Glossary of Music Tuning Definitions
A musical tuning dictionary for ethnomusicologists, early music buffs, xenharmonicists, and others.
interval of equivalence (IoE)
moment of symmetry (MOS)
quarter comma meantone
Similar to the asphyxia suffered by mountain climbers, acompositia is an illness that can afflict those who spend their lives exploring the frontier of sound. Symptoms are a feeling of peace, warmth and bliss, and a desire to never return from the mountain. Victims of acompositia never write back a postcard to their friends or family in the form of a new composition, essentially disappearing into the wilderness, their compositional potential vanishing like a lost martian space probe that just stops transmitting and no one knows what happened.
Also referred to as Generalized Keyboard.
An array keyboard has keys arranged two dimensionally rather than one dimensionally like a piano. A guitar neck could be seen as one sort of arrayed instrument. Array keyboards have been made for over 100 years and there are several companies that make them today, as well as a number of one-off designs that have been made by artisans over the years. Array keyboards can have the property of being isomorphic, meaning same shape, in which case chords and melodies are played with the same geometrical pattern no matter what key they start on. This differs from the piano’s Halberstadt keyboard layout which has three different hand shapes needed to play a major root position triad, depending on the starting key.
A bearing plan is a set of directions for tuning an instrument by ear.
A cent is a unit of relative pitch measurement. When you play two notes together, you can explain the difference in pitch between them by taking the ratio of their frequencies which is an exponential measurement, or you can take the logarithm of the ratio of their frequencies which is a linear measurement. The most common way to do linear measurements of pitch is to use cents. There are exactly 1200 cents per octave of 2/1. There are exactly 100 cents per equal tempered semitone. A pythagorean fifth (frequency ratio of 3/2) is 701.9550009 cents.
Abbreviation meaning cents Equal Temperament. Always used after a number. 88cET (cents equal temperament) refers to a nonoctave tuning consisting of equally sized steps which are 88 cents apart, which is 88/1200 of an octave since there are 1200 cents to an octave. Or alternatively, where the steps are 88/100 or 88% of the distance of the 100 cent steps of 12 note per octave equal temperament.
The keyboard of a musical instrument, usually referring to a Halberstadt style keyboard as is found on typical pianos, clavichords, harpsichords, organs and synthesizers. The term clavier might be preferable over keyboard because it distinguishes the musical clavier keyboard from the QWERTY or Dvorak style console keyboard used on typewriters and computers. Asking someone to “press the B key on your clavier” can be less ambiguous than asking them to “press the B key on your keyboard”.
If you have two intervals that are close to one another such to the extent that you might try to make one pass for the other, or want them to be equivalent under some use, the difference between the two intervals is called a comma.
For example, the difference between a stack of twelve fifths of 3/2 and a stack of seven octaves of 2/1 is (3:2)12 / (2:1)7, which is 531441:524288, or 23.46 cents, roughly an eighth of a whole tone. Much ado is made about the fact that these two intervals don’t match in a theoretical tuning of pure fifths called Pythagorean Tuning, and the western standard of twelve tone equal temperament is said to be the solution to this conflict.
This conundrum, considered an unsolvable puzzle that has lasted centuries because of unchangeable facts of unyielding mathematics, is nothing more than a case of small-minded thinking inside of a box of one’s own choosing. First, there are an infinite number of useful and marvelous tunings that don’t even need such constraints. Second, if you really must have all pure fifths in a conventional sounding tuning, it is a trivial matter to use the seventh root of 3:2 as your basic chromatic step instead of the 12th root of 2:1. This yields a slightly stretched octave of (3:2)12/7 or 1203.35 cents, which is not just only slightly and unnoticeably sharp of the 2/1 octave at 1200.0 cents, but it is basically the same octave that is used to tune all modern pianos anyway. Desiccate the comma by tempering the octave instead of the fifth and you have a great and useful tuning that you are already familiar with.
Two notes played together. Related to triad, which is three notes played together.
The complete set of notes that can be played on an instrument is the instrument’s gamut. For example, the gamut of a modern pianoforte consists of 88 notes, from A0 to C8. Gamut is a synonym for range. The popular expression to run the gamut means to span the complete range of possibilities.
The modern piano keyboard has seven white keys in a diatonic
scale pattern interwoven with five black keys in a pentatonic
pattern. This keyboard layout was first seen on the cathedral
organ in Halberstadt, Germany, in 1361. Because of this, the
formal name of this style of keyboard arrangement is the
Halberstadt keyboard. We use this term to help distinguish
between the many different types of keyboards, from array keyboards to typewriter keyboards.
Hastings Tunings describe cure-all tuning approaches and theories that are aggressively promoted by non-practitioners, and adopted by inexperienced folks who are subsequently driven to madness and compositional death in a futile attempt to follow the route laid out for them, unable to complete their adventuresome journey in microtonality by writing compelling music.
The spell of marketing materials promoted by marmalade eating cookie pushers with limited real life experience can be mesmerizing. Brochures and advocates often come across as confident, enthusiastic and well presented. Promises of a One True Tuning, or just a greater ease of composition can seem like a dream come true to someone struggling with the first steps.
Promising new paths will be accompanied with evidence that they have been completed by others and are real. This evidence is the presence of compelling music, of real life journeys completed using the advocated path. It is not sufficient just to find enthusiastic advocates.
Hastings Tunings are named in honor of the historical contribution of Lansford Hastings.
In 1845, Lansford Hastings, a wealthy attorney, published a book called “The Emigrants’ Guide to Oregon and California”, encouraging Anglo Saxons to embrace their manifest destiny and effectuate a bloodless revolution by taking over the west coast from Mexico through emigration and white population growth. He advocated saving 350 miles off the Oregon Trail by taking the Hastings Cutoff, a theoretical route across the Great Salt Lake. What he failed to tell anyone is that neither he nor anyone else had ever traveled this route.
The route was aggressively promoted by him and also an outpost merchant named Bridger who stood to financially gain from the alternate route, if it was ever developed.
The first time Hastings ever took the route himself, it was ahead of the Donner party. The Donner party was given a brochure advocating the Hastings Cutoff and, finding themselves ill prepared for travel, and falling behind as the last wagon train of the season, felt it was a miracule cure that would enable them to save time. As one member of the party wrote en route after receiving the marketing brochure: “Hastings Cutoff is said to be a saving of 350 or 400 miles and a better route. The rest of the Californians went the long route, feeling afraid of Hastings’s cutoff. But Mr. Bridger informs me that it is a fine, level road with plenty of water and grass. It is estimated that 700 miles will take us to Captain Sutter’s fort, which we hope to make in seven weeks from this day.”
The Donner party subsequently faced a route with no road that was not flat where they spent weeks at a time clearing trees and boulders to allow their wagons to pass. Eventually they succumbed to starvation, frostbite, cannibalism and madness.
Hertz is a unit of measurement of frequency it means cycles per second. When a conductor says that concert pitch is A-440, he means that the A above middle C is tuned to 440 Hz, or 440 cycles per second. This means that the sound you hear when you play A is vibrating 440 times per second.
Intentional microtonality is deliberate, chosen, conscious microtonality. This term is inspired by intentional communities, which are communes and other arrangements whereby the residents choose to create or join a community because of the community and not because it just happens to be where they live. Authentic historical revival music and ethnic music are often microtonal, but the microtonality comes as part of the environment. Players playing out of tune may sometimes be called microtonal, but is not intentional. Intentional microtonality is when the composer is aware of microtonality and chooses the tunings for a new piece of music.
Isomorphic is a Greek term which means same shape.
Isomorphic keyboards have consistent and regular key placements rather than irregular such as with a piano’s Halberstadt keyboard. On an isomorphic keyboard, assuming equal sized intervals, any identical physical relation between two keys will sound the same interval. Likewise chords with the same physical shape played plays the same chord.
Isomorphic keyboards are found in both one and two dimensional arrangements.
In two dimensions, keys may be in a grid or hexagon tiling. For two dimensional
keyboards, for the chords to be isomorphic, there would be equal intervals in each dimension, but each dimension can have a different interval, as in a two dimensional harmonic lattice. These keyboards can also be used to work with other sorts of non-equal scales. MOS scales, with two interval classes for each span of keys are one choice, but one can also go free form and have a wide
variety of intervals per shape, retaining the advantages of key density in a 2D keyboard, while setting aside the consistency possibilities from the isomorphy.
Some examples of isomorphic two dimensional keyboard layouts are the Bosanquet and Fokker arrangements of 31tET in a hexagonal keyboard (the Fokker has rectangular shaped keys arranged hexagonally: a hexagon can be represented as a grid where every other column or row has been offset); the Wicki-Hayden layout used in some Concertina accordians; the Janko layout used in some 19th century pianos; the Wesley Array system; and others.
Guitar fretboards can be considered a form of isomorphic keyboard as well, and isomorphic tunings are possible on the guitar such as by tuning all open strings to fourths, although other intervals are possible such as tuning open strings by the supermajor third (9/7) when working with an 88cET guitar.
Intervals which are related by a frequency ratio in which both the numerator and denominator of the ratio are small integer ratios like 2, 3, 5, 7 and 11 are called Just intervals. Thus, 3/2, 11/9, and 7/5 are just ratios. When we choose to work only with these kinds of intervals, that is called Just Intonation.
A linear scale is defined with two values: a Repeat Ratio, and a Base Interval. The Base Interval is often called the Generator. If you make a chain of stacked base intervals, such as fifths one on top of another, and then reduce the size of the resulting ratios that are larger than your Repeat Ratio by that Repeat Ratio until they fall within, then you have a linear scale.
You can also slightly stretch, compress, or knead the Base Interval in order to change the entire scale. This process of making a small change to the Base Interval is called tempering. If you have a base interval of a fifth (the ratio of 3/2) and a Repeat Ratio of the octave (2/1) and temper the Base by a quarter of a small interval called a syntonic comma (the ratio of 81/80), you have the famous historical tuning discovered by Pietro Aron in 1523, which gains pure thirds and sixths in trade for having flattened fifths.
Linear scales tend to generally sound good and are easier to work with because of their consistency and readily recognizable structure, which the human brain seems to understand instantly.
LMSO has a built in graphical editor called the Knead & Fold Appliance for instantly creating and exploring linear and MOS scales.
We define microtonal widely, to be scales that aren’t 12 tone equal temperament. This includes both xenharmonic scales and more conventional diatonic sounding scales. Scales may be based on just, equal, or other types of intervals.
Our definition is close to that used by composer Easley Blackwood, who stated simply, “Microtonal tunings are those that divide an octave in some other manner than into twelve equal parts.” We expand on Blackwood by not requiring division of an octave.
I consider all ethnic tunings to be microtonal. Some don’t agree but what can you do.
Whether historical western tunings are included is a matter of taste. I personally include them but have no conflict those who disagree. Most people readily agree that the Baroque harpsichords which were built with 16, 19 and 31 keys per octave in a variety of tunings are microtonal, but whether the 12 note harpsichords tuned to non-12 are microtonal is controversial to some. I propose that if a ordinary 5-limit just intonation major diatonic scale is microtonal, then certainly a 12 key version of Werkmeister III is microtonal as well.
People interested in stepping outside the conventional can accurately call themselves microtonalists, but to see where things start to get really interesting, you need to try out xenharmonics.
There is a controversy about this term. Etymologically, the term microtonal could be interpreted to mean pertaining to small tones. Presumably then, tone is assumed to refer to a whole tone of either a 9/8 ratio, or of the 200 cents found on a piano’s whole step (ie, from C to D). This definition is a bit quixotic since the term tonal when used in a musical context usually does not mean pertaining to whole tones, but instead means pertaining to tonality.
Some of those who favor the whole-tone based definition enjoy being particular, and proceed to conclude that microtonality can not include tunings such as 5 equal steps per octave because it has equal sized steps of 240.0 cents. They then may go on to define a cut off point of intervals that are at least smaller than a tone or semitone, or some other somewhat arbitrary choice which is then further debated.
These debates do not make useful distinctions because they don’t help to write music, nor do they have much practical meaning in terms of music practice or listening. Their main purpose seems to be to distract attention away from the practice of making music and focus instead or pre-compositional activities that seem to never culminate in a creative act.
Furthermore, these sorts of definitions eliminate arabic maqams, turkish maqamat, and even many Indian ragas from being microtonal, which is contrary to long established thought.
Some smart composers, exhausted by drama and division from pushy musicologists promoting precise pronouncements will avoid the ambiguity of term tonality by musing instead upon the mastery of microtuning. Microtuning as a verb means “to make minute adjustments to the intonation of musical instruments, thereby changing the character of musical melody and harmony.” As a noun it refers to the product of that activity. This may be a wise term to use tactically if you are at a convention comprising mostly of non-composers and wish to avoid counterproductive debates.
I regret that the definition of such a basic term spends so much time dealing with debate, but no matter how much is written, there are angry argumentative persons who are troubled by those who disagree with their perspectives. Please let this be the end of it. I do not wish to debate this. We essentially follow Blackwood’s definition. Creating artistic value does not come from devaluing others. It is perfectly fine and acceptable to prefer other terms.
Modulation is the change to and subsequent establishment of a particular tonality in music.
Note that I have sidestepped both defining tonality and explaining what it means to establish it, but maybe we can agree on this definition as is, and then move on to deciding what is tonality and what it means to establish it as a separate discussion.
Moment of Symmetry (MOS)
A MOS scale is a linear scale which also has something called Myhill’s property. MOS scales have the property that a given interval size played on a keyboard will have a more predictable pattern of intervals as mapped to a span of keys, or to a dyad shape on an isomorphic keyboard.
Myhill’s property means that for any given number of scale degree steps, when considering every possible starting point in the scale, the steps span no more than two different sizes of the interval. As an example, in quarter comma meantone with 12 notes, at every position in the scale, 7 scale degrees span one of two possible intervals. In this case both are fifths: the meantone fifth and the wolf fifth.
Most linear scales with two chromatic step sizes have Myhill’s property. Not all scales with two chromatic step sizes have it.
It is common in western european music to form tunings by arranging transposed copies of a scale end to end. Each scale spans an octave, and thus the scale’s pattern repeats at the interval of an octave to form a tuning. These are octave based tunings.
- If you repeat your scale at some other interval, or do not repeat your scale, you have a nonoctave tuning. Nonoctave tunings might have some incidental octave intervals within the tuning, but the octave is not the immutable constant element within the underlying structure of the scale, as it is with western tunings.
- Sometimes scales that are made up of stretched or compressed octaves are called nonoctave as well.
- In the strictest usage, a nonoctave tuning would not repeat at any octave or near-octave, and would not contain any intervals near an octave at all. These sorts of tunings can be more difficult to find, and you quickly run into the problem of defining what it means to be “near” an interval. A wily composer can stretch the limits of perception. One area where you’d find purely nonoctave tunings is certain macrotonal scales, such as the very xenharmonic 9th root of 7:3 which has a chromatic step size of 162.986 cents. However, even this tuning has intervals of 1141 cents and 1304 cents which in some compositional contexts could be made to sound like a flat and very sharp octave respectively. Likewise, 88 cent equal temperament’s 1232 cent interval can easily be made to sound like an octave, but 88cET is clearly a nonoctave tuning.
In microtonality, a numerologist is someone who writes articles about tuning which sound very impressive and involve complex operations but have no demonstrated recognizable musical value. Generally, a numerologist neither composes nor performs music. Some will generate pieces from fanciful algorithms or other methods which result in sounds which do not reflect known principles of music aesthetics except in the most abstract sense, such as to be symmetrical in some way.
An Oven is your basic document type in LMSO. It holds your Recipe, which is your the scale of your tuning. And it also holds the Anchor information, which is a key number and a frequency, enabling a fixed pitch reference to the tuning made from your scale so that it can actually be applied to an instrument, retuning it. Ovens also contain scale information such as the author, date of creation, keywords, references, and description of your scale. The Oven also holds information about how the scale will be mapped to your keyboard or other instrument in any of several different useful mapping patterns.
Pleng is an Indonesian gamelan term relating to the aesthetic value of avoiding beatless intervals, because these can sound dead and lifeless. Pleng is alive. A good tuning always has pleng. A tuning in which all intervals are so perfectly tuned so as to be beatless, has no pleng and is thus undesirable. Like many Indonesian words, pleng is onomatopoeic — it sounds like what it describes: plennnnng, the sound of a nice vibrant shimmer when notes are played together.
Some pleng comes from tuning instruments in an ensemble in pairs, each tuned slightly differently. Another element comes from using somewhat nonoctave repeat intervals. And another part comes from using timbrally derived variations of a scale in each register, keeping the notes in a given register in an ensemble tuned together, but not from register to register. Keeping pleng in a tuning is not arbitrary, but is a sensitive artistic adjustment done by the ear of the gamelan maker working together with the spirit of the gamelan to achieve harmony.
The aesthetic value of Pleng is something to keep in mind when using electronic tones that have harmonic overtone series, since using mathematically precise just intervals with perfectly harmonic timbres can make an instrument sound reedy and static like an organ.
Interestingly, a gamelan maker may call a dyad that beats at a pleasing rate to be in tune, and when tuned to not beat, to be out of tune. At the same time, a just intonation advocate who is listening to the same notes might pronounce the beating dyad to be out of tune and the beatless dyad to be in tune, exactly the opposite judgement. This shows that even the sense of notes being in or out of tune is subjective and dependent on cultural context, training, and artistic preferences.
1/4 Comma Meantone
1/4 Comma Meantone is a keyboard tuning which was immensely popular during the Late Renaissance and Baroque eras. It features pure major thirds and minor sixths at the cost of flattened fifths.
It’s 1523 and you are an Italian musician named Pietro Aaron. You know that if you tune an instrument using the Pythagorean tuning method, tuning through four consecutive fifths that are a pure 3/2 frequency ratio apart (and tuning down by a fourth each other time to octave reduce it), you will arrive at an interval that is a bright major third, at a frequency ratio of 81:64. This interval is the pythagorean major third.
You notice that this bright and somewhat forceful interval is sharp of a much more tranquil and placid third. This is because the Pythagorean major third is 81:64 / 5:4 = 81:80 sharp of the just major third at 5/4. This pitch difference of 81/80 is called a syntonic comma. You’d really like the major third in your scale to be the calmer 5/4. So you come up with the idea to flatten or temper each fifth by 1/4 of that syntonic comma. This spreads the difference out among all of the fifths it takes to stack to get to the pythagorean third, so now going up four of these flattened fifths hits 5/4 on the nose. The comma of 81/80 is a frequency ratio. The amount that each fifth is flattened is the 4th root of 81/80, or (81/80)(1/4). A scale based on chains of those fifths is quarter comma meantone. Each fifth is flattened, lowered by (81/80)(1/4). To lower a pitch, in this case the fifth 3/2, divide it by the ratio you are lowering it by.
A ratio is a fraction. In music, two relationship between two notes that are played together, a dyad, can be represented by the ratio of their fundamental frequencies. So if you play a note with a fundamental at 440 Hz together with a note with a fundamental at 660 Hz, the ratio between the two is 660 Hz / 440 Hz, which reduces to 3/2. (440 * 3/2 = 660). A ratio of 3/2 is a just intonation fifth, and the given example is that of Western Concert A (440Hz) being played together with the E that is a just fifth above it.
Sometimes when we mean to specify a relative ratio rather than an absolute pitch reference, we’ll use the ratio operator ‘:’ instead of the fractional operator ‘/’. So we might say that E is 3:2 above A. If A in our scale is also the tonic of the scale, which we often represent as a ratio of 1/1, then we could say the E is tuned to 3/2.
A rational intonation is a just intonation that uses higher order elements of the harmonic scale, elements which are normally disregarded as containing so little sound energy as to be irrelevant as far as western acoustics theory is concerned.
As an example, if a just intonation scale contains ratios involving numbers like 23 and 37, it is a rational intonation. There is no particular cut off point, though it would be reasonable to say that scales involving factors of 7 and less are just intonation and scales involving factors of 17 and above are rational.
Contrary to common belief, it is not difficult to identify ratios such as 24/23 by ear.
In LMSO, a Recipe consists of a Scale Pattern and a Repeat Ratio. A Scale Pattern which can be specified in any of a number of units, such as equal divisions called srutis, ratios of integers or frequencies, or cents. The Scale Pattern is repeated at the Repeat Ratio in order to create a tuning, which is
done when you “Bake” your Recipe in the Oven.
Repeat Ratio describes the interval which a repeating Scale spans. Scales can be repeated at that ratio up and down, and connected through an Anchor to specific physical frequencies and keys, then becoming a Tuning. The most common Repeat Ratio in modern tunings is the octave of 2/1, or a stretched or compressed version of it. Scales that do not repeat at the octave can be considered nonoctave scales.
Two other commonly used terms for Repeat Ratio are Interval of Equivalence (IoE) and Period. Period is most commonly used when discussing linear scales. Example usage: “The 12 note Pythagorean Scale is a MOS scale with a Period of 2/1 and a Generator of 3/2.”
None of these three terms are perfect. Each one implies particular meaning that a musician may not agree with, hence the favoring of different terms by different people.
Problems with each term
With Repeat Ratio, the use of the word Ratio is misleading since it does not have to be a pure fractional Ratio. It might be a stretched or compressed perceptible ratio, or something else. The more accurate term Repeat Interval seems more clumsy, and doesn’t phonically flow as well as Repeat Ratio.
With Interval of Equivalence (IoE), the presence of the word Equivalence suggests that Repeating the Interval across a melodic scale pattern makes it sound Equivalent harmonically. In my own hearing, repeated melodic scale patterns do sound equivalent, but it is that repeated pattern that creates the sense of equivalence and not the particular repeated interval it spans.
Regarding Period, a disadvantage of Period is that it’s an extremely common word with many meanings already. Layering another specific technical meaning on top can be more confusing than using a unique term that clearly has a specialized meaning. Period is a pretty nice term because it only suggests that something is repeating, without imparting any value to that repeating, or imposing requirements on its nature.
A single instance of a tuning’s underlying pattern, if any. For example, do re mi fa sol la ti do is the major scale. If you repeat that scale over and over again, you have a tuning.
Scale fetishism is an obsessive fixation on scales themselves as ritual objects. Scales become objects to accumulate and surround oneself with, rather than tools or friends to work with in harmony to accomplish something greater than either contributes alone.
Scale fetishism can be seen in the composer who amasses 10,000 microtonal scales and becomes obsessed with the task of to listening to all of them briefly before making a determination which of the scales are “good” and which are “bad”. These unfortunate scale addicts may feel that there are never enough scales and they need to acquire more. But acquisition of new scales only brings a brief euphoria before the cravings return.
Some victims have even been known to attempt to listen to every combination of scale with every patch or sound available from a similarly obese library of sounds, creating an impossibly large set of combinations, weighing themselves down with the Sisyphian task of mixturing them in every possible interaction.
Many of those seduced with this siren song never return from the sea of scales they have sailed into. The practice also tends to dehumanize the scales, reducing them to objects to be judged.
It is not really possible to find out what a scale is capable of so quickly. You have to get to know it, go out on dates, at the very least write a piece of music with the scale before you can even begin to have a hope of starting to understand its depth, complexity and nuances.
Sonomes are any musical instruments in the family of isomorphic hexagonal array claviers which use a default key mapping of Euler’s Tonnetz layout of vertical fifths and diagonal thirds; and have key dimensions such that a full 5 octave key range from C1 to C6 spans 6.5 inches, the same physical size of the octave span on a piano clavier. Sonomes are manufactured by various companies and were invented by British luthier Peter Davies in 1991.
A superparticular ratio is a fraction with the denominator one larger than the numerator: numerator/denominator = n/(n+1). Some examples are the ratios 3:2, 4:3, 5:4, 6:5, 7:6, and 21:20. The represent the interval between two adjacent harmonics. Ratios in this form are favored in ancient Greek music theory.
tET is an abbreviation that means -tone Equal Temperament. It is not usually used by itself, but as a suffix that comes after a number. So, 12tET means 12-tone Equal Temperament and 17tET means 17-tone Equal Temperament. Equal Temperaments (sometimes also called Equal Tunings) are tunings with every chromatic step the same size. tETs are a popular special case of Equal Temperaments which are equal divisions of the 2/1 octave interval.
EDO is an abbreviation meaning Equal Division of the Octave. It is a synonym for tET and has been preferred by many. Dropping the term temperament was motivated from observations that most tET tunings (12tET, 19tET, 31tET and 53tET are the notable exceptions), even if they do have approximations of many rational intervals, are not normally derived theoretically through a process of tempering those rational intervals. The term Octave was introduced to the abbreviation so that what was being divided equally would be explicit. Not all possible equal tunings are equal divisions of the octave, only a small subset are.
ED2 is an abbreviation meaning Equal Division of the 2/1. It is another synonym for the same idea. There was a concern that the term Octave can mean not only an interval of 2/1 precisely, but more generally refers to a perceptual or functional interval class that can range widely. To reduce ambiguity, ED2 can be used instead to indicate that an Equal Division of exactly 2/1 is being considered.
Equal Divisions of other intervals are common as well. This are usually referred to by using the exact name of the interval (ie: equal divisions of the quintave or fifth) or by specifying the ratio explicitly (ie: equal divisions of 4/1). To be more explicit, since 12 tone equal temperament’s semitone is mathematically the 12th root of 2, we can talk about 12th root of 2 tuning, which can be written 2^(1/12), meaning 2 raised to the 1/12 power. So a variation of 88cET tuning can be notated (3/2)^(1/8) and read as “the 8th root of 3/2”. This has the advantage of precision as the exact ratio intended is specified.
Tonality is a perception in the listener of a sense of a key center.
Tonality includes both an established tonic note (called the key center or tonal center) at a specific pitch, and the scale that is anchored to that pitch.
The composer uses the resources of the scale to support the perception in the listener of that pitch as the tonal center.
This definition defers issues of how that key center is perceived by the listener, how it is conveyed by the composer, and how modulation (whatever that is) works to establish a new tonality after a previous one was established.
A mapping of specific pitches to all the keys or actuators of a musical instrument.
A tuning can be structured as a stack of repetitions of an underlying pattern of intervals called a scale, or it could be a collection of pitches with some other structure, or no structure at all.
Sometimes a composer will become deeply involved with a number of tunings at once, maybe as few as a dozen. Exposure to new tunings leads to new neural interwiring within the brain. Harmless hallucinations are not uncommon, such as seeing colours more brightly, or food tasting different.
But on occasion, one will become agitated, or acquire a sense that the world is not real (this is also known as depersonalization). They may begin to hear voices or experience strange thoughts. They may become argumentative.
This is known as Tuning Psychosis. If a composer starts to become angry with the world after working with many tunings, have him pull back and just work with one tuning for a while, or take a break altogether. Give the brain time to integrate what it has learned into its neural interconnection matrix.
From the Greek ξενία (xenia, hospitality) and ξένος (xenos, foreign), Hospitable Harmony and also Alien Harmony. Refers to tunings that are hospitable to new harmonies, and thus do not sound like standard western tunings.
It is popular practice to create low limit just intonation scales that approximate or substitute for standard Western tunings. These scales are microtonal, but not xenharmonic.
If you find yourself writing a piece in 12 equal, and then after the fact you try out different tunings to what you have written and most listeners can’t tell the difference, then your piece is not xenharmonic. Which is fine. But the idea of xenharmonic is that has an intangible property in that it sounds noticeably different. After you work with tunings a while, you’ll know xenharmonic when you hear it. It doesn’t mean dissonant though, it just means different.
1. The use of unconventional keys and harmony within a musical composition.
2. The correlation of timbres and tunings within a musical composition.