Xenharmonic Wiki - User contributions [en]

Matrix echelon forms

2026-07-28T18:18:06Z

Fredg999: Use new redirect

The mathematical field of [[linear algebra]] has defined a number of echelon forms for matrices, many of them have been used in [[regular temperament theory]]. This article is intended as a reference and gathering place for relevant information about these matrix forms.

== Row echelon form ==
{{Wikipedia|Row echelon form}}

The most general form, with the fewest constraints, is simply called '''row echelon form''' ('''REF'''). Its only constraint is ''echelon form''<ref group="note">The name [[Wiktionary: echelon #English|''echelon'']] is a French word for a military troop formation with a similar triangular shape.</ref>: each row's pivot, or first nonzero entry, is strictly to the right of the preceding row's pivot. This single constraint is fairly weak, and therefore REF does not produce a unique representation. This constraint is shared by every matrix form discussed here.<ref group="note">Note that the definition of REF is inconsistent and sometimes it includes some of the constraints of RREF, discussed further below. See: [https://www.statisticshowto.com/matrices-and-matrix-algebra/reduced-row-echelon-form-2/ Statistics How to | ''Row Echelon Form & Reduced Row Echelon Form'']</ref><ref group="note">REF also requires that all rows that are entirely zeros should appear at the bottom of the matrix. However this rule is only relevant for [[rank-deficient]] matrices. We will be assuming all matrices here are [[full-rank]], so we do not have to worry about this.</ref>

In the below example, ''x''''ij'' represents any number.

{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | REF
|- style="width: 25px;"
| style="background-color: #6d9eeb;" | ''x''1
| style="background-color: #6d9eeb;" | ''x''2
| style="background-color: #6d9eeb;" | ''x''3
| style="background-color: #6d9eeb;" | ''x''4
| style="background-color: #6d9eeb;" | ''x''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #6d9eeb;" | ''x''2
| style="background-color: #6d9eeb;" | ''x''3
| style="background-color: #6d9eeb;" | ''x''4
| style="background-color: #6d9eeb;" | ''x''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #6d9eeb;" | ''x''4
| style="background-color: #6d9eeb;" | ''x''5
|}

== Integer row echelon form ==
'''Integer row echelon form''' ('''IREF'''), is, unsurprisingly, any REF which meets an additional ''integer'' constraint, or in other words, that all of its entries are integers.<ref>[https://people.sc.fsu.edu/~jburkardt/f_src/row_echelon_integer/row_echelon_integer.html John Burkardt's Home Page | <code>row_echelon_integer</code>]</ref> This is still not a sufficiently strict set of constraints to ensure a unique representation.

In the below example, ''n''''ij'' represents any integer.

{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | IREF
|- style="width: 25px;"
| style="background-color: #93c47d;" | ''n''1
| style="background-color: #93c47d;" | ''n''2
| style="background-color: #93c47d;" | ''n''3
| style="background-color: #93c47d;" | ''n''4
| style="background-color: #93c47d;" | ''n''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #93c47d;" | ''n''2
| style="background-color: #93c47d;" | ''n''3
| style="background-color: #93c47d;" | ''n''4
| style="background-color: #93c47d;" | ''n''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #93c47d;" | ''n''4
| style="background-color: #93c47d;" | ''n''5
|}

== Reduced row echelon form ==
{{Wikipedia|Row echelon form #Reduced row echelon form}}

'''Reduced row echelon form''' ('''RREF'''), takes REF in a different direction than IREF. Instead of stipulating anything about integers, it requires that the matrix is ''reduced'', i.e. that the pivots are exactly equal to 1. This may require dividing rows by a number such that some resulting entries are no longer integers. Actually, there's a second part to the "reduced" constraint: each pivot column (a column which contains any row's pivot) has zeros for all other entries besides the pivot it contains. Due to these strict constraints, the RREF of a matrix is the first one we've looked at so far here which does ensure uniqueness.

{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | RREF
|- style="width: 25px;"
| style="background-color: #d9ead3;" | 1
| style="background-color: #93c47d;" | 0
| style="background-color: #6d9eeb;" | ''x''3
| style="background-color: #93c47d;" | 0
| style="background-color: #6d9eeb;" | ''x''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #d9ead3;" | 1
| style="background-color: #6d9eeb;" | ''x''3
| style="background-color: #93c47d;" | 0
| style="background-color: #6d9eeb;" | ''x''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #d9ead3;" | 1
| style="background-color: #6d9eeb;" | ''x''5
|}

So IREF and RREF make a {{w|Venn diagram}} inside the category of REF: some IREF are RREF, but there are some RREF that are not IREF and some IREF that are not RREF. When we scope the situation to a specific matrix, however, because RREF is a unique form, this means that one or the other sector of the Venn diagram for RREF will be empty; either the unique RREF will also be IREF (and therefore the RREF-but-not-IREF sector will be empty), or it will not be IREF (and vice versa).

== Integer reduced row echelon form ==
'''Integer reduced row echelon form''' ('''IRREF'''): based on the name, one might expect this form to be a combination of the constraints for RREF and IREF, and therefore if represented in an {{w|Euler diagram}} (generalization of Venn diagram) would only exist within their intersection. However this is not the case. That is because the IRREF does not include the key constraint of RREF which is that all of the pivots must be 1. IRREF is produced by simply taking the unique RREF and multiplying each row by whatever minimum value is necessary to make all of the entries integers. Of course, this sometimes results in the pivots no longer being 1, so sometimes it is no longer RREF. It is always still REF, though,<ref group="note">Also, it will always still satisfy the second aspect of reduced, i.e. that all other entries in pivot columns besides the pivots are zeroes.</ref> and because it is also always integer, that makes it always IREF; therefore, IRREF is strictly a subcategory of IREF. And because the RREF is unique, and the conversion process does not alter that, the IRREF is also unique.

{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | IRREF
|- style="width: 25px;"
| style="background-color: #93c47d;" | n1
| style="background-color: #274e13;" | 0
| style="background-color: #93c47d;" | n3
| style="background-color: #274e13;" | 0
| style="background-color: #93c47d;" | n5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #93c47d;" | n2
| style="background-color: #93c47d;" | n3
| style="background-color: #274e13;" | 0
| style="background-color: #93c47d;" | n5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #93c47d;" | n4
| style="background-color: #93c47d;" | n5
|}

The multiplication of rows by whatever integer is required to clear the denominators of the RREF form is the action which is responsible for causing [[enfactoring]], whenever the IRREF produces an enfactored mapping from a defactored one.

It is not possible for an RREF to be IREF without also being IRREF.

== Hermite normal form ==
{{Wikipedia|Hermite normal form}}
{{Main|Normal forms #Hermite normal form}}

The '''Hermite normal form''' ('''HNF'''): this one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced (as discussed above; see [[Normal forms #Reduced row echelon form (RREF)]]); instead, it is ''normalized'', which – similarly to reduced – is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref group="note">The exact criteria for HNF are not always consistently agreed upon, however.</ref><ref group="note">We are using "row-style" Hermite Normal Form here, not "column-style"; the latter would involve simply flipping everything 90 degrees so that the echelon requirement was that pivots be strictly ''below'' the pivots in the previous ''column'', and that pivot ''rows'' are considered for the normalization constraint rather than pivot ''columns''.</ref> In other words, elements above the pivot have to be reduced modulo the pivot. The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.

So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.

[[Defactored Hermite form]] (which we will call "canonical form" here) is closely related to HNF, because the second step of finding the canonical form is taking the HNF. So the canonical form is always ''a'' HNF, and therefore it has all the same properties of being echelon, integer, and normalized, and in turn therefore it also provides a unique representation. However it is not necessary ''the'' same HNF of the original mapping, due to the first step being defactoring; it is the same as as HNF except when the original mapping is enfactored.

In the below example, ''p''''ij'' represents any positive integer, and ''a''''ij'' represents any nonnegative integer less than the ''p'' in its column.

{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | HNF (and canonical form)
|- style="width: 25px;"
| style="background-color: #e69138;" | ''p''1
| style="background-color: #ffe599;" | ''a''2
| style="background-color: #93c47d;" | ''n''3
| style="background-color: #ffe599;" | ''a''4
| style="background-color: #93c47d;" | ''n''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #e69138;" | ''p''2
| style="background-color: #93c47d;" | ''n''3
| style="background-color: #ffe599;" | ''a''4
| style="background-color: #93c47d;" | ''n''5
|-
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #e69138;" | ''p''4
| style="background-color: #93c47d;" | ''n''5
|}

For an example of working out a HNF by hand, see: [[Defactoring algorithms #Hermite decomposition by hand]].

== Smith normal form ==
{{Wikipedia|Smith normal form}}

There is also the '''Smith normal form''' ('''SNF'''), but we will not be discussing it in this context, because putting a mapping into SNF obliterates a lot of meaningful RTT information. SNF is also echelon, and integer, so like HNF it is also always IREF. But SNF requires that every single entry other than the pivots are zero, and that the pivots all fall exactly along the main diagonal of the matrix. The SNF essentially reduces a matrix down to the information of what its rank is and whether it is enfactored. For example, all 5-limit rank-2 temperaments such as meantone, porcupine, mavila, hanson, etc. have the same SNF: {{rket|{{map|1 0 0}} {{map|0 1 0}}}}. Or, if you 2-enfactor them, they will all have the SNF {{rket|{{map|1 0 0}} {{map|0 2 0}}}}. So while the SNF is closely related to defactoring, it is not itself a useful form to put mappings into.<ref group="note">Here is a useful resource for computing the Smith normal form manually, if you are interested: [https://math.stackexchange.com/questions/133076/computing-the-smith-normal-form] The fact that it involves calculating so many GCDs is unsurprising given its ability to defactor matrices.</ref>

The SNF is used for finding [[generator detempering]]s.

{| class="wikitable" style="text-align: center;"
|+ style="font-size: 105%;" | SNF
|- style="width: 25px;"
| style="background-color: #e69138;" | ''p''1
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
|-
| style="background-color: #274e13;" | 0
| style="background-color: #e69138;" | ''p''2
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
|-
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
| style="background-color: #e69138;" | ''p''3
| style="background-color: #274e13;" | 0
| style="background-color: #274e13;" | 0
|}

== Mapping form equivalence possibilities ==
[[File:Cases for temperament mapping forms3.png|300px|thumb|right]]
Considering only full-rank, integer mappings, we find three cases for a given temperament which is not enfactored. In all three cases, HNF is the same as canonical form (here abbreviated as DCF, for defactored canonical form<ref group="note">This was before the community decided on "defactored Hermite form", and I was too lazy to go back and update all these diagrams.</ref>):
# The RREF, IRREF, and HNF are all ''different''. Example: [[Porcupine family #Porcupine|porcupine]] with RREF of {{rket|{{map|1 0 {{frac|1|3}} {{map|0 1 {{frac|5|3}}}}}}}}, IRREF of {{rket|{{map|3 0 -1}} {{map|0 3 5}}}}, and HNF of {{rket|{{map|1 2 3}} {{map|0 3 5}}}}.
# The RREF, IRREF, HNF are all ''the same''. Example: [[Meantone family #Meantone|meantone]] with all equal to {{rket|{{map|1 0 -4}} {{map|0 1 4}}}}. This case is quite rare.
# The IRREF and HNF are the same, but the ''RREF is different''. Example: [[Kleismic family #Kleismic a.k.a. hanson|hanson]] with IRREF and HNF of {{rket|{{map|1 0 1}} {{map|0 6 5}}}} but RREF of {{rket|{{map|1 0 1}} {{map|0 1 {{frac|5|6}}}}}}.

And there are three corresponding cases when a temperament is enfactored. In all three cases, the key difference is that HNF is no longer the same as DCF, with the only difference being that the common factor is not removed. In all cases below, the examples are shown with a common factor of 2 introduced in their second row, which stays behind in the second row of the HNF:
# Now ''all four'' are different. Example: enfactored porcupine, e.g. {{rket|{{map|15 24 35}} {{map|14 22 32}}}} causes the HNF to be {{rket|{{map|1 2 3}} {{map|0 6 10}}}}.
# Everything is still the same now, ''except HNF''. Example: enfactored meantone, e.g. {{rket|{{map|5 8 12}} {{map|14 22 32}}}} causes the HNF to be {{rket|{{map|1 0 -4}} {{map|0 2 8}}}}. This case, like the corresponding defactored state, is also quite rare.
# The ''only match'' now is between IRREF and DCF. In other words, the HNF and DCF diverged, and it was the DCF which remained the same as IRREF. Example: enfactored hanson, e.g. {{rket|{{map|15 24 35}} {{map|38 60 88}}}} causes the HNF to be {{rket|{{map|1 0 1}} {{map|0 12 10}}}}.

There is also a final case which is incredibly rare. It can be compared to the #3 cases above, the ones using hanson as their example. The idea here is that when the HNF and DCF diverge, instead of DCF remaining the same as IRREF, it's the HNF that remains the same as IRREF. There may be no practical temperoids with this case, but {{rket|{{map|165 264 393}} {{map|231 363 524}}}} will do it,<ref group="note">AKA 165b4c19&231b6c24, which makes the 7.753{{cent}} comma {{vector|-131 131 -33}} [[vanish]]!</ref> with IRREF and HNF of {{rket|{{map|33 0 -131}} {{map|0 33 131}}}}, DCF of {{rket|{{map|1 1 0}} {{map|0 33 131}}}}, and RREF of {{rket|{{map|1 0 {{frac|−131|33}}}}}} {{map|0 1 {{frac|131|33}}}}}}.

That accounts for 7 of the 15 total possible cases for a system of equalities between 4 entities. The remaining 9 cases are impossible due to properties of the domain:
* If HNF equals RREF, then the pivots of HNF are all 1's, which means the temperament is not enfactored, which means HNF also equals DCF. This eliminates 3 cases: {{nowrap|HNF {{=}} RREF}}; {{nowrap|HNF {{=}} RREF|HNF {{=}} IRREF|RREF {{=}} IRREF}}; {{nowrap|HNF {{=}} RREF|IRREF {{=}} DCF}}.
* If RREF equals DCF, then RREF must be integer, which means RREF must also equal IRREF. This eliminates 3 cases: {{nowrap|RREF {{=}} DCF}}; {{nowrap|RREF {{=}} DCF|HNF {{=}} RREF|HNF {{=}} DCF}}; {{nowrap|RREF {{=}} DCF|HNF {{=}} IRREF}}.
* The case where the only equality is between RREF and IRREF would only be possible when the temperament is both enfactored and rank-deficient, such as {{rket|{{map|0 0 0}} {{map|2 4 6}}}} which gives to HNF, RREF, IRREF, and DCF of {{rket|{{map|2 4 6}} {{map|0 0 0}}}}, {{rket|{{map|1 2 3}} {{map|0 0 0}}}}, {{rket|{{map|1 2 3}} {{map|0 0 0}}}}, and {{rket|{{map|1 2 3}}}}, respectively.
* The case where the only equalities are between RREF and IRREF and between HNF and DCF is impossible, because if {{nowrap|RREF {{=}} IRREF}} that suggests that all entries are multiples of their pivots, which is easy if the temperament is enfactored, but if {{nowrap|HNF {{=}} DCF}} then it is not.

== Notes ==
<references group="note" />

== References ==
<references />

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]

Linear algebra

2026-07-28T18:17:44Z

Fredg999: Redirected page to Linear algebra formalism

#REDIRECT [[Linear algebra formalism]]

Mapping to lattice

2026-07-28T05:24:14Z

Fredg999: Categories

Many temperaments documented on the wiki are accompanied with a '''mapping to lattice'''. These use the same [[extended bra-ket notation]] as temperament [[mapping]]s but provide quite different information: they give the coordinates of the primes in a tempered lattice. The lattice for which they give coordinates does not necessarily correspond with the mapping otherwise provided.

== Example ==

Consider [[keemic]]. Its mapping to lattice is given as {{rket| {{map| 0 0 -1 3 }} {{map| 0 1 1 0 }} }}. We can read this in four vertical slices; the coordinate (0, 0) is for prime 2, (0, 1) is for prime 3, (-1, 1) is for prime 5, and (3, 0) is for prime 7.

Note that keemic is a rank-3 temperament, meaning it has three generators. Or said another way, it has one period and two (other) generators. This corresponds with the fact that it has three generator maps – or rows – in its mapping matrix, which is given as {{rket| {{map| 1 0 0 5 }} {{map| 0 1 0 3 }} {{map| 0 0 1 -3 }} }}. The reason the mapping to lattice has only two rows while the mapping has three rows is because [[octave equivalence]] has been assumed in the lattice. So the first generator, the special one, the period, which in this case is equal to the octave, is not visualized on this lattice; we use one axis for the first generator, and the other axis of the second generator.

The octave equivalence also explains why prime 2 is found at (0, 0).

It should not be surprising that prime 3 is found at a coordinate that is only one step away from (0, 0): because reading the column for prime 3 from the mapping, we get a [[generator-count vector]] of {{rket| 0 1 0 }}, meaning that one of the generators is an approximation of prime 3.

However, we may be a bit confused by the next coordinate. Why is the coordinate of prime 5 one step each in both directions, despite the fact that the mapping gives a vector of {{rket| 0 0 1 }} to prime 5, similar to the one it gives for prime 3? Well, this is simply evidence that the mapping to lattice does not correspond to the mapping given for this temperament. It means that on this particular lattice, we get to prime 5 by first moving one step of 3/1, and then one step of whatever else it takes to get from there to 5/1 (which is 5/3). In other words, this mapping to lattice appears to correspond to a different mapping (for the same temperament): {{rket| {{map| 1 0 0 5 }} {{map| 0 1 1 0 }} {{map| 0 0 -1 3 }} }}. The difference here is that the third row of the mapping has been added to the second row, and then the third row has been negated. So now if we read off prime 5's vector, we get {{rket| 0 1 -1 }}, meaning that the approximation of prime 5 is arrived at by moving up one of the first generator, and down one of the second generator. So the second generator is actually not a 5/1, but a 6/5. This is because moving down by that is multiplying by 5/6, and if we're starting at 3/2 (not 3/1, because of octave reduction), we need to go by 5/6 to get to 5/4 (again, not 5/1, because of octave reduction). Actually it may be the case that the mapping this lattice is for is {{rket| {{map| 1 1 1 5 }} {{map| 0 1 1 0 }} {{map| 0 0 -1 3 }} }}, so that the second generator is understood to be an approximation of 3/2 instead of 3/1; we cannot really know, because of the octave equivalence, but it seems likely that if it was important to move down from 3 to 5 within an octave under octave equivalence, that the author of this mapping would want all of the generators to be octave reduced.

Finally we have the coordinate for prime 7. Another simple one. It is just a move by 3 of the second generator. It is worth noting, though, that the generators of the mapping have their order swapped in the mapping to lattice. The first generator map (row) of the mapping corresponds with the second coordinate of the mapping to lattice, while the second generator map of the mapping corresponds with the first coordinate of the mapping to lattice. This is perhaps because the the mapping to lattice coordinates are sorted ascending by generator size, while the mapping is often sorted descending by generator size (because it is common for generators to be reduced within the size of previous generators; see [[Normal forms #Equave-reduced generator form]] and [[Normal forms #Minimal generator form]]).

So here is a small sample of this particular tempered lattice for this temperament, just enough to show what the mapping to lattice is indicating:

{| class="wikitable"
|+
! colspan="2" rowspan="2" |
! colspan="5" | g1
|-
! -1
! 0
! 1
! 2
! 3
|-
! rowspan="2" | g2
! 0
|
| 2/1
|
|
| 7/1
|-
! 1
| 5/1
| 3/1
|
|
|
|}

[[Category:Regular temperament theory]]
[[Category:Mapping]]
[[Category:Lattice]]

Map

2026-07-28T05:23:05Z

Fredg999: /* Maps in regular temperament theory */ Add mapping to lattice

{{Wikipedia|Map (mathematics)}}
A '''map''' or '''mapping'''<ref group="note">It is possible to distinguish the "mapping", which is the function, from the "map", which is its image. However, both terms are used interchangeably in most contexts.</ref> is an association of elements from a set to elements of another set. In mathematics, a map is a generalized function.

Various kinds of maps relevant to microtonal music theory are introduced below.

== Keyboard map ==
A '''keyboard map''' or '''keyboard mapping''' specifies the actions associated with each key of a keyboard.

=== Musical instruments ===
{{See also|Scale design software#Keyboard mapping}}
A [[keyboard]] mapping for a musical [[instrument]] ([[List of microtonal software plugins|software]] or [[List of microtonal hardware synths|hardware]]) associates notes or pitches from a scale file to the instrument's keys. Notable examples are [[Scala]] keyboard mappings (.kbm files) and [[Lumatone]] keyboard mappings (.ltn files).

=== Computer keyboard bindings ===
A keyboard map for a computer keyboard associates characters to certain keys or combinations of keys. It can be used to type special characters more quickly. An example is provided in [[Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations#WinCompose]].

== Maps in regular temperament theory ==
[[File:Temperament and tuning matrices 2.png|thumb|alt=An overview of maps in regular temperament theory.|An overview of maps in regular temperament theory.]]
In [[regular temperament theory]] (RTT), maps are generally assumed to be [[Wikipedia:Linear map|''linear'' maps]]<ref group="note">Since regular temperaments are defined on ℤ-modules, not on vector spaces, the term "linear map" is not quite correct and it would be preferable to say "[[Wikipedia:Additive map|''additive'' map]]", but "linear map" is often used by analogy since linear algebra terminology is more common in mathematics overall.</ref>, which, informally, can be thought of as a function that can be represented by a {{w|Matrix (mathematics)|matrix}}.

The most common maps in RTT are discussed below.

=== Temperament map ===
{{Main|Mapping}}
A temperament map or temperament mapping<ref group="note">[[Douglas Blumeyer]] and [[Dave Keenan]] recommend reserving the word "map" for a mapping with one row, so that all maps are mappings but not all mappings are maps; a simple tip to remember this usage is that the shorter word refers to the simpler object. In all occurrences at present on this wiki, as well as in Graham Breed's temperament finder, the term "map" (and not "mapping") consistently refers to a single-row mapping, so following this suggestion would be seamless moving forward. A "tuning map", which maps from generators to cents, is a map in "tuning space"; by analogy, a val is a map in "temperament space", and so it would be perfectly consistent with existing terminology to refer to a val as a "temperament map" as opposed to a "temperament mapping", and then when it is clear from the context that it is a ''temperament'' map, the qualifier "temperament" can be dropped, as is done with "temperament mapping matrix" being abbreviated to "mapping matrix". So the suggestion is equivalent to unqualified occurrences of "map" being assumed to be temperament maps, or in other words, synonymous with vals (except for the integer entry requirement), not tuning maps. Dave and Douglas recommend using "map" rather than "val", for two reasons. First, "map" is a basic linear algebra term with wide familiarity (being specialized for this purpose) while "val" is unnecessary jargon that creates a barrier to understanding by newcomers. Second, the coinage of "val" from the obscure mathematical term "valuation" is tenuous and unlikely to provide helpful insight: "p-adic valuation" is an obscure term for "prime count", which would be an element of a prime-count vector ("monzo"), not a map ("val").</ref> specifies the structure of a regular temperament. It transforms [[interval]]s, usually from a [[just intonation subgroup]] and expressed as prime-count vectors ([[monzo]]s), into [[mapped interval]]s ([[tmonzo]]s). It is often represented by a matrix, thus called a temperament mapping matrix<ref group="note">The terms "M-map" and "V-map" were also sometimes used to refer to temperament mappings and subgroup basis matrices, although these terms are rarely used nowadays.</ref>, or more simply a mapping matrix or even a mapping when the context is sufficiently clear.

Each row of a temperament mapping matrix, or the only row of a rank-1 temperament mapping matrix, is often called a [[val]].

A [[subgroup basis matrix]] is the dual of a temperament mapping matrix.

=== Tuning map ===
{{Main|Tuning map}}
A tuning map (or rarely a tuning mapping) specifies the tuning of a regular temperament. A (tempered-prime) tuning map transforms intervals into tempered [[interval size]]s. A generator tuning map transforms mapped intervals into interval sizes.

=== Projection map ===
{{Main|Projection matrix}}
A projection matrix (or rarely a projection map) uniquely identifies a specific tuning of a specific regular temperament.

=== Lattice map ===
{{Main|Mapping to lattice}}
A lattice map or mapping to lattice transforms intervals into their coordinates in a regularly tempered [[lattice]].

== See also ==
* [[Harmonic template]]

== Notes ==
<references group="note"/>

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
[[Category:Val]]
[[Category:Mapping]]

Keyboard map

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#REDIRECT [[Map#Keyboard map]]

Keyboard mapping

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Fredg999: Changed redirect target from Map to Map#Keyboard map

#REDIRECT [[Map#Keyboard map]]

Keyboard mapping

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#REDIRECT [[Map]]

Keyboard map

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#REDIRECT [[Map]]

Talk:Hardware Synths

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#REDIRECT [[Talk:List of microtonal hardware synths]]

Mapping

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Fredg999: About hatnote

{{interwiki
| en = Mapping
| ja = マッピング
}}
{{About|temperament mappings|3=Map}}
{{Beginner|Temperament mapping matrix}}

A [[regular temperament]] is more than simply a set of pitches. It's a set of notes together with a ''consistent rule'' that maps any pitch of the relevant [[just intonation subgroup]] to a specific note from that set. (In fact, an abstract regular temperament is not a set of definite pitches at all! The pitches can vary, and the rule mapping JI pitches to notes is the thing that uniquely characterizes the temperament.) This consistent rule is known as the ''JI mapping'' or simply '''mapping'''. The mapping answers the question "how do I play this JI pitch as a note of this temperament?". The answer will be the "tempered version" of that JI pitch, which may be a very close approximation or a very distant approximation depending on the circumstances.

Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this (non-linear) mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval.

== A note on mathematical terminology ==
In mathematics generally, "mapping" is synonymous with "map" and "function". In RTT, "mapping" has the more specific meaning of a {{w|Linear map|''linear'' mapping}}, which is a function that can be represented by a matrix.

== Equal temperament mappings ==
An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of
# A JI subgroup that is being represented, such as "5-limit JI", and
# A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer).

As an example, let's consider the familiar [[12edo]] considered as a 3-limit temperament. For concreteness, let's use A440 as the base note. In this case the JI subgroup is the [[3-limit]], that is, all pitches that form a JI interval with A440 whose prime factorization contains no primes other than 2 and 3. For people familiar with mathematical notation, this can be written as

<math>\left\{440\cdot 2^a\cdot 3^b\,\middle|\,a,b\in\mathbb Z\right\}</math>

Let's use integers to represent the 12edo notes, so that A440 is note 0, the B♭ above that is 1, the A♭ below it is −1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955… cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to {{nowrap|12''a'' + 19''b''}}.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 312/219 (the Pythagorean comma) is mapped to 0, the same note as 1/1. ''a'' and ''b'' are read from such prime factorization or [[monzo]].

=== Contrast with rounding ===
Now, consider the pitch 336/257. In JI, this pitch is 70.38… cents above A440, so the closest 12edo note to it is B♭. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (B♭). Why is this? The pitch 336/257is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since {{nowrap|0 + 0 + 0 {{=}} 0}} the pitch 336/257 must be represented by A, even though in JI it's closer to B♭. Mapping it to B♭ would require one of the three Pythagorean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.

=== Notation ===
In regular temperament theory there is a special notation for this kind of JI mapping. We notate the 3-limit 12edo temperament described above as "{{val| 12 19 }}"<ref group="note">Known as [[extended bra-ket notation]].</ref>, because the first prime (2) is mapped to 12 steps, and the second prime (3) is mapped to 19 steps. This mathematical object is known as a "mapping matrix" and it summarizes all the information in the mapping in a very compact form. Since this is an equal temperament, the mapping matrix contains only one row, and since it's a 3-limit temperament, the mapping matrix contains two columns, representing the primes 2 and 3. Such mapping with one row is called a [[val]], and a mapping matrix in multiple vals (described below) is also called a '''val list'''.

=== Many 12edo temperaments ===
Now, let's consider 12edo, not as a 3-limit temperament, but as a [[5-limit]] temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is {{val| 12 19 28 }}. It's important to keep in mind that this is, technically speaking, a ''different'' regular temperament than {{val| 12 19 }}, even though they would both be referred to as "12-tone equal temperament" in common parlance.

Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D♯". This choice results in two different mappings, {{val| 12 19 28 34 41 }} and {{val| 12 19 28 34 42 }}. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.

Strictly speaking, "5-limit 12edo" or even "3-limit 12edo" are also ambiguous, because {{val| 12 19 27 }}, for example, is a valid temperament even though it's much less accurate than {{val| 12 19 28 }}. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.

== Linear temperament mappings ==
Now let's consider a temperament that does not consist of a single chain of equally spaced notes. For example, consider conventional music notation ''without'' enharmonic equivalence. Every note of this system can be expressed as some combination of octaves and perfect fourths. For example:

* E5 = A440 + 1 octave − 1 perfect fourths
* B♭4 = A440 − 3 octaves + 5 perfect fourths
* A♯ = A440 + 4 octaves − 7 perfect fourths

In other words, every note can be represented as an ordered pair of integers {{nowrap|(''x'', ''y'')}} where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.

== Temperamental rank ==
A temperament's "rank" denotes how many independent chains of generators exist within the temperament. This is a mathematical term that's borrowed from the fields of group theory and linear algebra. It can also be viewed as the dimension of the temperament's mapping.

For example:
# An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.
# Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains.
# Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like).
# 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc.

A single [[val]] in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank 2 or higher, we simply need to use more than one val. In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.

== Example ==
At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:

{| class="right-all left-5 left-6"
|-
| [⟨ || ''a'' || ''b'' || ''c'' || ] || – period
|-
| ⟨ || ''d'' || ''e'' || ''f'' || ]] || – generator
|}

The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.

When mapping a prime in JI onto a rank 2 temperament, one must think about how many steps of each type of generator it takes to reach the final tempered prime interval. For an example, we'll look at meantone temperament, and we'll start by mapping 2/1. We'll assume that the period is 2/1, and the generator is 3/2. 2/1 maps to one period and zero generators, and hence we arrive at

{| class="right-all left-5"
|-
| [⟨ || 1 || _ || _ || ]
|-
| ⟨ || 0 || _ || _ || ]]
|}

3/1 is slightly more complicated – it requires one step along the 2/1 period chain, plus one step along the 3/2 generator chain, to get to 3/1. This is represented by the following mapping:

{| class="right-all left-5"
|-
| [⟨ || 1 || 1 || _ || ]
|-
| ⟨ || 0 || 1 || _ || ]]
|}

5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:

{| class="right-all left-5"
|-
| [⟨ || 1 || 1 || 0 || ]
|-
| ⟨ || 0 || 1 || 4 || ]]
|}

This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.

== Change of basis ==
In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament.

In the language of mathematics, you've simply '''changed the basis''' for your temperament, and the resulting temperamental spaces will be '''isomorphic''' to one another. This is just a fancy way of stating that they're the same temperament.

If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:

{| class="right-all left-5"
|-
| [⟨ || 1 || 2 || 4 || ]
|-
| ⟨ || 0 || -1 || -4 || ]]
|}

This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.

If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):

{| class="right-all left-5"
|-
| [⟨ || 1 || 0 || -4 || ]
|-
| ⟨ || 0 || 1 || 4 || ]]
|}

The [[normal lists #Normal val list|normal val list]] is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.

For more information on these sorts of operations, see [[Generator size manipulation]].

== Units ==
It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

== See also ==
* [[Reduced mapping]]
* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic
* [[Harmonic template]]

== Notes ==
<references group="note"/>

[[Category:Regular temperament theory]]
[[Category:Val]]
[[Category:Mapping| ]] 
[[Category:Terms]]
[[Category:Math]]

Map

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Fredg999: Major rewrite

{{Wikipedia|Map (mathematics)}}
A '''map''' or '''mapping'''<ref group="note">It is possible to distinguish the "mapping", which is the function, from the "map", which is its image. However, both terms are used interchangeably in most contexts.</ref> is an association of elements from a set to elements of another set. In mathematics, a map is a generalized function.

Various kinds of maps relevant to microtonal music theory are introduced below.

== Keyboard map ==
A '''keyboard map''' or '''keyboard mapping''' specifies the actions associated with each key of a keyboard.

=== Musical instruments ===
{{See also|Scale design software#Keyboard mapping}}
A [[keyboard]] mapping for a musical [[instrument]] ([[List of microtonal software plugins|software]] or [[List of microtonal hardware synths|hardware]]) associates notes or pitches from a scale file to the instrument's keys. Notable examples are [[Scala]] keyboard mappings (.kbm files) and [[Lumatone]] keyboard mappings (.ltn files).

=== Computer keyboard bindings ===
A keyboard map for a computer keyboard associates characters to certain keys or combinations of keys. It can be used to type special characters more quickly. An example is provided in [[Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations#WinCompose]].

== Maps in regular temperament theory ==
[[File:Temperament and tuning matrices 2.png|thumb|alt=An overview of maps in regular temperament theory.|An overview of maps in regular temperament theory.]]
In [[regular temperament theory]] (RTT), maps are generally assumed to be [[Wikipedia:Linear map|''linear'' maps]]<ref group="note">Since regular temperaments are defined on ℤ-modules, not on vector spaces, the term "linear map" is not quite correct and it would be preferable to say "[[Wikipedia:Additive map|''additive'' map]]", but "linear map" is often used by analogy since linear algebra terminology is more common in mathematics overall.</ref>, which, informally, can be thought of as a function that can be represented by a {{w|Matrix (mathematics)|matrix}}.

The most common maps in RTT are discussed below.

=== Temperament map ===
{{Main|Mapping}}
A temperament map or temperament mapping<ref group="note">[[Douglas Blumeyer]] and [[Dave Keenan]] recommend reserving the word "map" for a mapping with one row, so that all maps are mappings but not all mappings are maps; a simple tip to remember this usage is that the shorter word refers to the simpler object. In all occurrences at present on this wiki, as well as in Graham Breed's temperament finder, the term "map" (and not "mapping") consistently refers to a single-row mapping, so following this suggestion would be seamless moving forward. A "tuning map", which maps from generators to cents, is a map in "tuning space"; by analogy, a val is a map in "temperament space", and so it would be perfectly consistent with existing terminology to refer to a val as a "temperament map" as opposed to a "temperament mapping", and then when it is clear from the context that it is a ''temperament'' map, the qualifier "temperament" can be dropped, as is done with "temperament mapping matrix" being abbreviated to "mapping matrix". So the suggestion is equivalent to unqualified occurrences of "map" being assumed to be temperament maps, or in other words, synonymous with vals (except for the integer entry requirement), not tuning maps. Dave and Douglas recommend using "map" rather than "val", for two reasons. First, "map" is a basic linear algebra term with wide familiarity (being specialized for this purpose) while "val" is unnecessary jargon that creates a barrier to understanding by newcomers. Second, the coinage of "val" from the obscure mathematical term "valuation" is tenuous and unlikely to provide helpful insight: "p-adic valuation" is an obscure term for "prime count", which would be an element of a prime-count vector ("monzo"), not a map ("val").</ref> specifies the structure of a regular temperament. It transforms [[interval]]s, usually from a [[just intonation subgroup]] and expressed as prime-count vectors ([[monzo]]s), into [[mapped interval]]s ([[tmonzo]]s). It is often represented by a matrix, thus called a temperament mapping matrix<ref group="note">The terms "M-map" and "V-map" were also sometimes used to refer to temperament mappings and subgroup basis matrices, although these terms are rarely used nowadays.</ref>, or more simply a mapping matrix or even a mapping when the context is sufficiently clear.

Each row of a temperament mapping matrix, or the only row of a rank-1 temperament mapping matrix, is often called a [[val]].

A [[subgroup basis matrix]] is the dual of a temperament mapping matrix.

=== Tuning map ===
{{Main|Tuning map}}
A tuning map (or rarely a tuning mapping) specifies the tuning of a regular temperament. A (tempered-prime) tuning map transforms intervals into tempered [[interval size]]s. A generator tuning map transforms mapped intervals into interval sizes.

=== Projection map ===
{{Main|Projection matrix}}
A projection matrix (or rarely a projection map) uniquely identifies a specific tuning of a specific regular temperament.

== See also ==
* [[Harmonic template]]

== Notes ==
<references group="note"/>

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
[[Category:Val]]
[[Category:Mapping]]

Temperament map

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Fredg999: Redirected page to Mapping

#REDIRECT [[Mapping]]

MIDI

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Fredg999: /* See also */ Update link

{{Wikipedia|MIDI}}
'''MIDI''' ('''Musical Instrument Digital Interface''') is a standard communications protocol for connecting musical instruments, computers, and related audio devices for playing, editing, and recording music.

In the xenharmonic space, it is used for transmitting scores as <code>.mid</code> files, and for driving synthesizers from keyboards/instruments or scores.

Each MIDI link supports 16 channels (numbered 1–16, but encoded as 0–15), with 128 pitches each (numbered 0–127). In typical applications, these pitches are assigned to [[12edo]] pitches from [[Wikipedia:scientific pitch notation|C−1]] (8.2 Hz) to G9 (12.5 kHz), and the channels are used for different instruments. Pitch bend messages allow the pitch to be varied with more precision than a semitone, but apply the same bend to ''all'' notes sent on a channel.

There are a variety of hacks/extensions that allow for varying amounts of xenharmonicity:

*'''[https://ccrma.stanford.edu/courses/220b-winter-2006/cm/doc/dict/midi-stream-cls.html Channel Rotation]''' - This technique uses all 16 channels for a single instrument, rotating between them for subsequent notes, so that each note can have an independent pitch bend applied to it. For polyphonic music with long sustain of each note, this can still result in audible pitch bend collisions, however.
*'''[[Wikipedia:MIDI tuning standard|MIDI Tuning Standard]]''' ('''MTS''') - Allows for both a bulk tuning dump message, giving a tuning for each of 128 notes, and a tuning message for individual notes as they are played.
*'''[https://www.midi.org/midi-articles/midi-polyphonic-expression-mpe MIDI Polyphonic Expression]''' ('''MPE''') - Allows controlling multiple parameters independently for each note, including pitch-bend.
*'''[https://www.midi.org/midi-articles/details-about-midi-2-0-midi-ci-profiles-and-property-exchange MIDI 2.0]''' - Allows for setting the pitch of each note independently, using the "[https://amei.or.jp/midistandardcommittee/MIDI2.0/MIDI2.0-DOCS/M2-104-UM_v1-0_UMP_and_MIDI_2-0_Protocol_Specification.pdf#page=34 Pitch 7.9]" attribute (which overrides the default pitch).
([[Wikipedia:Open Sound Control|Open Sound Control]] (OSC) is sometimes suggested as a microtonal replacement for MIDI,<ref>[https://groups.google.com/g/microtools/c/Gh1fFbCb6XI OSC question]</ref><ref>linux-audio-dev: [https://ccrma.stanford.edu/mirrors/lalists/lad/2005/05/0016.html <nowiki>Re: [linux-audio-dev] Common synthesizer interface -or- microtonal alternative to MIDI?</nowiki>]</ref> however it is an open-ended communication protocol, without even a de facto standard for sending notes, so the protocol must be customized for each synthesizer/server.<ref>[https://github.com/fabb/SynOSCopy Home · fabb/SynOSCopy Wiki], GitHub, "one of the reasons OSC has not replaced MIDI yet is that there is no connect-and-play … There is no standard namespace in OSC for interfacing e.g. a synth"</ref><ref>[https://focusritedevelopmentteam.wordpress.com/2012/10/24/we-hate-midi-we-love-midi/ We hate MIDI. We love MIDI.] Ben Supper, October 24, 2012, Focusrite Development, "OSC lacks a defined namespace for even the most common musical exchanges, to the extent that one cannot use it to send Middle C from a sequencer to a synthesiser in a standardised manner"</ref><ref>[https://www.nime.org/proceedings/2014/nime2014_300.pdf OSC-Namespace and OSC-State: Schemata for Describing the Namespace and State of OSC-Enabled Systems], "there is no fixed set of messages, each participating server needs to know what messages it can send to the servers it intends to communicate with."</ref>)

= See also =

* [[New Tuning Method]]
* [[Mclaren-notation#Computer and MIDI formats]]
* [[Tuning per channel]] - [Is this one MTS scale per channel?]
* [[List of microtonal hardware synths]] - Lists devices that support MIDI Tuning Standard
* [[DAWs#Approaches to Microtonal Composition in a DAW]]
* [[List of microtonal software plugins]]
* [[Scala]] - can retune MIDI streams and files using pitch bend, supports MIDI sysex and file-based tunings
* [[402653184edo]] - a tuning system that allows for finer subdivisions within the standard 128-note MIDI scale based on [[12edo|12-EDO]], using 32-bit data.
* [[1536/1]] - the largest interval allowed by MIDI without pitch bends (using [[Pythagorean]] mapping)

== References ==
<references />
[[Category:MIDI]]
[[Category:Software]]
[[Category:Hardware]]

List of microtonal software plugins

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Fredg999: /* See also */ Update link

This is a list of software plugins that can be used in a [[DAWs|DAW]] for creating microtonal music.

== What are plugins? ==
Computer musicians typically use a DAW to create music. Plugins provide extra functionality within your DAW. For example, if your DAW doesn't allow microtuning then you can get that functionality via plugins.

Plugins featured on this page are typically either:

* '''Software instruments''' that have built-in tuning functionality (e.g. synthesizers, samplers)
* '''Tuning tools''' that alter the tuning of other instruments, plugins and/or audio clips but otherwise do not generate sounds themselves

Before purchasing a plugin, you should check your DAW's user manual to see which types of plugins it supports. In many cases you can try a free demo to make sure it works as expected. Plugin types include:

* [[Wikipedia: Virtual Studio Technology|VST3]] by [[wikipedia:Steinberg|Steinberg]], widely supported. VST2 is also by Steinberg, who discontinued it in 2018. But most Windows versions of non-Steinberg DAWs still support VST2 as of 2023.
* [[Wikipedia: Audio Units|AU]] by [[wikipedia:Apple Inc.|Apple]], widely supported on macOS only.
* AAX by [[wikipedia:Avid Technology|Avid]], used in ProTools, and [[wikipedia:Real Time AudioSuite|RTAS]], obsolete format which was used in ProTools until version 10.
* [[Wikipedia: LV2|LV2]], [[Wikipedia: CLever Audio Plug-in|CLAP]] ([https://cleveraudio.org/ site]) and [[Wikipedia:Disposable Soft Synth Interface|DSSI]] (obsolete, replaced by LV2), open-source cross-platform plugin formats.
* NKS by [[wikipedia:Native Instruments|Native Instruments]], used with their hardware.

== Instrument plugins with microtonal support ==
Further information about the list of microtonal software plugins is below the list.

{| class="wikitable sortable"
|-
! Publisher
! Name
! Description
! OS
! Plugin type
! Tuning ability
! Tuning method
! License / Cost
|-
| accSone
| [https://www.accsone.com/crusherx/aboutcrusherx crusher-X]
| Granular synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl or [[Anamark tuning file format|tun file]] import
| €289
|-
| amsynth contributors
| [https://amsynth.github.io amsynth]
| Subtractive / virtual analog
| Linux, macOS
| VST, AU, LV2, DSSI
| style="color: #007f00;" | '''Full-keyboard microtuning''' {{normal|(Linux version only)}}
| scl/kbm file import
| style="color: #007f00;" | '''Free software'''
|-
| Applied Acoustics
| [https://www.applied-acoustics.com/chromaphone-3/ Chromaphone 3]
| Physically modeled percussion
| Windows, macOS
| VST3, AU, AAX, NKS, standalone
| Partial Scala support (no keyboard mappings)
| scl file import
| $199
|-
| Applied Acoustics
| [https://www.applied-acoustics.com/lounge-lizard-ep-5/ Lounge Lizard EP-5]
| Physically modeled electric piano
| Windows, macOS
| VST3, AU, AAX, NKS, standalone
| Partial Scala support (no keyboard mappings)
| scl file import
| $199
|-
| Applied Acoustics
| [https://www.applied-acoustics.com/string-studio-vs-2/ String Studio VS-2]
| Physically modeled strings
| Windows, macOS
| VST3, AU, AAX, NKS, standalone
| Partial Scala support (no keyboard mappings)
| scl file import
| $199
|-
| Applied Acoustics
| [https://www.applied-acoustics.com/ultra-analog-va-3/ Ultra Analog VA-3]
| Virtual analog
| Windows, macOS
| VST3, AU, AAX, NKS, standalone
| Partial Scala support (no keyboard mappings)
| scl file import
| $199
|-
| Archie3d (Arthur Benilov)
| [https://archie3d.github.io/aeolus_plugin/ Aeolus]
| Pipe organ synthesizer
| Windows, macOS, Linux
| VST3, VST, AU, CLAP, LV2, standalone
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| style="color: #007f00;" | '''Free software'''
|-
| Arturia
| Augmented series ([https://www.arturia.com/products/software-instruments/augmented-brass/overview BRASS], [https://www.arturia.com/products/software-instruments/augmented-grand-piano/overview GRAND PIANO], [https://www.arturia.com/products/software-instruments/augmented-strings/overview STRINGS], [https://www.arturia.com/products/software-instruments/augmented-voices/overview VOICES])
| Wavetable, granular, virtual analog, sampling
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €99/$99 each
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/buchla-easel-v/overview Buchla Easel V]
| Physically modeled reproduction of the [[Wikipedia:Buchla Electronic Musical Instruments|Buchla Music Easel]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/clavinet-v/overview Clavinet V]
| Physically modeled reproduction of the [[Wikipedia:Clavinet|Hohner Clavinet]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/cmi-v/overview CMI V]
| Physically modeled reproduction of the [[Wikipedia:Fairlight CMI|Fairlight CMI]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/cs-80v/overview CS-80 V]
| Physically modeled reproduction of the [[Wikipedia:Yamaha CS-80|Yamaha CS-80]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €199/$199
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/cz-v/overview CZ V]
| Physically modeled reproduction of the [[Wikipedia:Casio CZ synthesizers#CZ-1|Casio CZ-1]] and [[Wikipedia:Casio CZ synthesizers#CZ-101|CZ-101]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/dx7-v/overview DX-7 V]
| Physically modeled reproduction of the [[Wikipedia:Yamaha DX-7|Yamaha DX-7]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/emulator-ii-v/overview Emulator II V]
| Physically modeled reproduction of the [[Wikipedia:E-mu Emulator#Emulator_II|E-mu Emulator II]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/jun-6-v/overview Jun-6 V]
| Physically modeled reproduction of the [[Wikipedia:Roland Juno-6|Roland Juno-6]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/jup-8-v/overview Jup-8 V]
| Physically modeled reproduction of the [[Wikipedia:Roland Jupiter-8|Roland Jupiter-8]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/korg-ms-20-v/overview KORG MS-20 V]
| Physically modeled reproduction of the [[Wikipedia:Korg MS-20|Korg MS-20]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €199/$199
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/op-xa-v/overview OP-Xa V]
| Physically modeled reproduction of the [[Wikipedia:Oberheim OB-Xa|Oberheim OB-Xa]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/op-xa-v/overview Piano V]
| Physically modeled piano
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| €249/$249
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/pigments/overview Pigments]
| Wavetable, granular, virtual analog, sampling
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| scl or tun file import, MPE, MTS-ESP client
| €199/$199
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/prophet-5-v/overview Prophet-5 V]
| Physically modeled reproduction of the [[Wikipedia:Prophet-5|Sequential Prophet-5]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €199/$199
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/prophet-vs-v/overview Prophet-VS V]
| Physically modeled reproduction of the [[Wikipedia:Prophet VS|Sequential Prophet-VS]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €199/$199
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/stage-73-v/overview Stage-73 V]
| Physically modeled reproduction of the [[Wikipedia:Rhodes piano|Rhodes Stage Piano]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/sq80-v/overview SQ80 V]
| Physically modeled reproduction of the [[Wikipedia:Ensoniq SQ-80|Ensoniq SQ-80]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €199/$199
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/synthi-v/overview Synthi V]
| Physically modeled reproduction of the [[Wikipedia:EMS Synthi AKS|EMS Synthi AKS]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| €149/$149
|-
| Arturia
| [https://www.arturia.com/products/software-instruments/vocoder-v/overview Vocoder V]
| Physically modeled reproduction of the [[Wikipedia:Moog Music#Vocoder (1978)|Moog Vocoder]]
| Windows, macOS
| VST, AU, AAX, NKS, standalone
| Partial microtuning support (no keyboard mappings)
| MPE, MTS-ESP client
| €149/$149
|-
| ash taylor?!
| [https://estrobiologist.gumroad.com/l/sflt SFLT]
| Soundfont player
| Windows, macOS
| VST3, CLAP, AU, native FL Studio plugin
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| MPE, scl/kbm file import
| style="color: #007f00;" | '''Freeware'''
|-
| Audio Damage
| [https://www.audiodamage.com/collections/plugin-instruments/products/ad051-continua Continua]
| Morphing synthesizer
| Windows, macOS, iOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $79
|-
| Audio Damage
| [https://www.audiodamage.com/collections/plugin-instruments/products/ad055-quanta-2 Quanta]
| Granular synthesizer
| Windows, macOS, iOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $129
|-
| Audio Damage
| [https://www.audiodamage.com/collections/plugin-instruments/products/ad038-phosphor-2 Phosphor]
| Additive synthesizer
| Windows, macOS, iOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $59
|-
| BigTick Audio Software
| [https://www.bigtickaudio.com/discontinued-plugins/ Angelina]
| Vocal formant synthesizer
| Windows
| VST (32-bit)
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| style="color: #007f00;" | '''Freeware'''
|-
| BigTick Audio Software
| [http://www.bigtickaudio.com/rhino/home Rhino]
| Hybrid FM
| Windows, macOS
| VST, AU
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| $49.00
|-
| [https://biptunia.com/?page_id=2070 BIPTUNIA SYNTHS]
| [https://biptunia.com/?p=10403 MICROTONE 5000] ([https://www.youtube.com/watch?v=eue7Gk4_nQE Demo video])
| Analog-style synth Replaces ''Simple Microtonal Synth'', ''Microtonal Polyphonic Shiny Dirt'', and ''Microtonal Poly Worms'', which were deprecated by maker in favor of ''MICROTONE 5000'', released April 2023.
| Windows 64-bit
| VST3, VST2
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| .mid files (5378 included)
| style="color: #007f00;" | '''Freeware'''
|-
| [https://biptunia.com/?page_id=2070 BIPTUNIA SYNTHS]
| [https://biptunia.com/?p=4040 Simple Microtonal Sampler] ([https://youtu.be/OoGHo2X6MFE Demo video])
| Sampler, .wav format Updated with fixes and more features March 2023. (Now stable, + rough and fine pitch control and ASDR)
| Windows 64-bit
| VST3, VST2
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| .mid files (5378 included)
| style="color: #007f00;" | '''Freeware'''
|-
| [https://biptunia.com/?page_id=2070 BIPTUNIA SYNTHS]
| [https://biptunia.com/?p=10610 FreakBone 9000]
| Analog-style synth, circuit bent. Released May 2023.
| Windows 64-bit
| VST2
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| Must be installed with either MICROTONE 5000 or FreakBone 9000, uses the .mid files from either of them.
| style="color: #007f00;" | '''Freeware'''
|-
| [https://github.com/brummer10 brummer10]
| [https://github.com/brummer10/Loopino Loopino]
| Sampler
| Linux, Windows
| Standalone, VST2, CLAP
|
| scl file import
| style="color: #007f00;" | '''Free software'''
|-
| Camel Audio
| Alchemy 2 [and above]
| Additive synthesizer/spectral sampler
| Logic Pro X bundled plugin
| N/A
| Limited to 12 note scales, each note ±100 cents from 12edo
| Via Logic Pro settings
| Part of [https://www.apple.com/logic-pro/ Apple Logic Pro X]
|-
| David Hillowitz
| [https://www.decentsamples.com/product/decent-sampler-plugin/ Decent Sampler 1.8 [and above]]
| Sampler, .dspreset/.dslibrary soundfont format; imports .sfz and .exs
| Windows, macOS, Linux 64-bit
| VST3, AU, AAX, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import with adjustable reference pitch (has a switch for frequency-aware sample mapping)
| style="color: #007f00;" | '''Freeware'''
|-
| Dexed contributors
| [https://asb2m10.github.io/dexed/ Dexed]
| FM synthesizer
| Windows, macOS, Linux
| VST, AU, LV2
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import, MTS-ESP client
| style="color: #007f00;" | '''Free software'''
|-
| Dillon Bastan
| [https://maxforlive.com/library/device/7027/swarmalators-t Swarmalators T]
| Various synthesizer, drone, FM
| Windows, macOS
| Max for Live device
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import (drag and drop scala file into Scale)
| style="color: #007f00;" | '''Freeware'''
|-
| Dillon Bastan
| [https://maxforlive.com/library/device/6946/fractal-filters Fractal Filters]
| Various synthesizer, drone, effect, filter
| Windows, macOS
| Max for Live device
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import (drag and drop scala file into Scale)
| style="color: #007f00;" | '''Freeware'''
|-
| FabFilter
| [https://www.fabfilter.com/products/twin-3-synthesizer-plug-in Twin 3]
| Virtual analog synthesizer
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, Midi Tuning Standard, MTS-ESP client
| € 129
|-
| Full Bucket Music
| [https://www.fullbucket.de/music/fb3300.html FB-3300] / [https://www.fullbucket.de/music/fb3200.html FB-3200] / [https://www.fullbucket.de/music/fb3100.html FB-3100]
| Virtual analog synthesizer
| Windows, macOS
| VST, AU
| Limited to 12 note scales, each note ±100 cents from 12edo
| User input, some presets, , MTS-ESP client
| style="color: #007f00;" | '''Freeware'''
|-
| Full Bucket Music
| [https://www.fullbucket.de/music/whispair.html Whispair]
| Wavetable synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl, kbm and tun file import, MTS-ESP client
| style="color: #007f00;" | '''Freeware'''
|-
| Fxpansion
| [https://fxpansion.com/products/cypher2/ Cypher2]
| Various synthesizer, MPE-optimised
| Windows, macOS
| VST, AU, AAX, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €179 (demo 30 min)
|-
| Fxpansion
| [https://fxpansion.com/products/strobe2/ Strobe2]
| Various synthesizer, MPE-optimised
| Windows, macOS
| VST, AU, AAX, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €165 (demo 30 min)
|-
| Fors
| [https://fors.fm/junior Junior]
| 4-bit synthesizer
| Windows, macOS, Linux
| VST3, AU, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| MTS-ESP client
| €39
|-
| Fors
| [https://fors.fm/pivot Pivot]
| 3OP FM synthesizer
| Windows, macOS, Linux
| VST3, AU, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| MTS-ESP client
| €39
|-
| Fors
| [https://fors.fm/Kit Pop]
| Various synthesizer
| Windows, macOS
| Max for Live device
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| User input for n-EDO
| €15 (Kit bundle)
|-
| Fuzzpilz
| [https://www.kvraudio.com/product/oatmeal-by-fuzzpilz Oatmeal] ([https://www.bestfreeplugins.com/download.php?p=Oatmeal download])
| Subtractive synthesizer
| Windows
| VST
| Limited to 12 note scales, each note ±200.00 cents from 12edo
| Manual entry
| style="color: #007f00;" | '''Freeware'''
|-
| J. Lang
| [https://hyperfield.jlang.io/ Hyper Field]
| Modular additive/spectral synthesizer
| Windows, macOS
| VST3i, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl or tun file import
| $200 (early pricing, no demo)
|-
| MakeMusic
| [https://www.garritan.com/powered-by-aria-player/ Garritan ARIA Player]
| Sampler
| Windows, macOS
| VST, AAX, standalone
| Partial Scala support (no keyboard mappings) (buggy with nonoctave scales)
| scl file import with adjustable reference pitch
| style="color: #007f00;" | '''Freeware''' (registration)
|-
| Image-Line
| [https://www.image-line.com/plugins/Synths/Harmor/ Harmor]
| Additive synthesizer
| Windows
| VST (no longer sold), FL studio plugin
| Partial Scala support (no keyboard mappings) Each note is ± 6000 cents from 12edo
| scl file import (doesn't adjust reference frequency properly) or user input, or from fnv preset files
| €89
|-
| Image-Line
| [https://www.image-line.com/plugins/Synths/Sytrus/ Sytrus]
| Subtractive FM/RM synthesizer
| Windows
| VST (no longer sold), FL studio plugin
| Each note is ± 4800 cents from 12edo
| User input, or from fnv preset files
| €169
|-
| Imoxplus
| [https://www.imoxplus.com/site/ Respiro]
| Physically modeled wind synthesizer
| Windows, MacOS
| VST3, AU, standalone
| built-in one-octave editor, also loads full-keyboard tuning maps
| .tun file import
| €165 + VAT
|-
| Klevgrand
| [https://klevgrand.com/products/tomofon Tomofon]
| Real Audio synth
| Windows, macOS, iPad
| VST, AU, AAX
| Full-keyboard microtuning
| tun file import
| $129
|-
| KORG
| [https://www.korg.com/us/products/software/kc_m1/ KORG Legacy Collection M1]
| Retro Wavetable / Sampler
| Windows, macOS
| VST, AU, RTAS
| Limited to 12 note scales, each note ±99 cents from 12edo
| User input, some presets
| $99.99 ($199.99 for 6 instrument Legacy Collection bundle)
|-
| KORG
| [http://www.korg.com/us/products/software/korg_legacy_collection/page_3.php KORG Legacy Collection Mono/Poly]
| Virtual Analog Synthesizer
| Windows, macOS
| VST, AU, RTAS
| Limited to 12 note scales, each note ±99 cents from 12edo
| User input, some presets
| $99.99 ($199.99 for 6 instrument Legacy Collection bundle)
|-
| KV331 Audio
| [http://www.kv331audio.com/ Synthmaster and Synthmaster One]
| Flexible semi-modular synthesizer
| Windows, macOS
| VST, AU, AAX
| Partial Scala support (no keyboard mappings)
| scl file import
| $99, $79
|-
| Madrona Labs
| [http://madronalabs.com/products/aalto Aalto]
| Semi-modular synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl file import
| $99
|-
| Madrona Labs
| [http://madronalabs.com/products/kaivo Kaivo]
| Semi modular synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl file import
| $129
|-
| Madrona Labs
| [http://madronalabs.com/products/virta Virta]
| Sound-controlled synthesizer and effects
| Windows, macOS
| VST, AU
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl file import
| $89
|-
| Mark Henning
| [http://www.mark-henning.de/am_downloads_eng.php AnaMark]
| Virtual analog/modulation synthesizer
| Windows
| VST
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| style="color: #007f00;" | '''Freeware'''
|-
| MeldaProduction
| [https://www.meldaproduction.com/MPowerSynth MPowerSynth]
| Versatile synthesizer
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €199
|-
| MeldaProduction
| [https://meldaproduction.com/MSoundFactory MSoundFactory]
| Semi Modular
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €299
|-
| Mike Moreno DSP
| [https://patchstorage.com/ep-mk1/ EP-MK1]
| VST, virtual Rhodes electric piano.
| Windows, macOS, Linux
| VST, AU
| Limited to equal divisions of the octave or a non-octave interval
| In tuning section, set number for "# of Divisions" and "Interval to Divide." Can actually work on non-integer decimal edos (!)
| style="color: #007f00;" | '''Free software'''
|-
| Modartt
| [https://www.modartt.com/pianoteq_overview Pianoteq 9]
| Physically modeled piano
| Windows, macOS, Linux
| VST, AU, RTAS, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import, MTS-ESP client
| €249-€399
|-
| Native Instruments
| [http://www.native-instruments.com/en/products/komplete/synths-samplers/fm8/ FM8]
| FM synthesizer
| Windows, macOS
| VST, AU, AAX, RTAS
| Limited to 12 note scales, each note ±50 cents from 12edo
| User input
| $149
|-
| Native Instruments
| [https://www.native-instruments.com/en/products/komplete/samplers/kontakt-6/ Kontakt 6]
| Sampler
| Windows, macOS
| VST, AU, AAX, RTAS, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| Kontakt script
| $399
|-
| Native Instruments
| [https://www.native-instruments.com/en/products/komplete/synths/reaktor-6/ Reaktor 6]
| Modular synthesizer
| Windows, macOS
| VST, AU, AAX, RTAS, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| macros, .txt file import
| €199
|-
| Oli Larkin
| [http://www.olilarkin.co.uk/index.php?p=virtualcz Virtual CZ]
| Virtual emulation of classic digital synthesizer
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €70
|-
| Plogue
| [http://www.plogue.com/products/chipsounds/ chipsounds]
| Emulation of classic arcade/console sound chips
| Windows, macOS
| VST3, AU, AAX, standalone
| Partial Scala support (no keyboard mappings) (buggy with nonoctave scales)
| scl file import with adjustable reference pitch
| $95
|-
| Plogue
| [https://www.plogue.com/products/chipsynth-c64.html chipsynth C64]
| SID synthesizer. Accurate emulation of the Commodore 64 and Commodore 128 sound chips.
| Windows, macOS
| VST3, AU, AAX, standalone
| Partial Scala support (no keyboard mappings)
| scl file import with adjustable reference pitch, MTS-ESP client
| $49.95
|-
| Plogue
| [https://www.plogue.com/products/chipsynth-md.html chipsynth MD]
| A four-operator FM synthesizer that accurately emulates the sound chips of the Sega Genesis console, also known as the Mega Drive.
| Windows, macOS
| VST3, AU, AAX, standalone
| Partial Scala support (no keyboard mappings)
| scl file import with adjustable reference pitch, MTS-ESP client
| $49.95
|-
| Plogue
| [https://www.plogue.com/products/chipsynth-ops7.html chipsynth OPS7]
| A six-operator FM synthesizer that accurately emulates the Yamaha DX7, DX1, DX5, TX81Z and OPL3.
| Windows, macOS
| VST3, AU, AAX, standalone
| Partial Scala support (no keyboard mappings)
| scl file import with adjustable reference pitch, MTS-ESP client
| $49.95
|-
| Plogue
| [https://www.plogue.com/products/chipsynth-portafm.html chipsynth PortaFM]
| A two-operator FM synthesizer that accurately emulates the OPLL (YM2413), OPL2 (YM3812) and OPK2 (YM7129).
| Windows, macOS
| VST3, AU, AAX, standalone
| Partial Scala support (no keyboard mappings)
| scl file import with adjustable reference pitch, MTS-ESP client
| $29.95
|-
| Plogue
| [https://www.plogue.com/products/chipsynth-sfc.html chipsynth SFC]
| A timbre layering synthesizer that accurately emulates the sound module of the Super Nintendo, also known as the Super Famicom.
| Windows, macOS
| VST3, AU, AAX, standalone
| Partial Scala support (no keyboard mappings)
| scl file import with adjustable reference pitch, MTS-ESP client
| $39.95
|-
| Plogue
| [http://www.plogue.com/products/sforzando/ Sforzando]
| Sampler, .sfz soundfont format
| Windows, macOS
| VST3, AU, AAX, standalone
| Partial Scala support (no keyboard mappings) (buggy with nonoctave scales)
| scl file import with adjustable reference pitch
| style="color: #007f00;" | '''Freeware'''
|-
|Renoise
|[https://www.renoise.com/products/redux/ Redux]
|Sampler, advanced and using variety of formats
|Windows, macOS,
Linux
|VST3
|Partial Scala support (no keyboard mappings)
|scl file import with adjustable reference pitch, MTS-ESP client
|$56,00
|-
| Reveal Sound
| [https://www.reveal-sound.com/plug-ins/spire Spire]
| Subtractive synthesizer with flexible oscillators
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl file import with adjustable reference pitch
| $189 + VAT
|-
| Rhizomatic Software Synthesis
| [https://rhizomatic.fr/ Plasmonic]
| Physical modeling + subtractive synthesis + few unique twists
| Windows, macOS
| VST3, AU
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl file import
| 149€/149$
|-
| rncbc
| [https://padthv1.sourceforge.io/ padthv1]
| Polyphonic synthesizer
| Linux
| LV2, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import
| style="color: #007f00;" | '''Free software'''
|-
| rncbc
| [https://samplv1.sourceforge.io/ samplv1]
| Sampler, .wav format
| Linux
| LV2, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import
| style="color: #007f00;" | '''Free software'''
|-
| rncbc
| [https://synthv1.sourceforge.io synthv1]
| Subtractive polyphonic
| Linux
| LV2, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import
| style="color: #007f00;" | '''Free software'''
|-
| Rob Papen
| [https://www.robpapen.com/albino-3-legend.html Albino-3 Legend]
| Hybrid synthesizer
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl file import
| $99
|-
| Rob Papen
| [https://www.robpapen.com/bit.html BIT]
| Analogue modeled synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import
| $99
|-
| Rob Papen
| [https://www.robpapen.com/BLADE-2.html Blade 2]
| Additive synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import
| $99
|-
| Rob Papen
| [https://www.robpapen.com/blue3.html Blue 3]
| Crossfusion (FM, subtractive, phase distortion, wave shaping) synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import
| $149
|-
| Rob Papen
| [https://www.robpapen.com/go2.html Go2]
| Additive/subtractive synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import
| $49
|-
| Rob Papen
| [https://www.robpapen.com/Predator-3.html Predator 3]
| Analog style synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import
| $149
|-
| Rob Papen
| [https://www.robpapen.com/Quad-vst.html Quad]
| Subtractive synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import
| $99
|-
| Rob Papen
| [https://www.robpapen.com/Vecto-vst.html Vecto]
| Vector synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import
| $99
|-
| Rob Papen
| [https://www.robpapen.com/RoCoder.html RoCoder]
| Vocoder
| Windows, macOS
| VST, VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €49
|-
| rs-met
| [http://www.rs-met.com/description_straightliner.html Straightliner]
| Subtractive synthesizer
| Windows
| VST
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €89
|-
| ryukau
| [https://github.com/ryukau/VSTPlugins Uhhyou]
| Various experimental synthesizers
| Windows, macOS, Linux
| VST3
| Limited to equal divisions of the octave
| In tuning section, set number for "ET"
| style="color: #007f00;" | '''Free software'''
|-
| Sonic Cat
| [https://sonic-cat.com/purity/ Purity]
| Rompler/synthesizer
| Windows, macOS
| VST, AU
| Limited to 12 note scales, each note ±99 cents from 12edo
| Manual entry, presets, scl file import
| €49.99
|-
| Spectrasonics
| [https://www.spectrasonics.net/products/keyscape/index.php Keyscape]
| Piano rompler
| Windows, macOS
| VST, AU, AAX, RTAS
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| €349
|-
| Spectrasonics
| [http://www.spectrasonics.net/products/omnisphere.php Omnisphere 3]
| Rompler/synthesizer
| Windows, macOS
| VST, AU, AAX, RTAS
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| $499, €399
|-
| Spectrasonics
| [https://www.spectrasonics.net/products/trilian/index.php Trilian]
| Bass rompler
| Windows, macOS
| VST, AU, AAX, RTAS
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| $299, €249
|-
| Steinberg
| [https://www.steinberg.net/vst-instruments/halion/ HALion 7]
| Sampler / Synth
| Windows, macOS
| VST, AU, AAX, standalone
| Limited to 12 note scales, each note ±99 cents from 12edo
| Midi module with manual entry, presets and SCL file import, limited to 5, 7 or 12 note scales. Also Lua script.
| $349
|-
| Stone Voices
| [http://stone-voices.ru/vst/polygas?lang=en PolyGAS]
| Granular synthesis
| Windows
| VST
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/tun file import
| style="color: #007f00;" | '''Freeware'''
|-
| Surge Synth Team
| [https://surge-synthesizer.github.io/ Surge XT]
| Subtractive hybrid synthesizer
| Windows, MacOS, Linux
| VST3, AU, LV2, CLAP, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import, MTS-ESP Client, user input
| style="color: #007f00;" | '''Free software'''
|-
| Surge Synth Team
| [https://surge-synth-team.org/tuning-workbench-synth/ Tuning Workbench Synth]
| Realtime tuning synthesizer & scl/kbm exporter
| Windows, macOS, Linux
| VST3, AU, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| Manual input, scl/kbm file import
| style="color: #007f00;" | '''Free software'''
|-
| SynthFont
| [http://www.synthfont.com/VSTSF_news.html SynthFontVSTi]
| Soundfont player (sf2, sfz, sfark, gig, dls, etc.)
| Windows
| VST
| Limited to 12 note scales
| Manual entry or choose from a list
| style="color: #007f00;" | '''Freeware'''
|-
| TAL Software
| [https://tal-software.com/products/tal-bassline-101 TAL-BassLine-101]
| Virtual analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €55, 80
|-
| TAL Software
| [https://tal-software.com/products/tal-j-8 TAL-J-8]
| Virtual analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $80
|-
|TAL Software
|[https://tal-software.com/products/tal-j8x TAL-J-8X]
|Virtual analog synthesizer
|Windows, macOS, Linux
|VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
|[[Anamark tuning file format|tun file]] import, MTS-ESP client
|$80
|-
| TAL Software
| [https://tal-software.com/products/tal-mod TAL-Mod]
| Virtual modular analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €68, 34
|-
| TAL Software
| [https://tal-software.com/products/tal-pha TAL-Pha]
| Virtual modular analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €79
|-
| TAL Software
| [http://tal-software.com/products/tal-sampler TAL-Sampler]
| Sampler/Synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $60
|-
| TAL Software
| [https://tal-software.com/products/tal-u-no-lx TAL-U-NO-LX]
| Virtual analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €55, €80
|-
| tiagolr
| [https://github.com/tiagolr/ripplerx RipplerX]
| Physically modeled synth
| Windows, macOS, Linux
| VST3, LV2, AU
| Partial microtuning support (no keyboard mappings)
| MTS-ESP client
| style="color: #007f00;" | '''Free software'''
|-
| u-he
| [https://u-he.com/products/ace/ ACE]
| Virtual modular analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €69
|-
| u-he
| [https://u-he.com/products/bazille/ Bazille]
| Virtual modular synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $129
|-
| u-he
| [https://u-he.com/products/diva/ Diva]
| Virtual analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $179
|-
| u-he
| [https://u-he.com/products/hive/ Hive 2]
| Wavetable synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €149
|-
| u-he
| [https://u-he.com/products/repro/ Repro]
| Virtual analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €149
|-
| u-he
| [https://u-he.com/products/tyrelln6/ TyrellN6]
| Virtual analog synthesizer
| Windows, macOS, Linux
| VST3, AU, AAX, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| MTS-ESP client
| style="color: #007f00;" | '''Freeware'''
|-
| u-he
| [https://u-he.com/products/zebra-legacy/ Zebra 2]
| Semi Modular
| Windows, macOS, Linux
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €99
|-
|u-he
| [https://u-he.com/products/synths/zebra3/ Zebra 3]
| Semi Modular
| Windows, macOS, Linux
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| €249
|-
| UVI
| [https://www.uvi.net/falcon.html Falcon 2]
| Hybrid synthesizer
| Windows, macOS
| VST3, AU, AAX, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl, kbm and tun file import
| €349
|-
| Unfiltered audio
| [https://www.unfilteredaudio.com/products/lion Lion]
| Hybrid synthesizer
| Windows, macOS
| VST3, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| tun file import, MTS-ESP client
| $199
|-
| Vienna Symphonic Library
| [https://www.vsl.co.at/manuals/synchron-player/introduction Vienna Synchron Player]
| Sample libraries
| Windows, macOS
| VST3, AU, AAX, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl, kbm and MTS-ESP client
| style="color: #007f00;" | '''Freeware'''
|-
| Vienna Symphonic Library
| [https://www.vsl.co.at/manuals/synchron-pianos/introduction Vienna Synchron Pianos]
| Sample libraries
| Windows, macOS
| VST3, AU, AAX, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl, kbm and MTS-ESP client
| style="color: #007f00;" | '''Freeware'''
|-
| Vital Audio
| [https://vital.audio/ Vital]
| Wavetable synthesizer
| Windows, macOS, Linux
| VST, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl, kbm and tun file import
| style="color: #007f00;" | '''Free software''' (registration) $25, $80 for additional features
|-
| Vitalium contributors
| Vitalium
| Wavetable synthesizer
| Linux
| LV2
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl, kbm and tun file import
| style="color: #007f00;" | '''Free software'''
|-
| The Wave Warden
| [https://www.thewavewarden.com/odin2 Odin 2]
| Subtractive synthesizer
| Windows, macOS, Linux
| VST3, AU, LV2, CLAP
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| scl/kbm file import
| style="color: #007f00;" | '''Free software'''
|-
| Xfer Records
| [https://www.xferrecords.com/products/serum-2 Serum 2]
| Wavetable synthesizer
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import, MTS-ESP client
| $189
|-
| ZynAddSubFX contributors
| [http://zynaddsubfx.sourceforge.net/ ZynAddSubFX]
| Various synthesizer
| Windows, macOS, Linux
| VST, LV2, standalone
| style="color: #007f00;" | '''Full-keyboard microtuning'''
| Direct input of values, or import scl/kbm file
| style="color: #007f00;" | '''Free software''' Free if installed via AUR or KxStudio repo or compiling from source $55 for pre-compiled binary
|}

; Notes
* "Full-keyboard microtuning" means that the plugin is capable of any arbitrary pitch, note or frequency on any given MIDI note. This is possible via .tun format, MIDI Tuning Standard, and .scl/.kbm pairs (but not .scl support on its own).
* All synths in the list are assumed to be 64-bit by default or offer an option of 32- or 64-bit. Synths that are 32-bit only are marked with "(32-bit)"
* For distinction of free software and freeware, see [[Wikipedia: Free software]] and [[Wikipedia: Freeware]].
* In some synths, portamento/glide doesn't work properly when microtuned. Examples are Xfer Serum and LinPlug synths. There may be others, so it is recommended to test this function before making a purchase, if it is important for you.
* There are plugins retunable by MTS-ESP ([[#Tuner plugins|see below]]) which aren’t in this table. See them on the official [https://oddsound.com/usingmtsesp.php MTS-ESP page].
** The list there seems incomplete as of January 26, 2023: Full Bucket Audio states that their plugins [http://music.fullbucket.de/mps.html MPS], [http://music.fullbucket.de/tricent.html Tricent mk III], [http://music.fullbucket.de/stigma.html Stigma], [http://music.fullbucket.de/fbvc.html FBVC], [http://music.fullbucket.de/monofury.html Mono/Fury], [http://music.fullbucket.de/blooo.html The blooo],[https://www.fullbucket.de/music/scrooo.html The scrooo], [http://music.fullbucket.de/qyooo.html The qyooo] now have MTS-ESP support, but these aren’t in the list yet.

== Discontinued instrument plugins with microtonal support ==
Plugins which are discontinued or no longer available are shown in the below table for reference.
{| class="wikitable sortable mw-collapsible mw-collapsed"
|-
! Publisher
! Name
! Description
! OS
! Plugin type
! Tuning ability
! Tuning method
! License / Cost
|-
| Cakewalk
| [https://www.cakewalk.com/Products/Dimension-Pro Dimension Pro]
| Combines real instruments with advanced synthesis
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | Full-keyboard microtuning'''
| scl file import
| Discontinued
|-
| Cakewalk
| [https://www.cakewalk.com/Products/Rapture Rapture Pro]
| Multisample synthesizer
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | Full-keyboard microtuning'''
| scl file import
| Discontinued
|-
| Cakewalk
| [https://www.cakewalk.com/Products/Z3TA Z3TA+2]
| Waveshaping synthesizer
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | Full-keyboard microtuning'''
| scl file import
| Discontinued
|-
| Camel Audio
| Alchemy 1.x
| Additive synthesizer / spectral sampler
| Windows, macOS
| VST, AU, AAX
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| Humanoid Sound Systems
| [http://www.humanoidsoundsystems.com/enzyme-vst-plugin-au-plugin-synth/ Enzyme]
| Scanned synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| LinPlug
| LinPlug Alpha
| Subtractive synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| LinPlug
| LinPlug CRX4
| Sample manipulation synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| LinPlug
| LinPlug MorphoX
| Subtractive synthesizer
| Windows, macOS
| VST, AU, RTAS (RTAS is macOS only)
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| LinPlug
| LinPlug Octopus
| Modulation synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| LinPlug
| LinPlug Organ 3
| Organ emulation
| Windows, macOS
| VST, AU, RTAS (RTAS is macOS only)
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| LinPlug
| LinPlug SaxLab
| Saxophone emulation
| Windows, macOS
| VST, AU, RTAS (RTAS is macOS only)
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| LinPlug
| LinPlug Spectral
| Additive/subtractive synthesizer
| Windows, macOS
| VST, AU
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| Native Instruments
| [http://www.native-instruments.com/en/products/komplete/synths-samplers/absynth-5/ Absynth]
| Virtual modular subtractive synthesizer
| Windows, macOS
| VST, AU, AAX, RTAS, standalone
| style="color: #007f00;" | Full-keyboard microtuning'''
| gly file import
| Discontinued
|-
| Rob Papen
| [https://splice.com/plugins/342-albino-3-vst-au-by-rob-papen Albino 3]
| Hybrid/subtractive synthesizer
| Windows, macOS
| VST, AU, RTAS
| style="color: #007f00;" | Full-keyboard microtuning'''
| tun file import
| Discontinued
|-
| SoundModeler
| [https://github.com/SoundModeler?tab=overview&from=2020-12-01&to=2020-12-31]
| VST, Additive synthesizer
| Windows
| VST
| style="color: #007f00;" | Full-keyboard microtuning'''
| The built-in choice for the temperaments: Equal, Pythagorean, Just and Mean-tone 1/4 comma. Main note flexible tune (from A=440), Supports each oscillator tuning by 0.01%
| style="color: #007f00;" | '''Freeware''' (discontinued)
|-
| VAZ
| [http://www.vaz-synths.com/vmIndex.html VAZ Modular]
| Virtual modular synthesizer
| Windows
| VST, standalone
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| VAZ
| [http://www.vaz-synths.com/vpIndex.html VAZ Plus]
| Virtual analog synthesizer
| Windows
| VST, standalone
| style="color: #007f00;" | Full-keyboard microtuning'''
| [[Anamark tuning file format|tun file]] import
| Discontinued
|-
| Xen-Arts
| Ivor2
| Subtractive synthesizer
| Windows
| VST (32-bit)
| style="color: #007f00;" | Full-keyboard microtuning'''
| MIDI Tuning Standard (MTS)
| style="color: #007f00;" | Freeware''' (discontinued)
|-
| Xen-Arts
| OneSF2
| Soundfont (sf2) player
| Windows
| VST (32-bit)
| style="color: #007f00;" | Full-keyboard microtuning'''
| MIDI Tuning Standard (MTS)
| style="color: #007f00;" | Freeware''' (discontinued)
|-
| Xen-Arts
| Xen-FMTS 2
| FM synthesizer
| Windows
| VST (32-bit)
| style="color: #007f00;" | Full-keyboard microtuning'''
| MIDI Tuning Standard (MTS)
| style="color: #007f00;" | Freeware''' (discontinued)
|-
| Xen-Arts
| XenFont
| Subtractive synthesizer with soundfont (sf2) oscillators
| Windows
| VST (32-bit)
| style="color: #007f00;" | Full-keyboard microtuning'''
| MIDI Tuning Standard (MTS)
| style="color: #007f00;" | Freeware''' (discontinued)
|}

== Tuner plugins ==
These plugins only tune other instruments and typically do not generate any sound themselves.

{| class="wikitable sortable"
|-
! Publisher
! Name
! Description
! OS
! Plugin type
! Tuning ability
! Tuning method
! License / Cost
|-
| Celemony
| [https://www.celemony.com/en/melodyne/what-is-melodyne Melodyne]
| Polyphonic pitch editing and autotune
| Windows, macOS
| VST, AU, AAX, RTAS, standalone
| todo
| scl file import
| €99-€699
|-
| Codenote
| [https://maxforlive.com/library/device/6713/microtone-toolkit-mpe-edition Microtone Toolkit MPE edition]
| Polyphonic MIDI retuning (pitch-bend based & MPE-based)
| Windows, macOS
| Max for Live device
| style="color: #007f00;" | Full-keyboard microtuning'''
| variety, including inputs for n-TET, just intonation ratios, detuning in cents, scl file import
| $40
|-
| Cycling '74
| [https://maxforlive.com/library/device/6713/microtone-toolkit-mpe-edition Max 7 Pitch and Time Machines] – ''Autotuna'' & ''Microtuner''
| Microtonal autotune
| Windows, macOS
| Max for Live device
| style="color: #007f00;" | Full-keyboard microtuning'''
| ''Autotuna'' – scl file import, ''Microtuner'' – graphic function
|
|-
| Dmitri Volkov
| [https://www.dmitrivolkov.com/projects/pivotuner/ Pivotuner]
| automatic real-time pure intonation and microtonal modulation
| Windows, macOS
| VST3, AU
| Limited to 12 interval sizes, but can microtonally modulate. More intervals possible with Chord Tuning.
| Manual user input
| $5 (monthly license), $47 (lifetime license)
|-
| Entonal Studio
| [https://entonal.studio/ Entonal Studio]
| Creates tunings and retunes softsynths
| Windows, macOS, Linux
| VST2, VST3, AU, standalone
| style="color: #007f00;" | Full-keyboard microtuning'''
| Direct input, scl file import, MTS-ESP master
| Intro :£49 Later: £79
|-
| Infinitone
| [https://infinitone.com Infinitone DMT]
| Creates tunings and retunes softsynths via MTS-ESP, MPE, MTS SysEx or MIDI pitch bend
| Windows, macOS
| ?, standalone
| style="color: #007f00;" | Full-keyboard microtuning'''
| Direct input, scl file import, MTS-ESP master
| $99 (introductory price)
|-
| Octarone
| [https://sites.google.com/site/octarone/code/midi-tools/key-tuner Key Tuner] (Broken link as of May 2023)
| MIDI note retuning
| Windows, macOS, Linux
| [http://www.reaper.fm/reaplugs/ ReaJS Reaper]
| style="color: #007f00;" | Full-keyboard microtuning'''
| scl file import
| style="color: #007f00;" | Free software'''
|-
| Oddsound
| [https://oddsound.com/mtsespmini.php MTS-ESP Mini]
| Retunes softsynths via MTS-ESP
| Windows, macOS Intel, macOS ARM
| VST, VST3, AU, AAX
| style="color: #007f00;" | Full-keyboard microtuning'''
| scl/tun import, MTS-ESP master
| style="color: #007f00;" | Freeware'''
|-
| Oddsound
| [https://oddsound.com/mtsespsuite.php MTS-ESP Suite]
| Creates tunings and retunes softsynths via MTS-ESP, MPE, MTS SysEx or MIDI pitch bend
| Windows, macOS Intel, macOS ARM
| VST, VST3, AU, AAX
| style="color: #007f00;" | Full-keyboard microtuning'''
| Direct input, scl/tun import, MTS-ESP master
| £79.99
|-
| PitchGrid
| [https://node.audio/products/pitchgrid PitchGrid]
| Parametrizes MOS Scales with parameters exposed to host DAW (i.e. supports automation for changing tuning), supports scale-aware pitch bend
| Windows, MacOS, Linux
| VST3, AU, standalone
| style="color: #007f00;" | Full-keyboard microtuning'''
| Direct MIDI/MPE to MIDI/MPE, MTS-ESP master
| £42
|-
| Surge Synth Team
| [https://surge-synthesizer.github.io/ Surge XT]
| Creates tunings and retunes softsynths via MTS-ESP
| Windows, MacOS, Linux
| VST3, AU, CLAP
| style="color: #007f00;" | Full-keyboard microtuning'''
| scl/kbm file import, tuning editor with equal-divisions of arbitrary intervals, or direct input of ratios or cents, MTS-ESP master
| style="color: #007f00;" | Free software'''
|-
| Pitch Innovations
| [https://www.pitchinnovations.com/products/fluid-pitch/ Fluid Pitch]
| Creates tunings and retunes softsynths via MPE or MIDI pitch bend
| Windows, macOS
| VST, VST3, AU, AAX, iOS
| todo
| todo
| $75
|-
| Ursine
| [https://www.maxforlive.com/library/device/3068/retune-for-live Retune for Live]
| Polyphonic MIDI retuning (pitch-bend based)
| Windows, macOS
| Max for Live device
| style="color: #007f00;" | Full-keyboard microtuning'''
| scl file import, csv file import
| style="color: #007f00;" | Freeware'''
|-
| VisualPlugin
| [https://github.com/Windows81/FL-Custom-Intonation Custom Intonation]
| Fruity Patcher preset; works on many of FL Studio's stock plugins
| Windows, macOS
| N/A
| Limited to 12 note scales, each note ±100 cents from 12edo
| scl file import or manual user input
| style="color: #007f00;" | Free software'''
|-
| Marcus Hobbs
| [https://www.wilsonic.co/ Wilsonic MTS-ESP]
| Dynamic microtuning. Retune the MPE software synthesizers in your DAW
| Windows, macOS, iOS
| AU, VST3
| style="color: #007f00;" | Full-keyboard microtuning'''
| Direct input, scl file import, MTS-ESP master
| style="color: #007f00;" | Free software'''
|}

== MIDI remapper plugins ==
These plugins are neither instruments nor tuners but are useful when working with those, remapping notes input to the plugin, presumed to be in one tuning or a mode of a tuning, to the notes the plugin outputs, presumed to be in another tuning. For example, making you use just a MOS subscale of an output equal-division tuning.

{| class="wikitable sortable"
|-
! Publisher
! Name
! Description
! OS
! Plugin type
! Tuning ability
! Tuning method
! License / Cost
|-
| Vincenzo Sicurella
| [https://github.com/vsicurella/SuperVirtualKeyboard SuperVirtualKeyboard]
| MIDI remapper, keyboard layout and scale explorer and visualizer
| Windows, macOS, Linux (on demand)
| VST, AU
| todo
| todo
| Free software
|-
| Joseph Sheasby (Userminusone)
| [https://github.com/Userminusone/Userminusone-s-Microtonal-Launchpad-tools Userminusone's Microtonal Launchpad Tools]
| MIDI remapper, Novation Launchpad MINI layout visualizer (utilizes text files for layout configuration and optional subset/remapping specifications, and relies on multiple plugins to work properly. This is explained in the video the GitHub repository links to.)
| Windows, macOS, Linux
| Reaper JSFX (JesuSonic FX)
| N/A
| N/A
| Free software
|}

== MPE-compatible software instrument plugins ==
A recent development in music technology is MIDI polyphonic expression (MPE). Instruments that support MPE are able to apply pitch-bend on a per-note basis. In theory, this allows a musician to create polyphonic microtonal music using any MPE-compatible synth. In practice, it is a laborious process to create microtonal music by specifying a pitch-bend on every single MIDI note. However some plugins shown in the above "Tuner plugins" table can intercept MIDI note data from the sequencer and systematically apply the correct per-note pitch-bend before that data is received by the instrument. Because of this, creating microtonal music with MPE-compatible instruments is now a viable approach.

{| class="wikitable sortable"
|-
! Publisher
! Name
! Description
! OS
! Plugin type
! Tuning ability
! Tuning method
! License / Cost
|-
| Ableton
| [https://www.ableton.com/en/packs/microtuner/ Ableton Microtuner]
| MIDI remapper, works only in Abelton DAW, can import .scl files, retunes any Ableton instrument or other MPE-compatible plugins.
| Windows, macOS
| VST, AU
| Full-keyboard microtuning
| .scl file import or manually setting pitches
| Included with Live 12 Suite
|}

== See also ==
* [[Software]]
* [[DAWs]]
* [[List of microtonal hardware synths]]

[[Category:Practice]]
[[Category:Software]]

Hardware Synths

2026-07-28T01:17:47Z

Fredg999: Update (again)

#REDIRECT [[List of microtonal hardware synths]]

Talk:List of microtonal hardware synths

2026-07-28T01:17:23Z

Fredg999: Fredg999 moved page Talk:List of hardware synths to Talk:List of microtonal hardware synths without leaving a redirect: Take 3... I suppose it should say "microtonal" like the software instrument list...

== Minilogue XD workarounds ==
I've been attempting to retune my Minilogue XD using the Korg Sound Librarian, which allows import of ''.scl'' / ''.kbm'' files. The process is poorly documented and the results don't match what is expected for files that correctly encode the Erv Wilson 1-3-5-7-9-11 eikosany. I have some workarounds, and I've opened an issue with Korg support. Is this wiki an appropriate place to document my findings?

M. Edward (Ed) Borasky (Znmeb) - 2023-01-10

: Absolutely! There is a ton of original work on this wiki, unlike Wikipedia for instance, and there is no plan to change that. It would be better to place any details on a separate page about the synth (create it if it doesn't exist), and you can link to it from this page so that people can find out about it when browsing through the list.

: By the way, you can sign your talk messages automatically by typing <nowiki>--~~~~</nowiki>. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 04:11, 11 January 2023 (UTC)

List of microtonal hardware synths

2026-07-28T01:17:23Z

Fredg999: Fredg999 moved page List of hardware synths to List of microtonal hardware synths without leaving a redirect: Take 3... I suppose it should say "microtonal" like the software instrument list...

This is a list of hardware electronic instruments that have support for microtonal tuning.

== Keyboards ==
{{Wikipedia|Korg Minilogue#Minilogue XD}}
{{Wikipedia|Korg Monologue}}
{{Wikipedia|ROLI Seaboard}}
* [https://www.ashunsoundmachines.com/ Ashun Sound Machines] (ASM) Hydrasynth Explorer - supports [[MIDI]] Tuning Standard
*[https://www.ashunsoundmachines.com/ Ashun Sound Machines] (ASM) Hydrasynth Keyboard (firmware 1.5 and later) - supports MIDI Tuning Standard
*[https://www.ashunsoundmachines.com/ Ashun Sound Machines] (ASM) Hydrasynth Deluxe - supports MIDI Tuning Standard
* [https://hpi.zentral.zone/tonalplexus Hπ Tonal Plexus]
* Korg Minilogue XD - All 128 MIDI notes can be tuned to anything (1 cent precision)
* Korg Minilogue (after firmware update)
* Korg Monologue - All 128 MIDI notes can be tuned to anything (1 cent precision)
* Korg Prologue - All 128 MIDI notes can be tuned to anything (1 cent precision)
* [http://www.modormusic.com/os13.html Modor NF-1] - Built-in quarter tones, Just Intonation, Harmonic series, many EDOs and a number of hexanies/dekanies + scales via Midi Tuning Standard.
* Novation Summit (firmware 1.2 and later) - All 128 MIDI notes can be tuned to anything (0.4 cent precision)
* ROLI Seaboard
* Starr Labs Microzone U-648
* [https://www.striso.org/ Striso board] - Generalized keyboard which supports [[fifthspan]] and [[7-limit]] JI tunings

== Sound modules ==
* [https://aodyo.com/anyma-phi-user-guide/ Aodyo Instruments Anyma Phi (User Guide - see page 45)]
* [https://www.ashunsoundmachines.com/ Ashun Sound Machines] (ASM) Hydrasynth Desktop (firmware 1.5 and later) - supports MIDI Tuning Standard
* Korg Minilogue XD Module - All 128 MIDI notes can be tuned to anything (1 cent precision)
* Novation Peak (firmware 1.2 and later) - All 128 MIDI notes can be tuned to anything (0.4 cent precision)
* [https://waldorfmusic.com/kyra-en/ Waldorf Kyra] (MTS sysex real time retuning)
* [https://waldorfmusic.com/iridium-en/ Waldorf Iridium] (SCL import)
* Yamaha TX81Z
== Eurorack ==
* [https://ornament-and-cri.me/ ornament & crime] - CV generator
* [https://tubbutec.de/%C2%B5tune/ Tubbutec μTune] - two-channel microtonal quantizer/mapper
* [https://www.expert-sleepers.co.uk/fh2.html Expert Sleepers FH-2 'Factotum'] - MPE and .scl/kmb MIDI to CV

== Other ==
* [https://hpi.zentral.zone/tbx2b Hπ TBX2b] - tuning unit that retunes GM (multichannel) synths, monophonic synths, MTS synths, and proprietary sysex synths.
* [https://sonoclast.com/products/plastic_pitch_midi_microtuner/ Sonoclast Plastic Pitch MIDI Microtuner] - Real-time interface for tuning multiple synths simultaneously. Transmits MIDI tuning for multitimbral synths, monophonic synths, MTS synths, and MPE synths. Also tunes MIDI received from MPE controllers. Supports Scala SCL.
* [[Wikipedia:Theremin|Theremin]]
[[Category:hardware]]
[[Category:synths]]
[[Category:todo:expand]]

Hardware Synths

2026-07-28T01:16:42Z

Fredg999: Update redirect

#REDIRECT [[List of hardware synths]]

Talk:List of microtonal hardware synths

2026-07-28T01:16:22Z

Fredg999: Fredg999 moved page Talk:Hardware synths to Talk:List of hardware synths without leaving a redirect: List page (should've thought of this during the previous move...)

List of microtonal hardware synths

2026-07-28T01:16:22Z

Fredg999: Fredg999 moved page Hardware synths to List of hardware synths without leaving a redirect: List page (should've thought of this during the previous move...)

Talk:Hardware Synths

2026-07-28T01:15:25Z

Fredg999: Fredg999 moved page Talk:Hardware Synths to Talk:Hardware synths: WP:NCCAPS (keeping redirect because it's a high traffic page)

#REDIRECT [[Talk:Hardware synths]]

Talk:List of microtonal hardware synths

2026-07-28T01:15:25Z

Fredg999: Fredg999 moved page Talk:Hardware Synths to Talk:Hardware synths: WP:NCCAPS (keeping redirect because it's a high traffic page)

Hardware Synths

2026-07-28T01:15:25Z

Fredg999: Fredg999 moved page Hardware Synths to Hardware synths: WP:NCCAPS (keeping redirect because it's a high traffic page)

#REDIRECT [[Hardware synths]]

List of microtonal hardware synths

2026-07-28T01:15:25Z

Fredg999: Fredg999 moved page Hardware Synths to Hardware synths: WP:NCCAPS (keeping redirect because it's a high traffic page)

Highly composite interval

2026-07-28T00:44:43Z

Fredg999: Merged into Harmonic

#REDIRECT [[Harmonic#Highly composite harmonic]]

[[Category:Highly composite]]

Prime interval

2026-07-28T00:44:25Z

Fredg999: Merged into Harmonic

#REDIRECT [[Harmonic#Prime harmonic]]

[[Category:Prime]]

Harmonic

2026-07-28T00:44:06Z

Fredg999: /* Prime harmonic */ Insert former see also links

{{Wikipedia}}
{{See also|Category:Harmonics}}
A '''harmonic''' is a whole-number multiple of the fundamental frequency of a sound. It is an element of the [[harmonic series]].

The timbre of a periodic sound, such as a bowed violin or the human voice, contains a nearly infinite amount of harmonic [[partial]]s, starting with 1''f'', 2''f'', 3''f'', 4''f''... where ''f'' is the fundamental frequency. Each of these harmonics has a distinct amplitude, generally decreasing as the 'height' of the harmonic increases. The [[span]] between any two of these harmonics is a [[just interval]]. If the harmonics are numbered such that the fundamental is number 1, the octave is 2, etc., then the interval's ratio is given by the two numbers. For example the interval between the 3rd and 4th harmonics is 4/3.

The ancient Greeks called these harmonics "multiples", and considered them to be a unique interval class separate from [[superparticular]] and [[superpartient]] intervals.

A '''subharmonic''' is a unit fraction of the fundamental frequency of a sound: ''f''/1, ''f''/2, ''f''/3, ''f''/4... It is an element of the [[subharmonic series]].

== Odd harmonic ==
An '''odd harmonic''' is a harmonic where the [[frequency ratio]] is an odd number. The first few odd harmonics are [[1/1]], [[3/1]], [[5/1]], [[7/1]], [[9/1]], [[11/1]], etc. Odd harmonics are significant in that they create distinct [[pitch class]]es, since any even harmonic is a whole number of [[octave]]s above an odd harmonic.

An [[odd limit]] is the set of all [[just interval]]s where the largest odd factor in the numerator and denominator both do not exceed a specified bound. For example, the [[5-odd-limit]] consists of all ratios where the only allowable odd factors are 1, 3, and 5; those being [[1/1]], [[6/5]], [[5/4]], [[4/3]], [[3/2]], [[8/5]], [[5/3]], and any whole number of octaves above those intervals.

We can also restrict the set of usable intervals to only allow odd harmonics. Such intervals include [[3/1]], [[5/3]], [[9/7]], [[27/25]], etc. Here, using [[3/1]] as the [[interval of equivalence]] (a "tritave") is natural, since the 3rd harmonic is the smallest odd harmonic after the fundamental. [[Edt]]s divide the tritave into a whole number of equal steps, analogous to how [[edo]]s divide the octave.

== Prime harmonic ==
{{See also|Category:Prime harmonics}}
A '''prime interval''' or '''prime harmonic''' is a harmonic which as a [[ratio]] of frequencies is a [[prime number]]; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1. It is an element of the [[prime harmonic series]].

Any interval of [[just intonation|just intonation (JI)]] can be expressed in terms of a product of prime intervals, allowing us to decompose a complex JI interval into simpler parts. A prime interval itself cannot be expressed by other prime intervals, so no prime intervals are redundant for reconstructing the entirety of JI. For those reasons and for the fact that prime intervals occur in [[harmonic series]], they form a very important [[basis]] (literally and mathematically) for JI.

For example, the [[octave]] is a prime interval whereas the intervals [[5/3]] or even [[1/1]] are not. In traditional ratio notation, the prime intervals are [[2/1]], [[3/1]], [[5/1]], [[7/1]], [[11/1]] etc.

The [[monzo]] notation of each prime interval consists of all-zeros except for a single entry equal to 1: (2: {{monzo| 1 }}, 3: {{monzo| 0 1 }}, 5: {{monzo| 0 0 1 }}, 7: {{monzo| 0 0 0 1 }}, 11: {{monzo| 0 0 0 0 1 }}, …)

Prime harmonics are often used to define [[harmonic limit]]s, also known as prime limits.

== Highly composite harmonic ==
{{See also|Category:Highly composite harmonics}}
A '''highly composite interval''' or '''highly composite harmonic''' is a harmonic which has a [[ratio]] of frequencies is a [[highly composite number]]; that is, a number such as 1, 2, 4, 6, 12, … each of which has more divisors than the number before it.

Highly composite intervals are significant for having the largest number of ways to be approached through [[harmonic]] [[chord]]s for its size.

Highly composite intervals can be thought of as the opposite of prime harmonics, since highly composite numbers are "antiprime", but they are not complementary sets, as they share a common element (2/1) and there are infinitely many harmonics which are neither prime nor highly composite.

== See also ==
* [[Mixed timbre]]

[[Category:Harmonic| ]] 
[[Category:Psychoacoustics]]
[[Category:Terms]]

Odd harmonic

2026-07-28T00:42:30Z

Fredg999: Merged into Harmonic

#REDIRECT [[Harmonic#Odd harmonic]]

[[Category:Odd limit]]

Harmonic

2026-07-28T00:41:08Z

Fredg999: Merge odd harmonic, prime interval and highly composite interval

{{Wikipedia}}
{{See also|Category:Harmonics}}
A '''harmonic''' is a whole-number multiple of the fundamental frequency of a sound. It is an element of the [[harmonic series]].

The timbre of a periodic sound, such as a bowed violin or the human voice, contains a nearly infinite amount of harmonic [[partial]]s, starting with 1''f'', 2''f'', 3''f'', 4''f''... where ''f'' is the fundamental frequency. Each of these harmonics has a distinct amplitude, generally decreasing as the 'height' of the harmonic increases. The [[span]] between any two of these harmonics is a [[just interval]]. If the harmonics are numbered such that the fundamental is number 1, the octave is 2, etc., then the interval's ratio is given by the two numbers. For example the interval between the 3rd and 4th harmonics is 4/3.

The ancient Greeks called these harmonics "multiples", and considered them to be a unique interval class separate from [[superparticular]] and [[superpartient]] intervals.

A '''subharmonic''' is a unit fraction of the fundamental frequency of a sound: ''f''/1, ''f''/2, ''f''/3, ''f''/4... It is an element of the [[subharmonic series]].

== Odd harmonic ==
An '''odd harmonic''' is a harmonic where the [[frequency ratio]] is an odd number. The first few odd harmonics are [[1/1]], [[3/1]], [[5/1]], [[7/1]], [[9/1]], [[11/1]], etc. Odd harmonics are significant in that they create distinct [[pitch class]]es, since any even harmonic is a whole number of [[octave]]s above an odd harmonic.

An [[odd limit]] is the set of all [[just interval]]s where the largest odd factor in the numerator and denominator both do not exceed a specified bound. For example, the [[5-odd-limit]] consists of all ratios where the only allowable odd factors are 1, 3, and 5; those being [[1/1]], [[6/5]], [[5/4]], [[4/3]], [[3/2]], [[8/5]], [[5/3]], and any whole number of octaves above those intervals.

We can also restrict the set of usable intervals to only allow odd harmonics. Such intervals include [[3/1]], [[5/3]], [[9/7]], [[27/25]], etc. Here, using [[3/1]] as the [[interval of equivalence]] (a "tritave") is natural, since the 3rd harmonic is the smallest odd harmonic after the fundamental. [[Edt]]s divide the tritave into a whole number of equal steps, analogous to how [[edo]]s divide the octave.

== Prime harmonic ==
{{See also|Category:Prime harmonics}}
A '''prime interval''' or '''prime harmonic''' is a harmonic which as a [[ratio]] of frequencies is a [[prime number]]; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1.

Any interval of [[just intonation|just intonation (JI)]] can be expressed in terms of a product of prime intervals, allowing us to decompose a complex JI interval into simpler parts. A prime interval itself cannot be expressed by other prime intervals, so no prime intervals are redundant for reconstructing the entirety of JI. For those reasons and for the fact that prime intervals occur in [[harmonic series]], they form a very important [[basis]] (literally and mathematically) for JI.

For example, the [[octave]] is a prime interval whereas the intervals [[5/3]] or even [[1/1]] are not. In traditional ratio notation, the prime intervals are [[2/1]], [[3/1]], [[5/1]], [[7/1]], [[11/1]] etc.

The [[monzo]] notation of each prime interval consists of all-zeros except for a single entry equal to 1: (2: {{monzo| 1 }}, 3: {{monzo| 0 1 }}, 5: {{monzo| 0 0 1 }}, 7: {{monzo| 0 0 0 1 }}, 11: {{monzo| 0 0 0 0 1 }}, …)

== Highly composite harmonic ==
{{See also|Category:Highly composite harmonics}}
A '''highly composite interval''' or '''highly composite harmonic''' is a harmonic which has a [[ratio]] of frequencies is a [[highly composite number]]; that is, a number such as 1, 2, 4, 6, 12, … each of which has more divisors than the number before it.

Highly composite intervals are significant for having the largest number of ways to be approached through [[harmonic]] [[chord]]s for its size.

Highly composite intervals can be thought of as the opposite of prime harmonics, since highly composite numbers are "antiprime", but they are not complementary sets, as they share a common element (2/1) and there are infinitely many harmonics which are neither prime nor highly composite.

== See also ==
* [[Mixed timbre]]

[[Category:Harmonic| ]] 
[[Category:Psychoacoustics]]
[[Category:Terms]]

Türk sent

2026-07-27T22:21:38Z

Fredg999: Fix double redirect

#REDIRECT [[10600edo]]

Turkish cent

2026-07-27T22:21:01Z

Fredg999: Merged into 10600edo

#REDIRECT [[10600edo]]

[[Category:Interval size measures]]
[[Category:Turkish music]]

10600edo

2026-07-27T22:20:26Z

Fredg999: Merge from Turkish cent page, reorganize sections

{{Infobox ET}}
{{ED intro}}

One step of 10600edo, or 1/100 step of [[106edo]], is known as the '''Turkish cent''' ('''Türk sent''' in Turkish), a term proposed by {{w|tr:Abdülkadir Töre|Abdülkadir Töre}} and {{w|tr:Ekrem Karadeniz|Ekrem Karadeniz}} (''links to Turkish Wikipedia'').

== Theory ==
=== Prime harmonics ===
{{Harmonics in equal|10600}}

== External links ==
* [http://www.tonalsoft.com/enc/t/turk-sent.aspx Türk-sent] on [[Tonalsoft Encyclopedia]]
* [http://web.archive.org/web/20111011133133/http://www.turkmusikisi.com/nazariyat/sistemler/ekrem_karadeniz_sistemi.htm Tabîatta mûsikî insanla birlikte doğduğu için mûsikînin başlangıcı hakkında araştırmalara girişecek değiliz] (archived) (in Turkish language)

[[Category:Equal divisions of the octave|#####]]

Prima

2026-07-27T21:46:40Z

Fredg999: Merge into 12276edo

#REDIRECT [[12276edo]]

[[Category:Interval size measures]]

12276edo

2026-07-27T21:46:18Z

Fredg999: Merge from Prima page, reorganize sections

{{Infobox ET}}
{{ED intro}}

12276edo's step size is sometimes called a '''prima''', a term proposed by [[Lillian Hearne]], when used as an interval size unit.

== Theory ==
12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps. It is the smallest [[atomic]] EDO inside its [[5-odd-limit]] [[diamond tradeoff]] tuning range.

=== As an interval size measure ===
The prima is useful for measurement of 11-limit intervals and commas. Given that 12276edo factors as 22 × 32 × 11 × 31, a prima is a whole number division of one degree of [[12edo]], [[22edo]], [[31edo]], [[99edo]] and [[198edo]]. The [[Pythagorean comma]] is represented by 240 prima, and the [[syntonic comma]] by 220 (and the [[schisma]] is therefore represented as 20 prima). A prima is almost exactly three [[tuning unit]]s. As one degree of [[12edo]] is 1023 prima, one [[cent]] is exactly 10.23 prima.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Subsets and supersets ===
12276edo factors into primes as {{nowrap| 22 × 32 × 11 × 31 }}, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s.

[[Category:3-limit record edos|#####]]

Purdal

2026-07-27T21:40:45Z

Fredg999: Restore 1 category

#REDIRECT [[9900edo]]

[[Category:Interval size measures]]

30103edo

2026-07-27T21:39:36Z

Fredg999: Headings

{{Infobox ET}}
{{ED intro}}

30103edo is consistent in the 11-odd-limit and is otherwise a strong 2.3.5.17 subgroup tuning.

== Theory ==
=== As an interval size measure ===
Since logarithm of 2 in base 10 is equal to 0.30102999..., one step of 30103edo comes exceptionally close to being one step of an otherwise perfectly decimal tuning system, [[100000ed10]], similar to heptameride being one step of [[301edo]] and savart being one step of [[1000ed10]]. It was named '''jot''' by Augustus de Morgan in 1864.

Any integer [[Gallery of arithmetic pitch sequences#APS of jots|arithmetic pitch sequence of ''n'' jots]] is technically a subset of 30103edo, since it is every ''n''th step of 30103edo.

=== Prime harmonics ===
{{harmonics in equal|30103}}

== External links ==
* [http://tonalsoft.com/enc/j/jot.aspx jot] on [[Tonalsoft Encyclopedia]]

Purdal

2026-07-27T21:38:28Z

Fredg999: Merge into 9900edo

#REDIRECT [[9900edo]]

9900edo

2026-07-27T21:38:10Z

Fredg999: Merge from Purdal page, reorganize sections

{{Infobox ET}}
{{ED intro}}

9900edo's step size is sometimes called a '''purdal''', a term proposed by [[Osmiorisbendi]], when used as an interval size unit.

== Theory ==
9900edo is consistent in the 9-odd-limit and it is otherwise a good 2.3.5.7.17.29 subgroup system.

In the 7-limit, it is a [[septiruthenia]]n system, setting 64/63 to 1\44, so that the septimal comma is 225 purdals. It is a member of the [[optimal ET sequence]] for the [[ruthenium]] temperament with an additional prescribed mapping for 5 in the 13-limit.

=== Prime harmonics ===
{{harmonics in equal|9900}}

=== Subsets and supersets ===
9900edo has subset edos {{EDOs|1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 15, 18, 20, 22, 25, 30, 33, 36, 44, 45, 50, 55, 60, 66, 75, 90, 99, 100, 110, 132, 150, 165, 180, 198, 220, 225, 275, 300, 330, 396, 450, 495, 550, 660, 825, 900, 990, 1100, 1650, 1980, 2475, 3300, 4950}}. Its abundancy index is around 2.42.

[[Category:Purdal]]

Diamond

2026-07-27T21:27:18Z

Fredg999: Disambiguation template

'''Diamond''' may refer to:
* the [[diamond function]];
* a [[tonality diamond]].

== See also ==
* [[Diamond-mos notation]] (not related to the above)

{{Disambiguation}}
[[Category:Diamond| ]]

User:Moremajorthanmajor/Map of 9&11 well temperaments

2026-07-27T21:23:58Z

Fredg999: Fredg999 moved page Map of 9&11 well temperaments to User:Moremajorthanmajor/Map of 9&11 well temperaments without leaving a redirect: XW:NG

[[User:Moremajorthanmajor/Map of Aviloid well temperaments]]

[[User:Moremajorthanmajor/Map of Carline well temperaments]]

User:Moremajorthanmajor/Map of 9&11 well temperaments

2026-07-27T21:23:46Z

Fredg999: Update links

[[User:Moremajorthanmajor/Map of Aviloid well temperaments]]

[[User:Moremajorthanmajor/Map of Carline well temperaments]]

Minthma

2026-07-27T21:18:02Z

Fredg999: Disambiguation template

The word '''minthma''' may refer to:
* Major minthma (352/351, 4.93¢) – see [[352/351]].
* Minor minthma (364/363, 4.76¢) – see [[364/363]].

{{Disambiguation}}

Interval space

2026-07-27T21:04:11Z

Fredg999: Redirected page to Monzos and interval space

#REDIRECT [[Monzos and interval space]]

Combination product set

2026-07-27T20:53:32Z

Fredg999: Separate external links

{{interwiki
| de =
| en = Combination product set
| es =
| ja = 組み合わせ積集合
}}
[[File:Wilson CPS names.png|400px|thumb|right|The names of all combination product sets up to 6 elements.]]

A '''combination product set''' ('''CPS''') is a [[scale|scale]] generated by the following means:

# A set ''S'' of ''n'' positive real numbers is the starting point.
# All the combinations of ''k'' elements of the set are obtained, and their products taken.
# These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitrary; if a canonical choice is required, the smallest element could be used).
# The resulting elements are [[Octave reduction|octave-reduced]] and sorted in ascending order, resulting in an octave period of a [[Periodic_scale|periodic scale]] (the usual sort of scale, in other words) which we may call CPS(''S'', ''k'').

This is sometimes called a ''k'')''n'' CPS, where the ''n'' denotes the size of the set ''S''. There are special names for special cases: a 2)4 CPS is called a [[Hexany|hexany]]; both 2)5 and 3)5 CPS are called [[Dekany|dekanies]]; both 2)6 and 4)6 CPS are called [[Pentadekany|pentadekanies]], a 3)6 CPS is called an [[Eikosany|eikosany]], etc. These are normally considered in connection with just intonation, so that the starting set is a set of positive rational numbers, but nothing prevents consideration of the more general case.

The idea can be further generalized so that the thing we start from is not a set but a [https://en.wikipedia.org/wiki/Multiset multiset]. A multiset is like a set, but the elements have multiplicities; there can be two, three or any number of a given kind of element. Submultisets of a multiset are also multisets, and the product over a multiset takes account of multiplicity. Hence the 2)4 hexany resulting from the multiset [1,1,3,5] first generates the two-element submultisets, which are [1, 1], [1, 3], [1, 5], [3, 5], and we obtain products 1, 3, 5, and 15. Reducing to an octave gives 5/4, 3/2, 15/8, 2 which can be transposed to 5/4, 4/3, 5/3, 2. In spite of being called a hexany it has only four notes.

CPS are closely related to [[Euler-Fokker genus|Euler genera]], since if we combine 0)''n'', 1)''n'', 2)''n'' ... ''n'')''n'' before reducing to an octave, and then reduce, we get an Euler genus.

CPS were invented by [[Erv Wilson]].

== See also ==
* [[Gallery of combination product sets]]

== External links ==
* [http://anaphoria.com/wilsoncps.html Wilson Archives - Combination Product Sets - CPS]

[[Category:Combination product sets| ]] 
[[Category:Erv Wilson]]

Pentatriandekany

2026-07-27T20:52:35Z

Fredg999: Regular redirect

#REDIRECT [[Combination product set]]

Enaeikosany

2026-07-27T20:51:58Z

Fredg999: Regular redirect

#REDIRECT [[Combination product set]]

User talk:Yourmusic Productions

2026-07-27T20:51:24Z

Fredg999: /* CPS vocabulary */ Re

== Linking wiki pages is simple ==

Hi Yourmusic Productions, 
I saw you adding links to other wiki pages the hard way (<code>[URL title]</code>). The most important birth helper of the wiki idea was the ability to link notes easily and quickly. You can link to another page in the wiki by putting its name in double square brackets (<code style="white-space:nowrap">[[page name]]</code>). The easy way has also the benefit to show by its color if the destination page already exists or not (you saw this when looking at [[just fifth]]) See [[Help:Editing]] for more information. '''BTW: Welcome to the wiki community!''' :-) 
In case you are wondering how to respond to messages, just write in the line below (with on or more <code>:</code> at the beginning of the line for indentation) and maybe sign your message with <code><nowiki>--~~~~</nowiki></code> (this will be auto-translated into user name plus timestamp). 
--[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 19:51, 31 May 2020 (UTC)

== the page you created ==

Hi Yourmusic Productions, 
Thanks for sharing your experiences. I read the introduction of ''[[Tuning various edos by ear]]'' as to see if the categories I was up to add fit. Maybe you'll find better categories or some more to add. It's currently only linked from you user page (short reminder: this can be done by placing the page title in double square brackets), hopefully there are other pages that can be linked to it. 
Best regards --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 05:56, 22 July 2020 (UTC)

: Thanks. Yeah, it probably still needs a bit of rewriting to fill in all the metadata and put it into objective language. But better out of my head and in a form where other people can contribute than staying in the drafts forever.

:: Please remember to change the internal link syntax, the [https://en.xen.wiki/w/Special:WhatLinksHere/Tuning_various_edos_by_ear WhatLinksHere] function automatically finds links using the internal syntax (double square brackets), but not the external syntax (single square brackets). --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 09:27, 22 July 2020 (UTC)

== Harmonize categories of interval pages ==

Hi Yourmusic Productions, we are interested in your opinion about [[Xenharmonic Wiki:Things to do #Categories of interval pages|Categories of interval pages]], thanks in advance for taking the time. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 21:21, 8 November 2020 (UTC)

== Ups and downs notation tables on the EDO pages ==

Hi Yourmusic Productions, we seem to have dueling edits. I'm the inventor of ups and downs and I created most of the ups and downs columns in the intervals pages. We seem to have a difference of opinion about the importance of minimizing the number of ups and downs needed. In 41-edo, one could notate 13\41 (~5/4) as either vM3 or d4. The latter minimizes that number, but IMO is inferior. I'd rather see C vE G than C Fb G. I'd rather call the chord a downmajor chord than a sus-flat4 chord. Likewise for 22-edo's very similar-sounding 5/4, C vE G not C D# G, and C downmajor not C sus-sharp2. This is an important advantage of ups and downs: you never have to spell an interval in the 300-400¢ range as a 2nd or 4th. And in general the notes column needn't zigzag madly back and forth among the 7 letters, but can for the most part march steadily C D E F G A B C.

There is of course the atonal approach where there is no key, there are no clear chords, there's just various combinations of notes. In that case, chord spelling isn't so important. But even then, one wouldn't want to zigzag too much.

Perhaps there is a misconception about the notes column in the table, as opposed to the interval names columns, which I believe we agree about. The notes column is meant to show how the notes would be named in the key of D. Because the note names vary depending on the key, and 22-edo's vE would indeed be D#, if you're in the key of E. But it's too much to write out all the possible names for each note. So the note names should match the interval names. In 34-edo, 10\34 is a mid 3rd not a dim 4th, so in D, the note isn't Gb but ^^F. (BTW I chose the key of D because it's symmetrical, and therefore easier to proofread.)

Of course, we microtonalists often disagree, and you may prefer to see ups and downs used differently. But as the creator of the system, I feel I have the right to make the note columns as I see fit. So rather than editing what I've written, please consider other options. The first thing to try is discussing your opinions directly with me and see if one of us can change the other's mind. The discussion tab of the ups and downs page might be a good place for that. Another is to create your own xenwiki page about your preferred ups and downs style, and link to it from the ups and downs page. Or maybe add a paragraph to certain edo pages. The beauty of the xenwiki is that it allows room for each of us to advocate for different approaches.

BTW, looking over the note columns, I see I haven't been totally consistent in choosing when to include enharmonic equivalents. I'll be updating those shortly.

Anyway, looking forward to hearing from you! :) --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 00:49, 16 December 2020 (UTC)

- Yeah, it seems that this is particularly a problem with superpythagorean systems, where if you follow the math strictly, it frequently looks like you're going the wrong way on staff notation. Guitar tab is so much less of a headache in that respect. Regarding ups and downs in general, I do think it might be helpful to distinguish between movements in systems where every interval is reachable by chains of 5ths, so you could just use sharps and flats if you want, and the ups & downs just give you different ways to notate the same interval, and edos that are contorted in the 3-limit, so a different type of notation becomes essential to move between chains. --[[User:Yourmusic Productions|Yourmusic Productions]] ([[User talk:Yourmusic Productions|talk]]) 08:57, 16 December 2020 (UTC)

: Good points. Very true about C# being above Db in those edos. About distinguishing between contorted vs. non-contorted use: in practice, any score that uses ups and downs must define them at the top of the page, e.g. "^1 = 1\15" or "^1 = 1\15 = 80¢" for 15-edo. That sort of clues you in to that. One could also write something more explicit there, e.g. "fifth = 9\15 = 720¢" or even "fifth = 3\5 = 720¢". --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 22:56, 16 December 2020 (UTC)

== Reduce comma tables on EDO pages ==

Please have a look at [[Xenharmonic Wiki: Things to do #Comma tables in EDO_pages]]. Thanks --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 09:09, 11 January 2021 (UTC)

== More pages to give yourself credit for ==

Should give yourself credit for the various Lumatone mapping pages you created. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 20:30, 1 January 2025 (UTC)

== Thanks for the temperament tip for 20edo, but . . . . ==

. . . Probably best to put such things on a user's Talk page instead of inserting them into the actual pages. (I had to go to work, so I didn't have time to try to look up the right temperament, so I had to leave it with a placeholder. Anyway, no harm done, but keep it in mind for the future. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 05:38, 12 August 2025 (UTC)

== Thank you for the hexany pages ==
Thank you for taking the time to write pages about so many common hexanies. The wiki has been needing more discussion about specific combination product sets for a while so it's really great to see someone making them. Thank you :) Keep it up :)
--[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 05:11, 3 October 2025 (UTC)
: It seemed like the next obvious thing to do after the lumatone mapping pages hit saturation. Another thing that I can do in little chunks over several years until all the simple combination product sets have been filled in and the remaining ones are above the point of practical usability. I just hope it inspires more people to actually make music using them as well. [[User:Yourmusic Productions|Yourmusic Productions]] ([[User talk:Yourmusic Productions|talk]]) 13:34, 3 October 2025 (UTC)

== CPS vocabulary ==

Hi there, I was wondering what were your sources for vocabulary terms of larger [[CPS]] such as [[Pentatriandekany]] and [[Enaeikosany]]. It might be useful to lay the vocabulary in a table or something alike to make it more obvious, because they quickly become complicated. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 20:14, 27 July 2026 (UTC)
:It's just basic conjugation of the greek number system, like the smaller ones. https://www.theonlinegreektutor.com/counting-in-greek/ [[User:Yourmusic Productions|Yourmusic Productions]] ([[User talk:Yourmusic Productions|talk]]) 20:20, 27 July 2026 (UTC)
::Thank you, I wasn't certain if Erv Wilson used these terms himself, but that makes sense anyway. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 20:51, 27 July 2026 (UTC)

Erv Wilson's Linear Notations

2026-07-27T20:37:42Z

Fredg999: Linking

The linear notations from [[Erv Wilson]] is a system of [[notation]] created to fit [https://en.xen.wiki/w/Tour_of_Regular_Temperaments linear temperaments] & [[equal-step tuning|equal scales]] on a [[wikipedia:Robert_Holford_Macdowall_Bosanquet|Bosanquet keyboard]]. They were first mentioned in the [http://anaphoria.com/xen2.pdf second volume] of the ''[[Xenharmonikôn]]'' in 1974, but discussed more in-depth in the [http://anaphoria.com/xen3a.pdf third volume], published in the spring of 1975. The system describes notations to be used for [[EDO]]s with different versions depending on the size of the [[perfect fifth]].

== ''A Doorway to Dialog'' ==
The earliest paper first mentions that the keyboard in question seems to imply linear temperaments over equal scales, due to the issues with fingering. Because of this, he gives three reasons as to why to equalize a scale:
# In equal scales, there is less math required.
# It simplifies the actual process of tuning [[instruments]].
# Some equal scales have [[generator]]s close to the original temperament, so a case could be made that it would make sense to use that instead of an unequal scale.
[[File:Eikosany Keyboard.png|left|thumb|Picture of the Eikosany keyboard, pg. 7|371x371px]]
Wilson makes an example of a scale that could be made into a linear temperament by presenting his [[Eikosany]] scale on a keyboard inspired by one for 41 notes. With the layout he chose, D'''\''' and β<big>ł</big> fall onto a homogeneous position from C (two keys down for D\, two keys up for β<big>ł</big>). Another detail shows that the 3, 9, & 3*9 keys are two [[schisma]]s off from the 5*7*11, 3*5*7*11, and 5*7*9*11 keys. With these two oddities, he suggests that a [[41edo|41EDO]] keyboard would be proper in hosting the Eikosany.

Wilson says that he's "''emphatically biased towards the positive systems''", where the fifth is greater than Pythagorean. He mentions that unlike systems like [[meantone]], where the fifth is damaged to make the third pure, a positive system would be the first time in Western history where a tuning had pure thirds and pure fifths. If the third was damaged, it would only be to help turn the [[7/4|harmonic seventh]] and [[11/8|eleventh]] pure, which Wilson calls "''far lesser apostles''". He also brings up that [[wikipedia:Raga|the ragas of India]] could be hosted in these positive systems as well.

On the next page, he gives a section of his thoughts on the original Bosanquet layout, noting things like key shape and the width of intervals. In the current & future papers, he suggests & uses smaller hexagonal keys, saying that it eliminates dead space & suits more scales, like meantone & [[just intonation]]. Along with the Eikosany keyboard, he demonstrates two other keyboards with this variation of the layout, those being the [[wikipedia:Shruti_(music)|22 ''shruitis'' of India]] and a traditional [[Arabic, Turkish, Persian|Arabic]] system of 17 notes.[[File:A Handy Guide For The Notation Of 12-, 22-, 31-, And 41-Tone Systems.png|thumb|366x366px|An example of Wilson's notation system]]On the last two pages of the document, Wilson makes mention of a system of [[notation]] he issued in 1965. In these pages, he makes two categories based on how many steps C is from C#, and how many steps B# is from C. If C-C# is one step, its singular, two steps is binary, three steps is ternary, etc etc; if B# is less than C, its negative, neutral if they're the same, positive if B# is greater by one step, 2bly positive if B# is greater by two steps, etc etc. He also gives examples for accidentals up to the quaternary system; by this point, he hasn't made accidentals for a quinary system.
{| class="wikitable"
|etc
|3bly Positive
|2bly Positive
|Positive
|Neutral
|Negative
|
|-
|
|
|
|[[5edo|5]]
|[[12edo|12]]
|[[19edo|19]]
|Singulary
|-
|
|
|[[10edo|10]]
|[[17edo|17]]
|[[24edo|24]]
|[[31edo|31]]
|Binary
|-
|
|[[15edo|15]]
|[[22edo|22]]
|[[29edo|29]]
|[[36edo|36]]
|[[43edo|43]]
|Ternary
|-
|
|[[27edo|27]]
|[[34edo|34]]
|[[41edo|41]]
|[[48edo|48]]
|[[55edo|55]]
|Quaternary
|-
|
|[[39edo|39]]
|[[46edo|46]]
|[[53edo|53]]
|[[60edo|60]]
|[[67edo|67]]
|Quinary
|-
|etc
|
|
|
|
|
|etc
|}

== ''On Linear Notations and The Bosanquet Keyboard'' ==
In this paper, Wilson elaborates more on the notation he first brought up in 1974. First, he brings up three families of fifths and their direction on the Bosanquet keyboard:
* (Vertical) doudecimally ''positive'' (>7\[[12edo|12]]): [[17edo|17]], [[29edo|29]], [[41edo|41]], [[53edo|53]], etc | ''negative'' (<7\12): [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], etc
* (Right-leaning) septimally ''positive'' (>4\[[7edo|7]]): 12, 19, [[26edo|26]], 31, etc | ''negative'' (<4\7): [[9edo|9]], [[16edo|16]], [[23edo|23]], [[25edo|25]], etc
* (Left-leaning) quintally ''positive'' (>3\[[5edo|5]]): [[8edo|8]], [[13edo|13]], [[18edo|18]], [[23edo|23]], etc | ''negative'' (<3\5): 7, 12, 17, 22, etc
Although the positive & negative notations of each family are part of the same system, they require different treatments in order to stay melodically consistent. Thus, there are different nominal systems for each kind of system. He gives four examples:
[[File:Quintally positive.png|thumb|407x407px|Quintally positive]][[File:Positive Septimal Notation.png|none|thumb|347x347px|Septimally positive]]
[[File:Septimally negative.png|thumb|351x351px|Septimally negative|left]][[File:Doudecimally positive notation.png|thumb|424x424px|Duodecimally positive]]
Wilson mentions that its possible to use a traditional staff with seven nominals for duodecimally negative systems due to tradition, though he mentions that it would be better to use the duodecimal system for more complex works.

[[File:Erv Wilson's Linear Accidentals.png|left|222x222px|The linear accidentals|thumb]]For accidentals, he suggests two pairs; one for positive systems, one for negative. Since there is three systems mentioned here, there are twelve total accidentals.[[File:A System of Fluctuating Nominal Systems.png|thumb|A graph to show the evolution of nominals in different systems|242x242px]]

Wilson made the accidentals mirror each other so that there would be no confusion in the sheet music, assuming one won't use positive & negative accidentals in the same piece. At this time, Wilson hadn't explored novenal (5\[[9edo|9]]) or tridecimal (8\[[13edo|13]]) accidentals, saying "''I need to experiment more with the septimally negative and quintally positive systems before expressing a view on this.''"

For the last few pages of the paper, he demonstrates six keyboards to demonstrate each unique system mentioned in the paper. Along with this, he also includes the notation for each system. In order of appearance, the EDOs used for the notation section are {{EDOs|41, 31, 23, 26, and 22}} EDO.

== Updates After Wilson ==
In the document for Wilson's ''On Linear Notations and The Bosanquet Keyboard'' from the [http://anaphoria.com/wilson.html Wilson Archives], [[Kraig Grady]] has been working on an appendix from [[Praveen Venkataramana]], showing not only two more systems of fifths (octal [5/[[8edo|8]]] & novenal) but also demonstrating the various ways you can notate EDOs with these systems up to [[72edo|72EDO]]. At the time of writing, there appears to be work showing that this can be applied to [[MOS|MOS scales]] and constant structures, though Grady has not added these yet.

{{Navbox notation}}

[[Category:Notation]]
[[Category:Erv Wilson]]

Just intonation

2026-07-27T20:36:44Z

Fredg999: Linking

{{interwiki
| en = Just intonation
| de = Reine Stimmung
| es = Entonación justa
| ja = 純正律
| ko = 순정률
| ro = Intervale raționale
}}
{{Wikipedia}}

'''Just intonation''' ('''JI''') is an approach to [[musical tuning]] where all [[interval]]s between two notes have [[frequency ratio]]s which are {{W|rational number}}s. For example, a perfect fifth in just intonation can have frequency ratio [[3/2]], a major third [[5/4]], a minor third [[6/5]], and so on. Just intonation is based off of the [[harmonic series]], which is the collection of tones found at integer multiples of a fundamental frequency, and is the set of [[overtone]]s of a note played on a string or pipe instrument. All just intervals can be found as the interval between two notes in the harmonic series; for example, [[5/3]] is the interval between the [[5/1|5th harmonic]] and the [[3/1|3rd harmonic]]. Just intervals with frequency ratios of small numbers, called [[low-complexity just intonation]] (LCJI) intervals, tend to be the most [[consonant]] in the sense that their sounds meld together.

In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit|5-limit tuning]], where the numerator and denominator of any ratio used has no prime factors greater than 5. ''Extended just intonation'', a term coined by [[Ben Johnston]], refers to any tuning in the harmonic series regardless of [[prime limit]].<ref>From Ben Johnston "A Notation System for Extended Just Intonation." ''Maximum Clarity'', 2006, p. 77</ref> In current usage, just intonation typically refers to extended just intonation. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].

Just intonation contrasts with [[equal temperament]]s in that equal temperaments include intervals with {{W|Irrational number|irrational}} frequency ratios, which are not intervals of just intonation. For example, [[12edo|12-tone equal temperament]] has a frequency ratio of 21/12, which is an irrational number, as a corollary of the {{W|rational root theorem}}. In fact, the only intervals in 12et which are also intervals of just intonation are multiples of the [[octave]], with a frequency ratio of 2/1. Equal temperaments are often used to approximate just intonation; for example, 12et approximates the perfect fifth [[3/2]], which is 702{{Cent}} in size, with the 7-step interval of 700{{c}}, only 2{{c}} flat. The major third with frequency ratio [[5/4]], which is 386{{c}} in size, is approximated by the 4-step interval of 400{{c}}, at 14{{c}} sharp. The ability of 12et to approximate simple ratios of just intonation is one of the reasons it became popular in the 20th century. Other equal temperaments, such as [[19edo|19et]], [[22edo|22et]], and [[31edo|31et]], also approximate various intervals of just intonation accurately, including higher-limit intervals not approximated well by 12et.

The structure of just intonation has several implications on music composition. Sequences of intervals that arrive back to the root in equal temperament may not do so in just intonation, and instead reach an interval a [[comma]] above or below the root. For example, going up four perfect fifths, and down a major third and two octaves, arrives back to the root in 12et {{Nowrap| (4 × 700{{c}} – 400{{c}} – 2 × 1200{{c}} {{=}} 0{{c}}) }}, but does not do so in just intonation {{Nowrap| ((3/2)4 ÷ (5/4) ÷ (2/1)2 {{=}} [[81/80]] ≠ 1/1) }}. The note reached is instead 81/80 (about 22{{c}}) above the root, rather than being equal to it. The 81/80 comma is known as the ''syntonic comma'', and occurs frequently in 5-limit just intonation. Modifying a simple ratio by a comma often produces a [[wolf interval]]; for example, 3/2 minus a syntonic comma is (3/2) ÷ (81/80) = [[40/27]], which is significantly less consonant than 3/2. Certain chord progressions may also become [[Comma pump|comma pumps]], which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as tools to use or as problems to be solved. Examples of approaches that try to solve these problems include pitch shifts, [[adaptive just intonation]] and [[temperament]]s.

== Consonance ==
[[File:Major triad 12et saw32.mp3|thumb|A major triad in 12-tone equal temperament.]]
[[File:Major triad ji saw32.mp3|thumb|The same major triad in 5-limit just intonation.]]
LCJI intervals achieve consonance through alignment of [[Partial|partials]] if the interval has [[Harmonic timbre|harmonic timbre]]. In fact, alignment of partials is a stronger effect with harmonic timbre: if partials align at frequency n, they will also align at every multiple of n; and in addition, two notes whose partials align with the same root note will also have partials aligning with each other. This allows for the construction of just-intonation chords of more than two notes where every comprising interval is a consonance.

Low-complexity JI intervals and chords also achieve consonance by being the ratios between harmonics of a (possibly unplayed) fundamental even if they do not have harmonic timbre.

Similar logic may be used for instruments with timbres not aligning with the harmonic series; see [[timbral tuning]].

== Ways of using JI ==
Here are multiple ways in which musicians and theorists have used just intonation.

; [[Free style JI]]
[[Lou Harrison]] used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music.

; Harmonic limits and subgroups
[[Harmonic limit]]s, also known as ''prime limits'', set a limit for the highest prime number in the factorization of any ratio used; for example, western music is based off the [[5-limit]]. Lower limits tend to be more familiar and consonant, while higher limits contain more exotic harmony. [[Subgroup]]s name a list of allowable prime numbers used. For example, the [[2.3.7 subgroup|2.3.7-subgroup]] consists of all intervals with only primes 2, 3, and 7 in the numerator and denominator. (A harmonic limit is also a type of subgroup, though they are less commonly stated as such.) Different subgroups each contain their own unique structures, including commas, temperaments, [[scale form]]s, etc.

; Restrictions on the denominator or numerator
Some approaches restrict "the denominator to one or very few values"<ref name=":0">From Jacques Dudon, "Differential Coherence", ''1/1'' vol. 11, no. 2: p.1).</ref> (the [[harmonic series]], [[isoharmonic chord]]s, [[AFDO]]s/[[overtone scale]]s, [[primodality]], [[Ringer scale|ringer scales]]), the "numerator to one or very few values" (the [[subharmonic series]], [[IFDO]]s/undertone scales), or both ([[Tonality diamond|tonality diamonds]]).

; Mediants
The use of harmonic and arithmetic [[mediant (operation)|mediants]] as was common with the [[ancient Greek music|Ancient Greeks]]. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.{{citation needed}}

; Approximations/alterations of tempered tunings
These are [[Detempering|detemperings]], including [[NEJI]] systems.

; Other approaches
Other approaches include [http://anaphoria.com/wilsonintroMERU.html Meru scales], [[tritriadic scale]]s, and [[combination product set]]s.

==Approximating JI with temperaments==
There are a lot of JI intervals, and it's difficult to keep track of all of them. As such, people often use simpler systems to approximate JI intervals, known as [[temperament]]s. A popular choice is [[equal temperament]]s; for example the predominant [[12edo|12et]], which is widely used to approximate [[5-limit]] JI. Other equal temperaments exist, for example [[19edo|19et]], [[22edo|22et]], and [[31edo|31et]]. Besides equal temperaments, other temperaments exist, such as [[regular temperament]]s and [[well temperament]]s.

Temperaments also create new structures not found in JI; for example, [[meantone]] temperament (which 12et [[support]]s) tempers out the syntonic comma [[81/80]], making [[5/4]] the same as the major third obtained by stacking four fifths, [[81/64]]; this structural feature is often assumed without thinking in western music. There are many temperaments in various limits, including the 5-limit, the [[7-limit]], and beyond, as well as many subgroups.

[[Microtemperaments]] are very accurate and temper out [[unnoticeable comma]]s, and add minimal damage to the intervals of JI, while also simplifying its structure. For example, [[schismic]] flattens the fifth by around 0.2 cents to equate the [[Pythagorean comma]] with the syntonic comma. Microtemperaments are also very useful in higher prime limits, where JI itself becomes too complex.

==Instruments==
{{todo|expand|comment=Expand the instruments section with more examples}}
*The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range.
*Most of [[Harry Partch]]'s instruments were designed to be for just intonation.
==Music==
{{Main|Music in just intonation}}

== Notation ==
There are various [[Musical notation|notation systems]] for just intonation, for example [[Helmholtz-Ellis notation]] and the [[Functional Just System]].
{{Todo|expand|inline=1}}

==See also==
{{todo|cleanup|inline=1}}
*[[List of approaches to musical tuning]]
*[[Gallery of just intervals]]
*[[Families of scales]]
*[[:Category:Just intonation]]

==References==
<references />

==Further reading==
*[http://www.tonalsoft.com/enc/j/just.aspx Just intonation] on the [[Tonalsoft Encyclopedia]]
*[http://nowitzky.hostwebs.com/justint/ Just Intonation] by Mark Nowitzky
*[http://www.kylegann.com/tuning.html Just Intonation Explained] by Kyle Gann
*[http://www.kylegann.com/Octave.html Anatomy of an Octave] by Kyle Gann
*[http://www.dbdoty.com/Words/What-is-Just-Intonation.html What is Just Intonation?] by David B. Doty
*[http://lumma.org/tuning/faq/#whatisJI What is "just intonation"?] by Carl Lumma
*[http://www.dbdoty.com/Words/werntz.html A Response to Julia Werntz] by David B. Doty
*[http://lumma.org/tuning/gws/commaseq.htm Comma Sequences] by Gene Ward Smith
*[https://casfaculty.case.edu/ross-duffin/just-intonation-in-renaissance-theory-practice/ Just Intonation in Renaissance Theory & Practice] by Ross W. Duffin

List of xenharmonic fixed pitch instruments

2026-07-27T20:36:03Z

Fredg999: Text replacement - "[[Combination_product_sets|" to "[[Combination product set|"

''See also: [[Instruments]]''

The Great Xenharmonic Fixed-Pitch Instrument Database!

Fill in this page with a list of every known xenharmonic instrument in existence today, to exist as a standard of compatibility for builders to voluntarily adopt when building new fixed-pitch instruments.

{| class="wikitable sortable"
|-
! Instrument(s)
! Tuning system
! Location
! Reference note name
! Reference pitch (Hz)
|-
| [http://www.afn.org/%7Esejic/DGinstr.html The Exotic Musical Instruments of Denny Genovese] (fixed include bass marimba, 2 xylophones, tubular bells, tubular chimes, fipple pipes)
| [[JI]]
| Gainesville, FL and Urbana, IL
| C
| 64
|-
| [http://www.newband.org/instruments.htm The NEWBAND Instrumentarium] (instruments by [[Harry Partch]], [[Dean Drummond]], et al.)
| [[JI]]
| Montclair, NJ
| G
| 392
|-
| [http://musikfabrik.eu/en/projects/stage-works/delusion-of-the-fury.html Ensemble musikFabrik] (replicas of all instruments by Harry Partch)
| [[JI]]
| Cologne, Germany
| ?
| ?
|-
| [http://anaphoria.com/musinst.html Kraig Grady's Ensembles (Vibraphones, Marimbas, Reed Organs, Hammer Dulcimer, Bowed Psaltery, Bass Meru Bars]
| [[Combination product set|cps, recurrent sequences]]
| various Anaphorian Embassies
| A
| 440
|-
| [http://www.huygens-fokker.org/instruments/fokkerorgan.html Fokker Organ]
| [[31edo]]
| Netherlands
| A
| 443 ([[The_Great_Xenharmonic_Fixed-Pitch_Instrument_Database|dependent on temperature]]!)
|-
| [http://www.ekmelic-music.org/en/instr/eog.htm Franz Richter Herf's Ekmelic Organ]{{Dead link}}
| [[72edo]]
| Salzburg (?), Austria
| ?
| ?
|-
| [http://eufonia.de/index.php/en/microtonal-composition/157-the-enharmonic-pipe-organ Martin Vogel's enharmonic pipe organ]{{Dead link}}
| [[JI]]
| Vienna, Austria
| ?
| ?
|-
| [https://sites.google.com/site/iboortgies/tableoforganswithsubsemitones Historical church organs with split semitones]
| Various
| various
| ?
| ?
|-
| 3-octave set of tubes
| [[17edo]]
| Urbana, IL
| A
| 440
|-
| 20-tone set of tubes
| [[88cET]]
| Urbana, IL
| B
| 240
|-
| [http://www.gayleyoung.net/Instruments_columbine.html Gayle Young's Columbine]
| [[JI]]
| Canada?
| ?
| ?
|-
| [http://www.sfoxclarinets.com/bpclar.html Bohlen–Pierce Clarinets by Steve Fox]
| [[Bohlen–Pierce]]
| various (U.S., Canada, Germany)
| A
| 440
|-
| [http://www.microtonaltrumpet.com/ 19edo trumpet (Stephen Altoft, Donald Bousted)]
| [[19edo]]
| ?
| ?
| ?
|-
| [[Keenan Pepper]]'s porcupine marimba
| Porcupine-based circulating 15-tone
| Berkeley, CA
| C
| 261.626
|-
| [http://www.ekmelic-music.org/en/instr/kk.htm Johannes Kotschy's Sound Cube]
| [[JI]]
| Salzburg (?), Austria
| ?
| ?
|-
| [[Conic_Bellophone_in_96edo|Conic bellophone]]
| [[96edo]]
| ?
| ?
| ?
|-
| [http://www.sauter-pianos.de/english/pianos/microtone.html Sauter 1/16 tone piano]
| [[96edo]]
| various
| ?
| ?
|-
| [[:purdal:Icositría|Tútim Dennsuul Instruments and Schemes]]
| [[23edo]]
| Arica, "Chile"
| '6#'
| '''409.6'''
|-
| [[Richie Greene]]'s [https://richiegreene.com/instruments/ metallophone]
| [[41edo]]
| Portland, OR
| C
| 261.626
|}

[[Category:Instruments]]
[[Category:Lists]]