Musical material is a set of musical symbols (and their meanings). Music system is a relation on the set.
We distinguish:
Order of the set is the cardinality of the set of representatives, k=#(F).
The order expresses in case of tone systems the number of different pitches in an octave, in case of rhythmic systems the number of elementary (jointiming) tics in tact.
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Tone systems with order r>12 are called microtonal systems.
PlatónPlatón, 427347 př.n.l., Greek philosopher, things are the "shadows" of ideas. Realized that semitone arose secondarily  as a product of division. Supported solely Doric and Phrygian mode. 
Tone system is determined by the relationship between the used tones. Discrete system consists of a set of tones on certain selected pitches.
Discreet systems, in which pitches are arranged into an arithmetic sequence, are called regular.
In regular systems makes frequencies corresponding to pitches (according to WeberFechner's law) geometric sequence.
Real music sound is a simple sequence of physical phenomena. Physical relationships are observed as a continuous functions.
Human discreet conceptualization of phenomena (the total number of tones, dynamic marks, .., border consonance) is projected to the aesthetic plane. It appears appropriate to use combinatorics for creation of structures.
Physical => Structural => Aesthetic
Between two worlds and incorporates the third (structural, logical, ..) world and it appears as discrete:
Suitable logical world (music system, modality, ..) must cover with tolerable inaccuracy requirements of a physical system (frequency ratio ...):
Properties of chords (sonance, potential, ...) depends primarily on the positions in logic diagrams (structure modality ...), wherein the properties of logic diagrams result from the physical (acoustic and psychoacoustic) regularities. 
Vicentino Nicola , 15111572, ? Italian composer and music theorist ?. Promoted chromaticism, recommended to divide the octave into 12 equal parts. He designed special instrument (harpsichord) for observing of the slight differences of natural tuning and testing of Ancient enharmonics. 
We call musical interval the difference of two tone heights.
Unit interval corresponds to one mark of the regular system.
In periodic systems are intervals between the immediately
following tones the same.
An impurity  disrespect to system or modality.
The ratio of closest tones in the regular system is
The smallest possible subjective interval in the system we call the unit interval. In the 12tone system the unit interval is the interval of a semitone.
As unit of subjective interval was introduced 1 cent, ie 1% of halftone.
Therefore 12 h= ln r 100 [ cents ] / ln 2 For a comparison with other systems reference interval is preferable.
Also other units can be used for the measurement (milioctave = thousandth of octave, ...) The interval d is the distance between two tones measured in pieces (divisions) of the given system. E.g. interval between tones c1 and e2 (in 12tone system) is 16 pieces, ie. 16 semitones. Interval d corresponds to the frequency ratio r = ξ^{d} = 2 ^{d/k}
Loguin A.Loguin A. , , modern music works with 12tones, tried to do a systematics.. 
Let us call formal system such an idealization of music system where tones with frequencies in the ratio 2: 1 are considered to be the same. It is assumed that all the properties associated with the sound frequencies are invariant relative to the multiplication by two, i.e. they are not significantly altered by changing octaves.
For each tone of regular system exists its representative in formal system.
Praetorius MichaelPraetorius Michael , 15711621, German composer, organist and music theorist, author of Early Baroque spiritual concerts and musical encyclopedia. He considered about equal temperament. 
E.g. representative of all tones C,c,c_{1},c_{2},c_{3},... is C. Representative represents a class of equivalent tones, the number of classes (ie. cardinality of representatives F), is called the system order, k=#F.
The order is a measure of diversity of the system, the higher it is, the more richer, but also more subtle and complicated are relations of tones.
Regular formal system is of order k, when it divides an octave into k equal pieces.
The aim of introducing of formal system is to provide basic relations. Musicians often speak in terms of formal systems e.g. they mean by chord A the chord ac#e regardless octave location of tones. Obtaining experimental data for research of informal systems is difficult and falls  rather than into the realm of musical theory  into areas of acoustics and psychoacoustics.
Werckmeister AndreasWerckmeister Andreas [], , German organist, initiated the introduction of tempered tuning (writing of the year 1691). 
The most widely used system is currently regular twelvesystem (k = 12). Its representatives are usually written by symbols: {C, C#, D, D#, E, F, F#, G, G#, A, A#, B}, respectively {C, Db, D, Eb, E, F, Gb, G, Ab, A, Bb, B} or by numbers F={0,1,2..11}.
Tuning of instruments face a number of challenges, the first of them (impossibility of current tuning fifths and octaves) was already pointed out by Pythagoras.
Unit interval (a mark) of tempered tuned systems makes the ratio of frequencies: ξ = _{ } = 2^{1/k.} . In 12tone system is ξ=_{ }= 2^{1/12}=1.059.
Division of 2:1 ratio to twelve pieces is advantageous because it allows approximate ratio of 3: 2 of by seven and ratio of 5: 4 of by four pieces. It's in fact nearly exactly 2^{7/12} = 3/2 (1.498) and approximately also 2^{4/12} = 5/4 (1.260).
By transformation of tones into formal systems also all the relationships of these tones adjust.
Median of the system is number κ =k div 2, parity of the system number k mod 2. Median constitute the greatest possible distance in the formal system, parity of systems affects the symmetry of intervals.
Formal interval abstracts from octave positions of tones. Every two tones of a regular system in distance given by interval i = 2 ^{s } become identical.
E.g. interval c_{1}e_{1} with 4 marks (halftones) also represents intervals c_{1}e_{2} with 16 marks, c_{1}e_{3} with 28 marks and so on. Formal interval h_{f}=4 is representative of class of equivalent intervals of musical system h=4,16,28.. (relation h => h_{f} is homomorfism).
p–q <=> f_{p}/f_{q} =2^{c}, cεZ.
Set F contains k formal pitches v_{f}, F = { 0,..,k–1}. Pitch v can be seen as a pair v = [o ,h_{f},].
Formal interval D thus represents entire class of intervals d = D+k∙z. So: D = d mod k, where d is interval of the regular system, D formal interval.
Intervals of 12 tone system
d 
Interval 
Ratio 
Aprox. 
Fraction 
0 
Prima 
1.00000 
1.00000 
1/1 
1 
Minor second 
1.05946 
1.06667 
(16/15) 
2 
Major second 
1.12246 
1.12500 
(9/8) 
3 
Minor third 
1.18921 
1.20000 
6/5 
4 
Major third 
1.25992 
1.25000 
5/4 
5 
Quart 
1.33484 
1.33333 
4/3 
6 
Triton 
1.41421 
1.40000 
(7/5) 
7 
Quint 
1.49831 
1.50000 
3/2 
8 
Minor sext 
1.58740 
1.60000 
8/5 
9 
Major sext 
1.68179 
1.66667 
5/3 
10 
Minor septime 
1.78180 
1.77778 
(9/4) 
11 
Major septime 
1.88775 
1.87500 
(15/8) 
12 
Octave 
2.00000 
2.00000 
2/1 
The table lists all formal intervals of 12tone systems ordered by size d.
Frequency ratio r = (1.05946)^{d} = 2^{d/12} corresponds to equally tempered tuning. Approximation of numbers r by fraction follows. For the minor second is used also equivalent name halftone, for the major second name whole tone. Distance 12 corresponds to octave interval, median  with distance 6  is so called tritonus.
Values of fractions in parentheses are quite complex for human ears, they does not "resonate". Nearby tones interact (by beats,...). Interaction then also  due to the existence of the overtones (aliquots)  influences the effect of tones, which are not very close (eg tritonus). These phenomena are closely observed by Helmholtz theory.
For conversion between systems of different orders we will use so called reference interval: ∆V_{ij}*k = ∆v_{ij} .
Between two tones with pitches i,j is subjective interval h = j – i ,
objective interval r = ξ^{h} and reference interval s = h / k.
It holds: h ln 2 = k ln r, s ln 2 = ln r
Subjective interval vij is difference of pitches: ∆v_{ij} = v_{j}– v_{i} , v_{j} > v_{i}.
We can express the integer remainders in two ways:
1/ smallest natural residues 0<D<k, e.g.: D=0,1,2,3...11
2/ absolutely smallest residues 1–κ(k)<D< κ (k), e.g. D=–5,–4,..,4,5,6
In the first case formal intervals get values h_{f} = 0..k–1 v druhém
m_{f}= <– κ, + κ+1> (centered formal intervals).
System of order k = 7 is odd parity with median κ = 4 and by range for centered intervals <–3, +4>.
Ellis John , , divided octave to 1200 cents. 
Relative r, reference s=log_{2}r=logr/log2
Formal relative r_{f}=2s_{f} Formal reference s_{f}=frac(s)
Sharp relative r_{s} = 2s_{s} Sharp reference s_{s}=min(s_{f},1s_{f})
E.g. for r=3.2(=16/5) is s=log_{2}3.2=log3.2/log2=1.678,
s_{f}=frac(1.678)=0.678, r_{f}=2s_{f}=1.6 (8/5),
s_{s} =min(0.678,10.678)=0.322 r_{s}=2s_{s}=1.25(5/4)
Subjective h_{f} = h mod k, objective r_{f} = ξh_{f} and reference s_{f} = h_{f} / k.
So s_{f} = frac s ( if r 1 ) s_{f} = 1 – frac s. ( if r< 1 )
E.g. for r= 0.750, r= 1.500, r= 3.000 is gradualy s=–0.415, s= 0.585, s= 1.585; f=0.585.
Distance d=k.s. Formal distance d_{f}=d mod k = k.s_{f}
Sharp distance d_{s}=d_{f} (for d_{f}<=k/2), d_{s}=d_{f}k (for d_{f}>k/2),
d_{s}=k.s_{s}. Relative interval r = 2^{d/k}.
Pitch h  distance from a given reference point h0 (tone c)
In the regular system is convenient to choose a reference tone. The height of each tone is expressed as a function of its ordinal number, i.e the ordinal number relative to the reference tone. Converting of tone with ordinal number d in regular discrete system to equivalent with ordinal number t in corresponding formal regular discrete system of order k is caltulated by relation: delta = d mod k , where d N0, t = 0,1,2...k–1. Two tones with ordinal numbers i, j are spaced by (j – i) div k octaves and (j – i) mod k portions (marks).
Tuning of instruments refers to the tone a1, which has for (for k=12) pitch ha1=21. Tone a1 has from the start c the reference distance s= 21/12 = 1.75, tedy r=21.75 =3.3636.
Frequency of basis is for tuning of tone a1 at 435440 Hz, hence 435/r440/r = 129.3130.8 Hz.
Midi pitch and also loudness 0..127, Halftone interval r_{1}=2^{1/12}. Halftones in interval p=logr/log r_{1} pitch h_{m}=h48 bending (necessary for k<>12) 64+64∆_{p}, where ∆_{p}=frac(p)
With the development of music, known groupings are enriched by new tones. Nonessential tones are gradually turning into essential part of chords [Filip] and sonance of harmonic stream changes. From the originally firmly built structures (triads) music (after the introduction of turnovers, sequences, alterations, ..) got to use the whole combinatorics of shapes. At the same time  like chords  also modalities are changing and their certainty, a possibility of modulation ... Restrictions of baroque harmonic style to a few basic harmonic functions allowed  in classical music  to escalate harmonious action.
With increasing numbers of used structures also the number of possible harmonic connections increase. Variety of possible different levels of potentials for multitone chords but rather decreases [Faltin]. So impressionistic music lost on the dynamics (against the nature of classical compositions) .
Two fourtone chords within the seventone modalities have necessarily one common tone. Common tone bring the chords closer and reduces tension between them.
Therefore, there are attempts for further breakdown and modification, of which the the most striking are tryings to establish a microtonal systems.. It appears that the regular 12tone system will be exhausted and replaced by systems of a higher order.
Aristoxenos z Tarentu , 4.stol př.n.l, Greek music theorist. He was interested in music theory and in acoustic problems. His writings on ancient music makes the basis of science. He considered as harmonic interval in addition to 1/2 tone also 1/3, 1/4 and 1/8 of tone. 
Harmonic system with the order k > 12 are usually called microtonal systems. Their beginning can be found in the music of ancient Greece, where microintervals were used (" chroai "). Quartertone music was met with acclaim in the Arab countries, where it has parallels in folk music.
Descartes, RenéDescartes, René [dekart], 15961650, French philosopher and mathematician regarded as the founder of analytical geometry. He dealt also with physics, optics, meteorology and music theory. Music inspires affection, music theory helps to control these affects. 
There are also other systems in some cultural areas – from simple oriental tunes in 5 or 7tone scales to finely articulated passages of Arabic songs in a 24tone system. Such systems, which have their origins in human (irrational) feeling, will be called natural.
Order 
Interval d (2^{d/k}) 
Occurence 

2:1 
3:2 
5:4 
6:5 
7:4 

5 
5(2.000) 
3(1.516) 
 
 
4(1.741) 
Malaysia, Indonesia, China, Japan 
7 
7(2.000) 
4(1.486) 
 
2(1.219) 
 
Indonesia, Burma, Thailand, Siam, Cambodia,.. 
12 
12(2.000) 
7(1.498) 
4(1.260) 
3(1.189) 
 
Practically the whole world 
16 
16(2.000) 
9(1.477) 
5(1.242) 
4(1.189) 
13(1.756) 
Arabia 
17 
17(2.000) 
10(1.503) 
 
 
 
Arabia, Persia 
22 
22(2.000) 
13(1.506) 
7(1.247) 
6(1.208) 
18(1.763) 
India 
24 
24(2.000) 
14(1.498) 
8(1.260) 
6(1.189) 
 
Ancient Greece, Arabia, Persia 
Ratios of tone frequencies for selected intervals are calculated assuming that systems are regular. In fact, there are slight variations  eg observed in the research of 5and 7tone music instruments etc.
Given systems support the strongest continuity binding (fifth) 3:2 in the range 1.471.51.
(Precise tuning of fifths in 5tone and 7tone systems leads to modalities of 12tone systems, e.g. 2323(2) a 212221(2)).
For some systems is observed binding 5:4 (natural third) in the range 1.241.26.
In the 16tone and a 22tone system natural septime (7:4) appears.
Some systems are based on pure theoretical considerations  or at least to some extent. We'll talk about artificial systems. Often arise together with proposals for new musical instruments, or with proposals for a new tuning of existing instruments. System  arranged in a certain way. Hierarchy ...Derivation (microintervals, continued fractions ..). Several such systems are for illustration in the following table (complete list would require detailed studies from many sources).
Order 
Interval d (2^{d/k}) 
Designed by 

2:1 
3:2 
5:4 
6:5 
7:4 

19 
19(2.000) 
11(1.494) 
6(1.245) 
5(1.200) 
 
M.Mersenne, R.Descartes 
24 
24(2.000) 
14(1.498) 
8(1.260) 
6(1.189) 
 
I.Newton, A.Hába 
29 
29(2.000) 
17(1.501) 
9(1.240) 
8(1.211) 
 
I.Newton 
31 
31(2.000) 
18(1.496) 
10(1.251) 
8(1.196) 
25(1.749) 
L.Euler, C.Huyghens, A.D.Fokker 
36 
36(2.000) 
21(1.498) 
12(1.260) 
9(1.189) 
29(1.748) 
I.Newton, A.Hába 
41 
41(2.000) 
24(1.500) 
13(1.246) 
11(1.204) 
33(1.747) 
I.Newton 
48 
48(2.000) 
28(1.498) 
15(1.242) 
13(1.207) 
39(1.756) 

53 
53(2.000) 
31(1.500) 
17(1.249) 
14(1.201) 
43(1.755) 
I.Newton, J.Petzval 
The first possible suitable candidate is 19tone system. (Její zavedení by nevedlo vzhledem k překrývání znamének # a b v 12ti tone system  large changes in marking tones).
31tone system covers with great precision major third (5: 4) and the natural septime.
Busoni Ferruccio Benvenuto , 18661924, Italian composer, pianist, conductor, music theorist and esthetician. Promoted the usage of new scales and attempts to quartertone music. He suggested thirdtone intervals. 
The higher the order of system k is, the more diverse the system is, but also more difficult to manage.
Systems with order, which is a divisor of numbers 12 (i.e. 2,3,4,6) are only subsets of 12th tone systems. Systems with order, which is a multiple of numbers 12 ( 24,36,48,60,72..144…) are its supersets (comminution).
The new tone system does not necessarily involve the older system, it is sufficient as it can evoke similar energy contexts. E.g. maintaining of fifths is important for the transition between systems (5,3)=>(7,4) => (12,7) =>(19,11), lower systems remind modalities of higher systems.
The extent of possible comminution is limited by the pitch resolution of human hearing, we can expect also possible shift of the boundary. (like the shifting of boundary of consonance of chords…)
Microtonal systems anticipate music of the future. But so far none is too widespread  due to the limited possibilities of man (10 fingers ...) New world for microtonal systems opens up the advent of computers and electronic music. In the moment when we manage to penetrate into the deeper patterns of 12tone systems, we gets us opportunities to try to extrapolate the results  to apply regularities which we know from the 12tone system.
The existing failure of composers in mikrointerval systems does not prove that these systems are unusable in the future. The system of 12 tones, as we know it today has undergone thousands of years of evolution. In its course not only suitable chords but also 'effective' modalities were sought. If we can using computers identify 'effective modalities' also in microinterval systems, we come soon to a new and undiscovered possibilities of music.
Hába AloisHába Alois , 18931973, Czech composer and music theorist and promoter of microintervalic music. He designed and enforced changes of musical instruments for quartertones (harmonium, piano, clarinet, trumpet) and production of sixthtone harmonium. He started experimenting with twelfthtones and fifthtones. For microintervals introduced symbols. 
Besides the modalities from compositions in the 12tone system, of which the whole range was recognized and named, there are countless microtonal modalities about which we know very little. Greeks introduced quartertone inserts (called pykna) between tones of halftone intervals for affirmation of the melodic tones . Greek modality 118411(8) evolved from Doric 122212(2) i.e. 244424(4).
Modality of 22tone indian music 432434(2) (so called Svaragrama) reminds our natural modality (major scale). The same applies also for 985998(5) in 53tone system (5halftone, 8minor whole tone, 9 major whole tone) .
E.g. in 24tone Persian and Arabic music can be found modalities 352424(4) and 433433(4). Similar modalities (e.g.3343124(4),..) arise from modifications of 12ti tone systems (by halving of certain intervals) in Indian music (Šur and Afšāri). In 72tone ArabPersian music natural modality: 663663(6) is used ana further e.g. modalities 436654(6), 436636(6), 474647(4) (modalities are abbreviated by numbers 2 to 36tone system).
Keybords  efective struktures of modalities e.g. in 12 tone system: white keys.
Christian modes + 1 black key => harmonic major and minor black keys  the Chinese pentatonic
+ 1 white key => blues
Guitar  to place as much as possible major and minor chords into the range of 4 sleepers, minimalize number of pressings on strings; according to it  tuning of the strings.
Even after the introduction of tempered tuning problems not definitely disappeared. When tuning the piano  perfect fifths are tuned, inaccuracy (23 cents) is transferred to octaves.
Assuming fifths tuning, an extension of octaves is necessary, what also some guitarists make (by tuning of string basses lower).
In addition the human ear does not behave linearly in the edge zones (of frequencies and intensities)  at higher altitudes, some interval seems to be narrower.
By amplification of a higher (lower) tone we acquire a sense that it is higher (lower). But we hear short high tones (due to faster decay) as rather lower.
Tuning  of system  trying to tune all used tones (tempered, Pythagorean ...)  of modality  tuning of only one specific modality (natural tuning...)  of harmony  fine tuning of harmonic stream