecimal schisma, which is the difference between
19¹ [= 19/16 = 2.98 Semitone = 297.513+ cents] and the standard Pythagorean "minor 3rd"
3^-3 [= 32/27 = 2.94 Semitone = 294.134+ cents], and which also has an interval size of approximately 0.04 Semitones [3³19¹ = 513/512 = 3.378+ cents].

I feel that since the prime-factor or ratio notations give precise measurements, and 1/1200th of an "octave" is approximately the limit of human pitch discrimination, more precision than this is not ordinarily needed, and I prefer to use the decimal point so that the interval may be related immediately to the familiar 12-EQ scale. I use cents on occasion, when I feel that more precision is valuable.

In either case, the Semitone is calculated as the 12th root of 2, or 2^(1/12), an irrational proportion with the approximate ratio of 1:1.059463094359.

The ratio 64/63, in JustMusic prime-factor notation designated as 3²7¹, with an interval size of approximately 0.27 Semitones [= 27.264+ cents].

It is the difference between the Pythagorean or 3-Limit "7th" of approximately 9.96 Semitones [= 16/9 = 3^-2 = 996.089+ cents] and the "harmonic 7th" of approximately 9.69 Semitones [= 7/4 = 7¹ = 968.825+ cents].

The relative consonance/ dissonance of an interval.

Rather than use this phrase (as Partch did), I have adopted the single term sonance, because I agree with Schoenberg's (and Partch's) assertion that rather than describing two diametrically-opposed sensations, consonance and dissonance refer instead to the opposite poles of a single continuum of sensation.

The ratio 81/80, in JustMusic prime-factor notation designated as 3⁴5^-1, with an interval size of approximately 0.22 Semitones [= 21.506+ cents].

It is the difference between the Pythagorean or 3-Limit "major 3rd" of approximately 4.08 Semitones [= 81/64 = 3⁴ = 407.82+ cents] and the 5-limit so-called "just" "major 3rd" of approximately 3.86 Semitones [= 5/4 = 5¹ = 386.3137+ cents]. Also known as the Comma of Didymus, who was the first theorist to specify the use of 5/4 in music theory; frequently referred to simply as comma.

a tuning which is not a just-intonation; that is, the intervals are not small-integer ratios.

The various kinds of temperament are equal, meantone, well, and non-just non-equal.

a psychological phenomenon having as its chief characteristic a tonal polarity around a 1 Identity; the sounding of various of the identities -- either Odentities or Udentities -- with a Numerary Nexus will create this polarity, and the smaller the odd-number identities played the stronger the polarity.

Acoustically, the identities of a tonality represent the maximal consonance of any given number of stipulated identities because of the Numerary Nexus:

the tones of the triad 8/7 - 10/7 - 12/7 -- with the Numerary Nexus 7 -- are in the relation 8:10:12, or 4:5:6 (Identities 1-5-3), and create a polarity around 8/7, the 1 Odentitiy. This is the maximal consonance that three different tonal identities in music can attain.

Were the triad 10/7 - 12/7 - 7/7 (= 1/1) chosen, with the relation between the tones of 5:6:7 (Identities 5-3-7), the polarity around 8/7 would still be created, though it would not be as strong.

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 75]

an arbitrary arrangement of the Monophonic ratios designed to constitute prima facie proof of the at least dual identity of each ratio, and consequently of the capacity of a Monophonic system of Just Intonation for providing tones that may be taken in more than one sense each.

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 74-75]

Partch's Incipient Tonality Diamond (5-Limit):

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 110]

(for a different view of this diamond, see the Monzo 5-Limit Lattice)

a chord which contains three different notes.

one of the Utonality correlatives...; example: "the 5 Udentity of 7/4 unity is 7/5".

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 75]

An equal temperament uniquely articulates JI in some odd limit if all just intervals of that odd limit are approximated by different numbers of steps in the ET.

For example, 12-tET does not uniquely articulate the 7-limit because 7:5 and 10:7 are both approximated by 6 steps and 6:5 and 7:6 are both represented by 3 steps. The data for this case are tabulated in the last column of this table.

Although this definition is perfectly applicable to non-octave equal temperaments, uniqueness may also be defined with respect to an "integer limit". The data for this case are tabulated in the second-to-last column of this table.

[from Paul Erlich]

one of those tonalities expressed by the under numbers [denominators] of ratios having a Numerary Nexus -- in current musical theory, "minor" tonality.

[from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 75]

a system of tuning in which the intervals between degrees are unequal, and in which the members of various chords approximate just ratios to various degrees of accuracy, depending on the "root" of the chord.

This produces a sound which gives a different "color" or affect to the different chords and keys.

Modern research has reached somewhat of a consensus that J. S. Bach's infamous Well-Tempered Klavier was written for keyboards tuned in a well-temperament (hence the name of the work), thus exploiting the different characters of the 24 major and minor keys in which the pieces are written, in contrast with the formerly-held opinion that this work demonstrated the "usefulness" of 12-EQ.

A well-temperament is generally named after the theorist who first wrote about it. Two of the most famous are Werckmeister III and Valotti & Young.