What is the extended scale
A system of diatonic scales
›Scales‹, ›scales‹ or ›modes‹ are central constructs of »Tonsysteme [n]«, like these themselves, »on the one hand tone constituents, on the other hand complexes of tone relationships« , which are frequently referred to in composition and musical analysis becomes. But a comprehensive system for the order of scales is and remains a music-theoretical desideratum. The following essay presents a system of scales based on the diatonic heptatonic and the alteration possibilities of the harp, which as an ordering system of pitch material is largely independent of forms of tonal context or the historical development of tonal systems (such as the major-minor tonality or the modus System of vowel polyphony).
The pedal harp is assumed because it is characterized by the diatonic heptatonic system and its tuning in C flat major makes it possible to shorten each string by two semitones by means of pedals or to alter the corresponding tones upwards , so all twelve semitones to reach the octave. Referring to this has the advantage for the systematization of scales that the alterations can be thought of in one direction (from ›Ces major‹ upwards by two excessive primes) and thus two interval steps arise that can be used for classification . (Starting with ›C major‹, two alternating procedures would have to be thought of in different directions.) That is, for systematic reasons, a technical instrument construction circumstance is chosen as a reference for a music-theoretical concept , but the systematic to be developed is based on the established notation and the corresponding alteration operations.
After an introductory clarification of terms and the presentation of previous approaches to the order of scales, a system of diatonic scales is developed in four steps, then the limits of this system are shown and finally an outlook is given.
Scale, scale, mode
The terms “scale”, “scale” and “mode” are used differently in the literature, partly synonymously, partly to denote different facts. The following text uses the term ›scale‹, which should therefore be defined in more detail as a distinction to the two related terms.
»The scale, especially the major scale  applies [resp. was] as a paradigmatic model for the representation of the key and its tonal references […]. […] In the theory of the 18th century the key was explained and defined with the relationships in the scale «. Because the systematics to be presented is not supposed to be about a key-determining moment, but about the abstract classification of sound material, the ascending ›ladder‹ is taken up as an established and descriptive form of notation, but the term ›musical scale‹ is avoided.
The term ›mode‹ is currently perceived as a complex and controversial category; its historical and systematic significance is discussed intensively.  Markus Jans writes: "What mode is can neither be determined for unison nor for polyphony in general" , "but only according to time and style, genre and form" . The comprehensive article "Mode" in the second edition of the New Grove Dictionary gives the best overview of the historical change in meaning of the term.  Among other things, it is negotiated to what extent and in what contexts modes or mode systems were and are used as "theoretical [.], Prescriptive [.]" Or as "pragmatic [.], Descriptive [.]"  constructs. Interestingly, both the prescriptive and the descriptive character of modes can be found in Olivier Messiaen's concept of mode: Messiaen's well-known system of "modes with limited transposability"  has on the one hand a normative character, insofar as it is about the abstract setting of the sound material to be used prior to the composition process acts; on the other hand, in his Traité de rythme, de couleur et d’ornithologie, Messiaen presents these modes as a music-theoretical concept, describes their (color) properties and uses them descriptively by applying them analytically to the music of other composers.  In a terminological tradition that goes back to Messiaen, the term ›mode‹ could now be used for the project pursued here of a systematic, decidedly ahistorical order of pitch material, but the historically diverse and therefore dazzling differentiation of the term speaks against it. The aspects of the concept of mode that go beyond a pure concept of scale also speak against taking it up in the present context: “What essentially defines the concept of modality is the repetition of patterns. These may be a group of pitches, resonances (the recurrence of a particular frequency) or rhythmic motives that are organized hierarchically. « The development of structural patterns, for example by weighting individual tones (e.g. finalis, repercussa) or motivic repetition and variation is, according to this view, at the center of the term ›mode‹.
Now, when there is talk of an abstract compilation, order or classification of sound material, the mathematically shaped term set (“set”, “group”) is often used, based on the ›musical set theory‹ , and it would offer It is therefore advisable to use the term “quantity” or “pitch quantity” here as well. However, a set is a number of elements that are put together in a potentially different manner. For the systematisation pursued here, however, the ladder-like arrangement of diatonic material will be essential, which is why the use of the term “quantity” is dispensed with and the term “scale” is preferred instead.
The difficulty with the term ›scale‹ is that for him the meaning ›ladder, stairs‹ (lat. Scala) and not ›quantity‹ or ›material‹ is fundamental, and therefore there is a connotative tendency towards level thinking, linearity or interval thinking - as in the case of the term scale.  The etymology of the term makes it clear that ›scale‹ is to be viewed as a metaphorical-conceptual model which, like other metaphors, has a paradoxical quality, circularity and openness.  If the scale model is related to music, then the creative “conflict” inevitably arises as a “moment [.] Of metaphoricity” that surrounds every “transfer of a scheme”, every “migration of concepts” . The ›scale‹ model therefore has a blurring effect.  The general term ›scale‹ is understood today as a step-like arrangement, as a ladder-like form of representation  or as a measure (cf. ›temperature scale‹). Other, especially music-related aspects of understanding populate an intersection of the two terms “mode” and “scale”.
In the following, I understand scales in a pragmatic sense as "forms of elementary material disposition"  and use the term independently of the cultural location  and the historical, linear, harmonic, spatial, color, emotional, spectral or performance-related implications of what it describes Materials; in short: It is about diatonic tone sets, which are represented in the form of a scale.
The word 'diatonic' is derived from the Greek: diá literally means' through ', tónos' that which is used to tension something, rope, cord, belts, [...] string' or 'tension, emphasis, strength , […] Elevation of the voice, tone, […] color ” or also“ whole tone ”. Against this linguistic background , in ancient Greek music theory ›diatonic‹ initially denotes one of the three »melodic tone sexes«  alongside ›chromatic‹ and ›enharmonic‹; in the descending diatonic tetrachord, two whole tones are followed by a semitone (1-1-½).  Two disjoint diatonic tetrachords form an octave. In a broader sense, the term 'diatonic' denotes that form of the overall tonal system extending over a double octave, which is composed exclusively of (disjoint and conjunct) diatonic tetrachords.  A large part of the different aspects of the definition of 'diatonic', which are still used by various authors to this day, can be related to this ancient basis:  Diatonic in the sense of a pentatonic or heptatonic »tone stock , in the sense of an interval arrangement irregularly alternating whole-tone and semitone steps , in the sense of a system of tone relationships , in the sense of the basic scale, series of stem tones, non-alteration , as "the epitome of modes" . The stratification of pure fifths is usually used to explain the inner order of diatonic relationships. 
In the following, I use the term 'diatonic' to refer to those scales that are based on the key sequence ces-des-es-fes-ges-a-flat-b, so that their origin has a minimal internal fifth, consists of a maximum of seven tones and the semitone -Equal tuning can be transposed. I assume that the tones can be altered without losing their "level value" , but that tones can be confused enharmonically at the same time and, in the equivalence, intervals that contain enharmonically confused tones (e.g. de; d-fes) included, are interchangeable and that the octave identity applies. In the following I will use the term “transheptatonic scales” to denote scales that have more than seven tones.
In order to be self-determined, the term 'diatonic' must now be related to complementary terms such as 'chromatic'.  The basis for this enterprise is problematic insofar as the terms with different meanings in different contexts were set differently in relation and moreover were and are related to different things. The Greek term chroma (color) is used in music today to refer to tones (chromatic displacements of a diatonic tone), intervals (diatonic and chromatic intervals), chords (chromatic chords) or scales (chromatic scale). I am content here with a discussion of individual aspects of the chromatic scale. For more information on the topic of chromatics, I refer to relevant specialist publications. 
The decisive question for our context is now: Is there only one chromatic scale or several?  Mostly, ›the chromatic scale‹ is spoken of as a single scale with twelve equal semitone steps that has no fundamental and can therefore be viewed as a material scale , from which rows or other useful scales can be formed.  However, it would now also be possible to designate certain scale sections derived from this chromatic scale as chromatic scales. 'Diatonic' and 'chromatic' would then be qualifying terms that would describe a certain quantity such as heptatonic in more detail. Anatol Vieru determines the qualities of a scale with the help of a measure of the ›diatonic and chromatic capacity‹ (“diaton [ic] and […] chromatic capacity” ). He calls the measure »DIACRO«  and develops it as a fraction CRO / DIA: in the numerator: ›Chromaticity‹ (number of uninterrupted second rows), in the denominator: ›Diatonic‹ (number of uninterrupted fifths). Vieru is based on set theory, in a similar way to the American ›musical set theory‹, and relates it to sets of tones and intervals.
I want to stick to the usage of a single chromatic scale, keep the possibilities of the qualifying terms in mind and develop a classification that does not start with pitches or intervals, but with the accidents - just as they appear in the scalar arrangement of the harp and work in our notation. This makes it clear that the sound sources generated below are diatonic in terms of their origin and, in line with this perspective, can generally be referred to as ›diatonic scales‹.
Music lexicons such as The New Grove Dictionary, Music in Past and Present or the Lexicon of Systematic Musicology largely lack a systematization of scales. There are mainly descriptions of terms that contribute to the musical understanding of scales.  Approaches to an orderly representation of scales have the character of drafts, groupings or collections.
In the first edition of the music, Carl Dahlhaus breaks down various understandings of the term ›tone system‹ as a tone stock, mood, scheme of tone relationships or melody structures, points out confusions and inadequacies of terminology and explains the term ›tone system‹ with regard to it European music.  In a table depicting non-European tone systems, Marius Schneider shows, within the framework of the same article, list-like arranged scales and melody examples using the twelve-tone chromatic scale.  This ordered list of scales written as unsigned as possible, starting from the root tones, makes it clear that a wide variety of scales can be formed by selecting tones from the chromatic of the same level. However, Schneider does not elaborate on this systematization option, its advantages and difficulties.
Even in jazz, where scales are used in a very differentiated way, efforts are made to present scales in an orderly manner. For example, in the first volume of his jazz theory, Andreas Pilsenbeck shows how scales can be determined and used in relation to chords. He goes into the scales common in jazz improvisation, including "bebop scales"  and "blues scales". As "modes" of "harmonic minor (HM)", "melodic minor (MM)" and "harmonic major (HD)", Pilsenbeck describes the rotations of these scales beginning on the different scale tones, e.g. HM1 to HM7.  Then he depicts symmetrical scales such as the "Messiaen scales"  in order to finally go into "Exotic scales" . He emphasizes: »Every scale has a tonal characteristic. It is essentially determined by its interval structure (including tritone content) as well as its internal fifth width [...]. " And further:" The assignment of a scale to a chord clarifies the tonal meaning of all twelve tones in relation to the respective harmonic situation «. With this, Pilsenbeck provides useful instructions for practical jazz improvisation, but apart from forming groups, he does not undertake any systematic classification of scales.
Eric Krüger analyzes and describes scale groups similar to those of Pilsenbeck, but not as a guide to jazz improvisation, but with systematic approaches, particularly with regard to the »number of levels«, the »interval relationships« and the »location of the different level intervals« . Each analysis is accompanied by a "scale diagram [.] For visualization"  of the respective scale architecture. This representation makes up the main part of the book, which is informative for the topic dealt with here. A tonal comparison of the scales completes the description, but there is no elaborate systematization here either.
In the article »Diatonic / Chromatik / Enharmonik« in the Lexicon of Systematic Musicology, Stefan Rohringer shows the draft of a comprehensive systematization option for scales: a »simple-tritonic heptatonic«  F-c-g-d1-a1-e2-H2 forms a tritone between the bottom and top notes. “This single-tritonic heptaton becomes the double-tritonic (e-b, f sharp-c) [F sharp-c-g-d, as a result of the symmetrical rearrangement of the middle tone by two diminished fifths1-a1-e2-b2] or triple tritonic (es-a, f-h, g-cis) [F-cis-g-d1-a1-it2-H2] Heptaton. If the middle tone of the heptaton is determined as the starting tone of a basic scale ”, different scales can be formed, including the“ church scales ”and the“ melodic minor scale ”. "Tone reserves in which individual fifths of the heptaton are reduced in an asymmetrical distribution can be regarded as borderline cases of diatonic scales."  On the one hand, these provisions can be taken up because they provide an indication that scales are classified by changing the tritone content or by alterations can be; The fact that there are "borderline cases of diatonic" is also an indication that is absolutely worth considering; on the other hand, the procedure described is not carried out up to a systematic classification.
The compositional attempt by Urmas Sisask to demonstrate the tonal character of a single scale in various concretizations should be mentioned here as a special case.His 24 hymns for choir are based exclusively on the ›planetary scale‹ c sharp-d-f sharp-g sharp-a.  On the basis of this and through the differentiated use of rhythm and dynamics, he creates a world of sound that is reminiscent of a single color or a single-colored room. As far as I know, Sisask did not work out a classification of scales.
I borrow from the diatonic tuning of the pedal harp in C flat major (seven strings per octave) and its possibility of increasing the sound of each string by one or two excessive primes (in the tonal result: semitone intervals) using one of the seven pedals As already mentioned, the idea of systematizing diatonic scales through an initially subtractive and then additive process. The accidentals used are numbered according to their positioning in the pitch space or their appearance in the conventional master signature (b and f sharp as 1st, e flat and c sharp as 2nd. - or. -Sign etc .; Ex. 1).
Example 1: Numbering the signs
First, the seven -Signs of the C flat major scale successively ›deleted‹ (I call this method ›subtractive‹), which tonally is synonymous with the successive increase of the scale tones by a semitone, so that the C major scale is at the end of this process. This results in a total of 128 subtractive scales (Ex. 2 and Appendix / Ex. 12).  These 128 scales include all “church modes” and other scales established in jazz such as the “altered scale” (= MM7)  or the ›whole tone scale‹, but interestingly also many scales that do not have a name. These scales can be subdivided into continuously altered scales (one sign after the other is deleted) and discontinuously altered scales (e.g. first and fifth accidentals are deleted). In the European tradition, the scales based on continuous alteration were mainly used: This is how the 'church modes' arise when the -Signs are subtractively deleted from the C flat major scale one after the other.  The scales with discontinuous alteration can be associated with Arabic or Asian scales due to the excessive seconds or minor thirds that arise - neglecting mood differences; however, according to the definition above, they are also diatonic scales. It is noteworthy that the subtractive process creates hemitonic pentatonic scales in addition to the well-known anhemitonic pentatonic scales (e.g. by deleting the first, second, fourth and fifth -Sign; 22.214.171.124 .: ces-d- [e] -fes-g-as- [h]). The ›basic shape‹ of the diatonic scale can be considered to be ›Ionic‹ or the major scale, because its interval relationships are identical to those of the main tone series.
Scales that (based on the C flat major scale) only -Sign and Characters included are, as follows from the foregoing, subtractive scales; they can also be called first-degree scales or first "semitone increase" scales.
Example 2: Subtractive scales 1–10 (complete listing of all 128 scales in the appendix / example 12)
But there are other diatonic scales such as the so-called ›Gypsy minor‹ (cd-es-fis-g-a-flat-b), the ›Gypsy major‹ (c-des-efg-a-b-b), the ›enigmatic one Scale ‹ (c-des-e-f sharp-g sharp-a sharp-h) or scales that can be assigned to Klezmer music such as› Mi-Shebach ‹(cd-es-f sharp-sharp) , which are not alone can be formed in a subtractive way, but require a second step, the addition. Addition here denotes the addition of -Sign after subtracting -Sign. The additive procedure corresponds to a ›whole tone transposition‹ based on the C flat major scale (see example 3 and below the section on nomenclature 1). For example, the result is: ›Gypsy minor‹ (c-d-es-fis-g-a-flat-b) through the alteration process 126.96.36.199.7.1. The scale levels show the following alterations as a result: 1,2,3,4,5,6,7; That means: The resulting scales would have to be called ›subtractive-additive scales‹, but for the sake of simplicity should be called ›additive scales‹, ›scales of the second degree‹ or ›scales of the second semitone increase‹ (example 3). Each of the subtractive scales can theoretically be used with the seven -Signs can be processed additively, resulting in an almost unmanageable number of possible scales (a total of 128x128 = 16,384). Some of these scales are identical to a subtractive scale with regard to the interval relationships (e.g. whole-tone scale as a subtractive scale: ces-des-es-fga- [h] or cde- [fes] -ges-as-b; as an additive scale: cde-fis-gis-ais- [his]), others are new (Ex. 3).
Example 3: Additive scales (selection)
Due to the fact that identical scales (as in the case of the whole-tone scale) can be formed in different ways, it becomes important to establish rules for the representation, naming and categorization of a scale: If two different representations of a scale (quasi transpositions) are possible , the 'simpler' one, which can be reached with fewer operations from the basic C flat major scale, should apply to the naming. The question arises as to how many different scales can actually be formed against the background of this systematic restriction with the aid of the two types of alteration. A mathematically exact calculation shows that after the exclusion of ›duplicates‹, a total of 977 scales remain, with each root note either with -, - or -Sign is provided. 359 of these scales are heptatonic; Altered root tones, which coincide with immediately adjacent scale tones when confused enharmonic, also result in 415 hexatonic, 178 pentatonic and 25 tetratonic scales. 
Nomenclature 1: Aspects of the subtractive or additive alteration process
One of the most important aspects of the presented subtractive or additive method is that it operates on the level of the signs and not on the level of the stages. A procedure was chosen that allows two 'semitone transpositions' upwards in succession and not one that assumes three different states of a 'level' which overlap with those of the neighboring levels. The consequence is that not only the specific scale as the result of an alteration process of stages, but also the process of alteration itself must be named.
A first nomenclature therefore relates to the process of aging. It can be represented by first one and then the numbers of the -Signs, separated by dots, must be noted. The numbering of the signs is based on the order in which they appear in the general preliminary drawing (example 1). For example, Phrygian c-des-es-f-g-as-b is created by: 5.6.7. (the fifth, sixth and seventh were deleted, i.e. by a -Sign replaced). For additive scales are -Please note the sign where previously there was one had been struck away. To designate the additive process, the -Sign added accordingly, e.g .: 7.1. (the seventh was deleted first [the tone fes resulted in an f] and then the first at the same level added so that there is an F sharp); the scale that emerges is: ces-des-es- [fis] -ges-as-b. One difficulty with this nomenclature is the different assignment of digits to the individual scale levels according to the different order in which - and -Signs appear in the general signature (Ex. 1).
A second nomenclature describes the result of the alteration process: The easiest way to do this is to write down the scale with its pitch names, e.g. Phrygian: c-des-es-f-g-a-flat-b. (Alternatively, the levels or scale positions could be marked individually or collectively with the corresponding sign: 1,2,3,4,5,6,7 or 2367145. The numbers following one another without a point denote the stages.)
Another example (example 3): ›Gypsy minor‹ c-d-e-flat-f sharp-g-a-flat-b. First, the scale has to be written from c, because from ces this can only be done in a less ›simple‹ way, with doubling of a pitch (ces-cis-d ... or ces-des-d ...) or double alternation (ces-des- eses ...) would be possible. This scale is a second degree scale, an additive scale because it -Sign, Characters and -Sign contains: 1,2,3,4,5,6,7 (or. 3612574), created by the aging process 188.8.131.52.7.1.
Up to this point we have received a system of diatonic scales that includes familiar things such as major, minor, church keys, pentatonic, whole-tone scales, scales with excessive seconds such as the ›enigmatic scale‹ or previously unnamed scales. Other scales such as the ›Blue Scales‹ have not yet appeared in the representation, and the question arises as to how they relate to the system developed so far. How can the system be further differentiated in order to enable a classification of these scales as well?
A clue for answering this question is provided by those subtractive scales in which tones have dropped out due to the alteration, because they unified tonally with an adjacent tone in the equal tuning, e.g. e with fes, h with ces. This union of the tones leads to a reduction in the number of scale levels , which is why the corresponding scales should be called ›reduced scales‹ (Example 4). But there are now reduced scales not only among the subtractive, but also among the additive scales. Here, for example, the ›planetary scale‹ c-des-fg-as is to be mentioned: It is a hemitonic pentatonic scale that was so named by Urmas Sisask , but already in antiquity , in the Icelandic [ 70] and in Japanese music (as "kumayoshi" ) is known. In the system, it cannot be notated from ces, but from c and cis (since from ces a pitch would have to be doubled or double alternation would have to be used, which would be less 'simple' and thus the method not preferred by the system). It corresponds to five tones of the Phrygian scale, a heptatonic subtractive scale. The two omitted tones cannot be eliminated by the subtractive process alone, but an additive step is required for their omission: e and h must be altered upwards. The resulting scale is c-des- [eis] -f-g-as- [his]; ice and his are omitted. The ›planetary scale‹ is consequently an additive scale with eliminated tones within our system and can therefore be understood as a reduced diatonic scale of additive origin.
A similar example is the “blues minor” hexatonic scale, usually represented as c-e-flat-f-f sharp-g-b; it can also be understood as a reduced scale of additive origin: c-dis-cis-fis-g- [ais] -b - created by the alteration process 184.108.40.206.220.127.116.11.5.6.
Another example of a reduced scale is the Do pentatonic scale c-d-e-g-a.  Its five tones can be found in several subtractive scales, e.g. in major or in the through process 18.104.22.168.6. resulting scale c-d- [e] -fes-g-a-b.  The 'Do pentatonic' is 'ambiguous' in this sense and it lacks those tones (fes and his) that were eliminated by half-tone and whole-tone alteration. It can also be represented in a second form using ces: ces-des-es- [fis] -ges-as- [h]. Both forms are an additive scale with eliminated tones, but the second form is ›simpler‹ according to the rule set out above (i.e. can be reached with fewer operations starting from the basic scale in C flat major); it is therefore preferable.
Within the tradition of our notation system there is now, contrary to the conditions of the pedal harp, the possibility of double signatures. This should therefore be taken into account when explaining the reduction of a scale: A double sign with the consequence that a scale tone is omitted, is to be called a “deletion” of a diatonic tone level. Scales with deleted tones are therefore also reduced scales, e.g. c-d-e-f- (g = asas) -h or c- (disis = e) -f- (g = asas) -h-c. The ›planetary scale‹ from c sharp would also have to be written with a double sign: cis-d- [eses] -fis-gis-a- [heses]; however, the version c-des- [eis] -f-g-as- [his] presented above is ›simpler‹ in the sense mentioned and should therefore apply. Sign combinations that exceed the double signing (i.e. three or more than one sign) - or. - preliminary drawings) should be excluded from the system presented here.
Among the reduced scales we find: subtractive scales with eliminated tones, additive scales with eliminated tones or scales with tones deleted by double signatures. Reduced scales could therefore be viewed as a subclass of both the subtractive and additive scales, but as 'reduced' they form a separate third category, the third-degree scales. Within the class of the reduced scales, the pentatonic from the hexatonic, the subtractive scales from those of additive origin, and scales with eliminated from those with erased tones can then be distinguished. It will be shown below that expanded scales with less than seven tones are also grouped here, whereby reduced scales can generally be defined as a class of scales with less than seven tones (see section Extended scales). A consequence of this determination of reduced scales is that the classes of the subtractive, additive and the expanded scales to be described below become inversely to classes of heptatonic scales.
It is also clear that heptatonic scales - understood as broken clusters  - can in principle be reduced seven times. In addition to hexa- and pentatonic scales, the subtractive-additive process also creates tetratonic scales, and tritonic scales (corresponding to broken three- and four-notes, cf. for example the scale ces-d- [e] -fes- [gis]) by double signatures. -as- [h]). A further reduction is then only possible by deleting scale levels, so that there is first an interval, then a tone and finally no tone (a scale becomes a form of nothing): categories that could in principle still be understood as scales, however, since they themselves represent building blocks of scales, for logical reasons they should be excluded from the concept of scales.
The reduced scales can be subdivided according to what has been said as follows:
- Tritonic scales / broken triads
- Tetratonic scales / broken tetras
- Pentatonic scales
a) subtractive origin
b) of additive origin
c) of extended origin (see below)
with deleted notes (double signature)
with deleted and added tones
- Hexatonic scales
a) subtractive origin
b) of additive origin
c) of extended origin (see below)
with deleted notes (double signature)
with deleted and added tones
At this point it should be noted again that elimination and erasure cause tones to drop out due to alteration. Initially implicitly, then by definition (›reduced scales‹), the present argument made a transition to operations on the level of the number of scale levels. The potential of the procedure to delete or add signs has now been exhausted within the framework of the system presented here. This leads to the next scale class, the extended scales.
Example 4: Reduced scales (selection) 
Expanded scales and reduced scales of expanded origin
There are now scales that consist of seven tones, but cannot be formed solely by the subtractive-additive process, but only with the help of double signing (s) or by deleting and adding tones - they should therefore be called ›extended scales‹ . ›Blues major‹, often given as c-es-e-f-f sharp-g-b, is such a case (Ex. 5).  It can also be represented as c-dis-e-f-ges-asas-b. The tones c, e, f can be explained subtractively; the tone dis is created additively; Gs and Bb are still part of the main tone scale of C flat major; the tone g must be formed as asas by double signing. So there are heptatonic scales in which tones, which otherwise could only appear as an added second form of an already existing scale level, have to be formed by enharmonic confusion and double signatures. Such extended scales also include ›harmonic minor7‹(HM7) c-des-es-fes-ges-a-flat and ›Harmonic major7‹(HD7) c-des-es-f-ges-as-a: Apparently the h has been dropped in both cases and a has been added. This can be explained in the context of the system presented here by the fact that a tone became a second form of its neighbor through double signatures and enharmonic confusion (heses became a).
At this point it becomes clear that from ces a double prefix with -Sign appears as the third ›semitone transposition‹ upwards (ces-c-cis-cisis), a preliminary drawing with double but only as a ›semitone transposition‹ downwards (ces-ceses). Generally speaking, a double signature in our context has the consequence that either a tone level is omitted (deletion) or a double existing one is created (addition). Double marking should therefore ultimately be assigned to the area of the extended scales. Scales that are created with the help of double signatures should be referred to as (seven-tone) ›extended scales‹ or as (made with six or fewer tones) ›reduced scales of extended origin‹. Double signatures should basically be regarded as less ›simple‹ compared to the subtractive-additive approach, but as ›simpler‹ compared to mere deletion or addition of tones.
There are now also such heptatonic scales that cannot be formed by means of double signs, but only by deleting and adding tones. This is explained using the example of the scale c- + cis-d- + dis-e-f-ges- [as] - [his].  Here the tones c, d, e, f can be explained subtractively, Ges still belongs to the base tone scale of C flat major, his can be explained additively, but c sharp and d flat are added - they cannot be constructed by double signing, rather they each show a second state of alteration of a root note already included in the scale. The omission of the root a flat can neither be explained by subtraction, addition nor by double signatures.
Generally speaking, this means: There are scales of diatonic origin to which a new tone has been added for each note that has been dropped (I use the term ›dropped‹ as a generic term for ›eliminated‹, ›deleted‹, ›deleted‹). The expanded scales are thus firstly close to the transheptatonic scales, which are characterized by more than seven tones, but, in contrast to these, remain within the heptatonic scale. Second, they are related to the chromatic scale through the doubling of pitches.
With regard to their number, extended scales can be regarded as those heptatonic scales which correspond to the difference between the 792 seven-tone groups that can be constructed from the twelve pitches of the chromatic total and the 359 subtractive and additive scales (792–359 = 433).
As an example of an ›extended scale‹ with less than seven tones (reduced scale of extended origin), c- + cis-d- + dis-e-f- [geses] - [as] - [his] should be mentioned. It is the cluster-like heptatonic scale just discussed without a sat. By double signatures, f and geses are also combined here, and a hexatonic scale is created. As a result, a further subclass already noted above can be identified among the reduced scales: the expanded scales with fewer than seven tones. In order to keep the systematics and nomenclature simple, all scales with fewer than seven tones should now be called ›reduced scales‹, even if they contain added tones.  The decisive characteristic of the class of reduced scales does not explain operations at the level of alteration such as subtraction, addition, elimination or deletion, and also not expansion, but the low number of tones, although the reduced scales are of course characterized by the operations mentioned. The reduced scales, also known as 'third-degree scales', thus become a reservoir for all kinds of scales.
Extended scales are thus now the fourth scale class as heptatonic scales with dropped and added tones, which is why they are to be called ›fourth degree scales‹. But because added tones also play a central role in the area of the transheptatonic scales, a limit range of the diatonic scales and thus a limit of the efficiency of the systematics presented here is reached at the latest with the extended scales. 
Example 5: Extended scales (selection)
Nomenclature 2: Aspects of the reduced and expanded scales
The nomenclature already presented must now be supplemented: notes that have been omitted are indicated by square brackets, added notes by a + (the associated noteheads are framed by a square in the note examples). ›Blues minor‹ can be represented process-wise as follows: 22.214.171.124.7.4.5. + F sharp; the result is the scale: c- [dis] -es-f- + f sharp-g- [a sharp] -b. or 1,,3,4,+4,5,,7.
A second example: The ›So-Pentatonic‹ ces-des- [e] -fes-ges-as- [h] must be written from ces, because the spelling cdfga would only be possible additively (with ice) and therefore less › simply ‹failed. It is therefore the subtractively reduced scale created by the aging process 1.2. arises; the result can be calculated with 1,2,,4,5,6, represent.
With regard to the nomenclature, this means: Step into the notation of subtractive scales (scales of the first degree) -Sign and -Signs on, with additive scales (scales of the second degree) as well -Sign; reduced scales (scales of the third degree) have fewer than seven tones, so that square brackets  appear in each case; the extended scales (fourth-degree scales) again contain seven tones, and for all of the signs mentioned so far there are also + (plus sign) or double signs (,) necessary. 
In order for a scale to be analytically classified in the system, it must first be noted as ces, if possible. If this does not work because a tone level would have to be used twice, the same process must be tried on c or then also on c sharp. In this way, you can gradually work through all four scale degrees until the best possible - in other words: in the ›simplest‹ system - notation form is found and the scale can be grouped in this way.
Even with the reduced scales it became clear that this system, like every attempt at classification, has its limits: Above all, ambiguities and classification alternatives call for decisions that have to be disclosed and substantiated by arguments. Another critical objection at this point is that with the reduced and expanded scales, the alteration method, as it was used in the first two scale classes, was abandoned - precisely because it had reached its limit - and an ambivalent method was used, on the one hand although it still uses alterations of levels, on the other hand the number of levels is actually reduced or increased, as the terms ›reduced‹ and ›expanded‹ suggest. For these reasons, it is now necessary to look more closely at the limits of the system, to represent and to sound out. This is done in the following by describing the peculiarities of important scale groups which, unlike in the usual representation, cannot be recorded as homogeneous groups by the systematics developed here. This clarifies and differentiates what has been developed so far.
Modes with limited transposability
The extended scales reached the limit of the diatonic scales because, according to the above definition, an extension of a diatonic-heptatonic scale, strictly speaking, creates a transheptatonic scale - a scale with more than seven tones. Olivier Messiaen's »modes with limited transposition possibilities«  - these are ›symmetrical scales‹, which are characterized by rotationally symmetrical figures in the circle of fifths, which makes their limited transposability clear (ex. 6)  - now define this limit more closely as a mode or scale field (Ex. 7), which, according to the nomenclature of the present system, contains both diatonic-heptatonic and transheptatonic scales: Messiaen represents a reduced scale of subtractive origin (the whole-tone scale ces-des-es-fgah, created by the alteration 126.96.36.199 .; in Messiaen the 1st mode) with a reduced scale with deleted tone - double signature, therefore reduced scale of extended origin - (5th mode in Messiaen) and five transheptatonic scales due to the common property of limited transposability as a group of ›modes‹. As such, however, it goes beyond the grouping and classification rules of the system presented here and thus points to their limits.
Messiaen's system does not, of course, cover all scales of limited transposition possibilities. Mention should be made, for example, of the »8. Mode with limited transposition options «. It is the reduced scale of additive origin ces-d-es-fis-g- [ais] -b (Kleinterz-semitone scale) or c-des-ef-gis-a- [his] (semitone-Kleinterz Scale). 
Overall, the ›modes with limited transposability‹ characterize the area between the scale classes of the systematics presented here and between the features diatonic and chromatic as a transition field on which Messiaen specifically composed ›polymodal‹. 
Example 6: Rotational symmetry using the whole-tone-semitone scale as an example
* * *
Example 7: Symmetrical scales
Anhemitonic pentatonic scale
Another limit of the system presented here is shown by the anhemitonic pentatonic scales.  The scales of the ›Re-, Mi-, So-, and La-Pentatonic‹ appear in the present system as subtractive scales with eliminated tones (reduced scales of subtractive origin), even with similar series of numbers in their origin: 1.2.; 1.2.3.; 188.8.131.52.; 184.108.40.206.5 .; it is therefore a question of continuously altered scales. The ›Do-Pentatonic‹ c-d-e- [fes] -g-a- [his], on the other hand, cannot be achieved simply by subtractive alterations, but only with the help of an additive step: 220.127.116.11.5.6.7. or ›easier‹ 1.7.1. (ces-des-es- [fis] -ges-as- [h]) (see above, section Reduced scales). This means that the affiliation of the anhemitonic pentatonic scales as a group cannot be represented with the system developed here. However, this limitation can be compensated by considering the reduced scales, as described above, as a separate class and not as subclasses of the subtractive, additive and extended scales. In this way every pentatonic belongs to the reduced scales, the scales of the third degree.
›Harmonic minor‹ (HM), ›Melodic minor‹ (MM), ›Harmonic major‹ (HD)
A similar problem - the breaking up of a group of scales, which is usually represented as a unit - also arises with those ›modes‹ that Andreas Pilsenbeck created by rotating the ›harmonic minor‹ scales (HM1: c-d-es-f-g-a-flat-b), ›Melodisch-Moll‹ (MM1: c-d-es-f-g-a-h) and ›Harmonic Major‹ (HD1: c-d-e-f-g-as-h) - where he is also HM1, MM1 and HD1 himself calls "modes" (Ex. 8, 9 and 10).  None of these mode groups - in the parlance of this essay: scale groups - can be placed as a whole in one of the four scale classes developed above. (Only the ›church modes‹, which are also used in jazz, are all subtractive scales - according to the linguistic usage just quoted, which is not used here, they would be ›modes‹ of Ionian, I.1-7). In the case of ›harmonic minor‹ e.g. only HM2, HM3, HM5 and HM6 a group of subtractive scales, HM4 is an additive scale and HM7 an expanded scale. The break-up of this scale group (s) arises because in the system developed here, based on the C flat major scale, each note is granted two ›semitone increases‹, but both ›semitone increases‹ are assigned to different scale classes (subtractive and additive). It was also assumed that as soon as a third alteration by double or double is necessary to explain a scale, this scale then falls into the category of the extended scales or, if the number of tones is less than seven, into the class of the reduced scales.
This means that the representation of scales by rotation and their classification in the system presented here, which starts with alterations, often cannot be brought into congruence. In jazz theory, however, astonishingly, both possibilities are used in combination when naming scales: scales are named there on the one hand using the names of the ›church modes‹ (obtained through rotations of the Ionic scale) and, on the other hand, by naming the alterations (of chord and option tones) specified (e.g. HD5 = »Mixolydian9 «with 7 /9/11/13).  It is precisely this combination of designation that shows the break between the two possible forms of representation of scales: with the help of rotations or based on alterations (e.g. 8, 9 and 10) .
Example 8: ›Harmonic minor‹
* * *
Example 9: ›Harmonic major‹
* * *
Example 10: ›Melodic minor‹
The aforementioned break-up of traditional scale groups within the presented system is also evident in the "modes" that Annie Gilchrist introduced in 1911 to explain Gaelic folk songs (Ex. 11).  Gilchrist's illustration shows hexatonic scales that arose from the pentatonic scale by adding tones. It was based on the five semitoneless pentatonic scales (Gilchrist modes 1–5; Gilchrist counted them differently than in the tradition listed above ) and each filled one of the two minor thirds with a non-altered tone of the diatonic root series. Thus, two different hexatonic scales emerged from each of the pentatonic scales, which Gilchrist then referred to as A and B variants. In the linguistic usage pursued here, these are all reduced scales, but of different types. These types can be differentiated as follows: Gilchrist's mode 1 (ces-des- [e] -fes-ges-as- [h]) is a subtractive scale with two tones eliminated by the alteration 1.2., It corresponds to the ›So-Pentatonic‹. Gilchrist's mode 1A (ces-des-es-fes-ges-as- [h]) corresponds to that subtractive scale with an eliminated tone that is created by the alteration 1. has arisen; consequently it is a hexatonic reduced scale of subtractive origin. Gilchrist's mode1B (ces-des- [e] -fes-ges-as-b), created by 2., is also reduced by one tone. Mode2A (c-d-es-f-g- [a sharp] -b), on the other hand, can only be explained as an additive scale with an eliminated tone; the 6th level is missing, the tone a sharp is tonally eliminated by combining it with b; consequently, this is also a hexatonic reduced scale, but of additive origin.  The Gilchrist mode2B also belongs to this category. Mode3B is again a hexatonic reduced scale of subtractive origin: 1.7. The Gilchrist modes4A and 4B are hexatonic reduced scales of additive origin. Modes 5A and 5B are of additive origin. The break-up of this group of scales within the system presented here happens because Gilchrist has chosen a method that adds tones, operates on the level and not on the alteration level, starting from the pentatonic scale - a method that is contrary to the line of thought chosen here.
Example 11: Gilchrist modes
The system presented here offers a classification of scales in which simple diatonic scales are initially created using a subtractive-additive alteration process. Simple diatonic scales are called subtractive and additive scales, the scales of the first and second degree. Various types of omission of tones (elimination, deletion or deletion) result in scales that are missing individual tones, i.e. less than seven tones. They form a third class, the reduced scales or scales of the third degree. When tones other than those that have been dropped are added to these reduced scales, expanded scales, the fourth-degree scales, are created. Extended scales with less than seven tones are also included in the reduced scales. As a generic term for reduced and expanded scales, in contrast to ›simple scales‹, the term ›complex scales‹ can be used.  The extended scales characterize the transition field to the transheptatonic scales, which are characterized by a supply of more than seven tones.
Diatonic scales can be formed within the system from the C flat major scale by four operations: firstly by increasing individual scale tones by an excessive prime (scales of the first degree, subtractive scales), secondly by increasing individual tones by a doubly excessive prime (scales second degree, additive scales), thirdly by eliminating tones (third degree scales, reduced scales) and fourth by adding tones after others have dropped out (fourth degree scales, expanded scales; if more tones are dropped than added, they are reduced ones Scales of extended origin).
For systematic naming, scales must be transposed to ces if possible, otherwise to c or c sharp. The scale levels are denoted by the base tones c-d-e-f-g-a-h and the accidents and (or if necessary with the double sign and ). With the help of the number of tones and the additional symbols for alteration, omission and expansion, it becomes clear to which scale class a scale belongs: Subtractive scales are heptatonic and through - and - marked, additive scales are also heptatonic and also contain -Signs, reduced scales contain fewer than seven tones, which is indicated by square brackets  (but all other characters also appear in this class), extended scales are heptatonic again and you can also find + signs in addition to the characters mentioned above and / or double signature. The alteration process is marked by specifying the signs that have been deleted or added (each with a point), the resulting scale by specifying the type of alteration of the respective levels (e.g. in the case of Phrygian c-des-es-fg-as-b: alteration process starting from C flat major 5.6.7 .; Result 1,2,3,4,5,6,7 or 2367145).
The procedure is based on the pedal harp and its possibilities of alteration or the common equilibrium and heptatonic oriented notation and thought conventions. It offers a possibility to classify all semitone equal scales up to the heptatonic scale (and a bit beyond): as simple, i.e. subtractive or additive scales, as well as complex, i.e. reduced and expanded scales. It should be noted that in the course of the above discussion of this scale system, transition areas between the scale classes have become clear, in which the boundaries between categories / classes do not result from the system itself, but could only be defined and formed argumentatively. The dichotomy of diatonic and chromatic was relativized by introducing 'diatonic' and 'chromatic' as quality terms.  The concept of the “chromatic scale” as a single one was retained. The limits of the developed system were shown.
The classification can now be further differentiated if one removes the requirements defined at the beginning: If, for example, the fact of the octave identity is no longer accepted as a limitation, there are possibilities to form combined scales that extend beyond the octave space, with the scale excerpts in the first and second octave are different from each other. If the equal tuning is left, further scales can be systematized such as Arabic maqamāt, Indian rāgas, slendro and pelog in gamelan music or empirically documented forms of the Chinese pentatonic scales mentioned. The boundary between tuning systems and scale or level systems would become fluid and, under certain circumstances, a continuum: One could discuss whether or not 'church modes' represent different scales in the mean-tone tuning than in the equal-level tuning. The same question arises, for example, with Arabic scales - Abraham Z. Idelsohn has already provided an answer that comes from practice for the Jewish tradition.  It follows from this that all the scales discussed so far must be called ›equal levels‹ and others ›non-equal level‹.
Taking into account such possible extensions, one finally arrives at the following classification:
Equal scales based on quarter, sixth, eighth tone, etc.
Equal scales on a semitone basis
Simple scales (diatonic origin)
Heptatonic subtractive scales (first degree scales)
Scales with continuous alteration
Scales with discontinuous alteration
Heptatonic additive scales (second degree scales)
Complex scales (diatonic origin)
Reduced scales (third degree scales)
a) subtractive origin
Hexatonic scales (simply reduced scales)
a) subtractive origin
Heptatonic Expanded Scales (Fourth Degree Scales)
Scales with added tones
Scales with complemented tones (double signature)
Equal scales on a three-quarter tone basis
Scales that are not equal
Table 1: Scale system (overview) 
The presented type of systematization does not claim to displace established and proven ways of forming and dealing with scales in musical practice. However, it offers a systematically consistent method of classifying (even previously unnamed) scales and sharpening one's awareness of how scales can be used in compositional and analytical practice. For further research and composition, it would be left to investigate scales and their overtone structure as spectral colors and to work with these aspects in a targeted manner , a process that could perhaps be described as "new modality" based on the practices of electroacoustic and acousmatic music, where this term has already been used in connection with "sound objects as modes" and "sonic magnification" .
Example 12: Subtractive scales (complete listing)
I would like to thank Isabel Mundry, Peter Leu, Jonas Labhart and Christian Strinning for the interesting, benevolent technical discussions during the creation process of this article as well as the editorial staff of ZGMTH for the intensive editing. I would like to thank Klaus Frieler for his substantial assistance with the necessary calculations.
Dahlhaus 1998, 624. "In order to separate the two elements, the material and the structure, some authors refer to the tonal system as the tone system and the› principe régulateur des rapports ‹(Fétis) as the tonality." See also Hyer 2001a, 509: "key as a musical container".
With the harp, the basic position is C flat major with -Sign marked (e.g. C), the first heightening of a string by an excessive prime (tonally corresponding to a semitone) with natural signs (e.g. C), the second increase, i.e. a total increase by a doubly excessive prime (tonally corresponding to a whole tone) with -Sign (e.g. C).
This process and conclusion by analogy is justified in so far, and especially with the harp, as it has been one of the relevant scaled instruments that have shaped Central European music theory since ancient times.
Cf. Heygster / Grunenberg 1998, 10: »The scale is a systematic arrangement of the tension relationships. The diatonic tension relationships are put together in a melody according to artistic principles. A solmized major scale represents the structure of the tension relationships in the diatonic in general. "
Moßburger 2012, Vol. 2, 1016 (addition by the author). On the similarity and difference between the terms ›modes‹, ›psalm tones‹, ›church tones‹ and ›church tones‹, also as ›key order‹ or ›key system‹, Ziegler 2009 (based on music in the 17th and early 18th centuries on the way to the major-minor tonality).
See Calella 2005, 87; Wiering 2001; Brieger 2010a; Brieger 2010b, 18ff. Gissel uses the German word ›Tonart‹ for the translation of the term ›modus‹, referring to the time »from Ockeghem to Palestrina« (2013, 11). See also Gissel's introductory remarks on understanding the terms “mode” and “tonus” at that time.
Jans 2013, 42.
Powers / Wiering / Porter / Cowdery / Widdess / Davis / Perlman / Jones / Marett 2001. See also Schmidt-Beste 1997.
Calella 2005, 87.
Messiaen 2012, 458-476; The term "modes with limited transposition possibilities" is also in use (Messiaen 1966, vol. 1, 56).
See Messiaen 2012, 459-465. Compare with "The Physics of Music and Color" Gunther 2012.
Richards 2001, . In addition, the general usage of the term ›mode‹ today is less based on the classic meaning of »measure, measure, manner, manner« (Hüschen 1961, 402; see the expression ›modus vivendi‹), but more on the meaning of ›condition‹ (cf. ›Stand-by mode‹ for electrical devices; cf. “Modes of Vibration and Harmonics”: Gunther 2012, 20), “Permutationsform” (Rambold 2012, 52f .; cf. also Pilsenbeck 2007, 104) or “ Form of appearance [.] ”(In Hüschen 1961, 413, who quotes Josef Rufer, introduced as an equivalent to“ Modus ”; in Rufer himself [1966, 78] but actually directly related to“ series ”). In the more recent psychotherapy theory, especially with regard to mentalization-based therapies, further mode concepts are current that can also be applied to acoustic phenomena: e.g. mode as level (cf. Schrader 2017, 54), structure (cf. Kruse 2017) or medium (cf. . Scharff 2010, 14; Schultz-Venrath 2017).
See introductory Scheideler 2005; the presentation in Rambold 2012, 20–28 for more in-depth information. Reference should be made to the little-known Book of Modes (1993) by the Romanian composer and music theorist Anatol Vieru.
See the following definition of the term “scale”: “A sequence of notes in ascending or descending order of pitch. As a musicological concept, a scale is a sequence long enough to define unambiguously a mode, tonality, or some special linear construction "(Drabkin 2001, 366). See also Holtmeier 2010a.
See Heinemann 2010, 221.
Thorau 2012, 60f., There as part of a presentation of Nelson Goodman's symbol theory.
See Bühler 2014, 7f. and 18-20.
Cf. Schneider 1997, 316, on "Basic principles of any [musical] scaling". See also Thompson 2013, 127: “The concept of a scale can be defined from physical, mathematical, and psychological perspectives. From a physical perspective, it refers to the set of pitches that can be produced on a musical instrument given a certain tuning system. From a mathematical perspective, one can use a group theoretical description of pitch sets […]. From a psychological perspective, a scale refers to a mental representation of regularities in pitch that is activated when one listens to music «.
Eybl 2010, 485.
For an understanding of 'scale' as a 'modal ladder' in relation to Indian rāgas see Michaels 2005, 40; on Arabic maqamāt see Powers 2006. On the “Problematic of Equidistant Moods” see Schneider 1997, 332–399 (quotation: 332).
Gemoll 1937, 194.
Cahn 1995, 1214.
Cf. Rohringer 2010, 82: »› Diatonik ‹(Greek diátonos: going through whole tones)«.
See Cahn 1995, 1214 and 1218.
See ibid. 1219. See also Vetter 1956, 852–856.
See Cahn 1995; Dahlhaus 1998, 625; Moßburger 2012, Vol. 2, 947-959.
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