Scales catalog Ia.
Scales catalog of my published scales, demonstrated using both the Cséffai Method and the binary way
Special thanks to Dávid Maczkó and Tamás Tóth
Brother basic scales (+ melodic minor scale degrees)
Harmonic minor scale degrees (+Phrygian dominant scale)
Inverse harmonic minor scale degrees
Published however not yet cataloged scales
Special thanks to Dávid Maczkó and Tamás Tóth
It is a very joyful thing, particularly for me, that Pénzes Guitar School is becoming a major knowledge center for guitar, only one of its kind. This could only be achieved through not just teaching my own methodology but also through always having an open ear for the thoughts and comment of my students and readers, and publishing them. I will publish them because I think reader's forums, where everyone can comment anything on a specific topic, are not the right place to display these thoughts; some comments are just inappropriately put or twist the intended message of a certain thought.
Dávid Maczkó sent me his observations on scales. I knew the second I had read thorough it that I would certainly have to publish it and also try to merge it into Pénzes Methodology, which was very easy to do. Dávid apparently comprehended globally my methodology, which made him capable of making thorough observations and come up with something new.
So let me introduce to you Dávid Maczkó, from Vértesszőlős, in his Halloween dress:
He thought this picture would best introduce him. Even though I publish the picture, I disagree with that. He reminds me of the brilliant yet extravagant Nigel Kennedy...
Source: www.guardian.co.uk
...who dresses up the way Dávid did, and plays classy classical music along with other classy musicians. Dávid has extraordinary creativity; I am confident he will find a profession that suits his need for always creating something new. I can't say it enough that Dávid came up with this idea of demonstrating the scales in a binary way.
Demonstrating the scales in a binary way is a turning point in not just Pénzes Merhodology but also in general music theory; the 4096 notes combinations virtually cover all notes, intervals, note groups and scales that are part of the twelvetone equal temperament.
Now all I have to do is demonstrate this enormous amount of scales in a binary way. I want to thank Tamás Tóth for programming this in php language.
A huge binary thanks to Tamás (in ASCIIcoding)!
010010111100001110110110011100110111101011000011
101101100110111011000011101101100110110100100001
In Pénzes Guitar School we have tons of guitar scales, many of which created by us or first published in Hungary by us, to show. Needless to say, this huge amount needs to be sorted.
At the same time, you can still notice the gaps, i.e. there are scales not yet cataloged.
Since the scales catalog starts with scales cataloged according to the Cséffai Method, make sure you read through 'The mnemonic method of Norbert Cséffai' in the section 'Onestring scales practices'.
Reminder: the inner structure of the basic scales and the additional seven scales, which are derived from the basic scales and are called brother basic scales, as a reference to the close relation, can be described with the Cséffai Method, that is three combinations of thirds (trichords).
1
2
3
The harmonic minor scales degrees involve two new scale structures
4
5
There is one more combination that is inherent of the Gipsy scales
6
These six trichord combinations can make a whole lot of scales.
We should take the basic scales as a reference point, and not scales created mathematically, because the basic scales are the fundaments of today's music of the Western culture.
The sequence 3123122 represents the basic scales. There is an another sequence, the consecution of the previously mentioned, which is described in the subsections 'The invention of the brother basic scales with Cséffai Merhod' and 'Brother basic scalesâ' within section 'Basic scales III'.
Besides, it has become evident that the ascending melodic minor scale degrees are the equivalent of the brother basic scales.
Consequently, the harmonic minor scale degrees also have some sort of a brother scale group, as the studies of Sándor Marosi and Tamás Farkas suggest in the section 'Harmonic minor III' (Inverse harmonic minor).
The harmonic minor scale degrees, however, determine another oftenused mode, the Phrygian dominant, which is described in the section 'Phrygian dominant scale'.
Gergely Csaba invented his own sevendegree scales, which I will explained in the section 'Other sevendegree scalesâ'.
Furthermore, I will keep cataloging any scales that can be made up using the above trichord structure.
The scales are cataloged according to the Cséffai Method sequence, which replaces the trichord structure. Besides, there are many cases other cataloged scales that are not yet cataloged; they are known scales, however I donâ't indicate them, because it would be too long to read through once. I separate the major number groups with (...) signs.
The cataloged scales all have seven degrees, that is consist of seven notes and have a range of an octave. Consequently, the pentatonic scales are not included in the catalog.
The scales cataloged so far:

Basic scales  Basic scales I. (7 scales)

Brother basic scales  Basic scales III. (7 scales)

Harmonic minor scale degrees  Harmonic minor I. (7 scales)

Inverse harmonic minor scale degrees  Harmonic minor III. (7 scales)

Melodic minor scale degrees  Melodic minor I. (7 scales)

Phrygian dominant scale  The Phrygian dominant scale (7 scales)

Gipsy scales  Gipsy scales I. (7 scales)

The scales of Gergely Csaba (Graegorius scale)  Other sevendegree scales (7 scales)

Fullstep scale  Other scales (1 scale)

122223(6)  Graegorius scale  Second degree

122231(3)  Brother basic scale  Lukácsi scale + melodic minor  Second degree

122312(3)  Basic scale  Phrygian

123122(3)  Basic scale  Locrian

123134(5)  Inverse harmonic minor  Third degree

123451(3)  Harmonic minor  Second degree + Phrygian dominant  Fifth degree

131222(3)  Brother basic scale  Jene scale + Melodic minor  Seventh degree

131234(5)  Harmonic minor  Seventh degree + Phrygian dominant  Third degree

134512(3)  Inverse harmonic minor  Sixth degree

134564(5)  Gipsy scale  Third degree

...

222222(2)  Fullstep scale

222236(1)  Graegorius scale  Third degree

222313(1)  Brother basic scale  Szabó scale + Melodic minor  Third degree

222361(2)  Graegorius scale  Fourth degree

223123(1)  Basic scale  Lydian

223131(2)  Brother basic scale  Cséffai scale + Melodic minor  Fourth degree

223612(2)  Graegorius scale Fifth degree

231223(1)  Basic scale  Major

231231(2)  Basic scale  Mixolydian

231312(2)  Brother basic scale  Gasztonyi scale + Melodic minor  Fifth degree

231345(1)  Inverse harmonic minor  Fourth degree

234513(1)  Harmonic minor  Third degree + Phrygian dominant  Sixth degree

236122(2)  Graegorius scale  Sixth degree

...

312223(1)  Brother basic scale  Szemerszky scale + Melodic minor  First degree

312231(2)  Basic scale  Dorian

312312(2)  Basic scale  Minor

312345(1)  Harmonic minor  First degree + Phrygian dominant  Fourth degree

313122(2)  Brother basic scale  Vedres scale + Melodic minor  Sixth degree

313451(2)  Inverse harmonic minor  Fifth degree

345123(1)  Inverse harmonic minor  Seventh degree

345131(2)  Harmonic minor  Fourth degree + Phrygian dominant  Seventh degree

345645(1)  Gipsy scale  Fourth degree

361222(2)  Graegorius scale  Seventh degree

...

451231(3)  Inverse harmonic minor  First degree

451312(3)  Harmonic minor  Fifth degree + Phrygian dominant  First degree

451345(6)  Gipsy scale  First degree

456451(3)  Gipsy scale  Fifth degree

...

512313(4)  Inverse harmonic minor  Second degree

513123(4)  Harmonic minor  Sixth degree + Phrygian dominant  Second degree

513456(4)  Gipsy scale  Second degree

564513(4)  Gipsy scale  Sixth degree

...

612222(3)  Graegorius scale  First degree

645134(5)  Gipsy scale  Seventh degree
I recently received some critics on the Cséffai Method not being the optimal tool to demonstrate the scales. They may be right; since this method only applies indirect tools, which are subjectively numbered trichord group references, the whole system is not transparent.
That raises a question: how could the scales system be more transparent? Here comes the answer: the binary way of demonstrating them. (I have to add that the original idea popped up in Dávid Maczkó's mind)
By the binary way I mean showing the scales structures (and making calculations) using the binary numeral system. This approach has been successfully used in all sections on scales patterns, particularly in 'Scale patterns II' section. Nevertheless, the good old Cséffai Method can still remain in practice.
The twelvetone equal temperament, as the name suggests, uses twelve semitones. If this were illustrated on piano keys, it would be seven white keys and five black keys, something like this below:
These twelve tones replaced with digits:
000000000000
The keyboard of the piano is actually linearly converted into digits, i.e. regardless of whether they are black keys or white. In another way of describing, it would look like this:
0 0 0 0 0
0 0 00 0 0 0
...or like this...
000000000000
...but let's stick to the last version. Here comes the question: How do I plot a major scale with these numbers: The answer is simple:
101011010101
The scale I have just plotted is actually a C Major scale, as that's what the position of black keys and white keys would look like on the piano. However, what you need here is something that is fit fot the guitar. So letâ's just remove the black marks from the figure:
101011010101
The above figure is a scale independent of pitch, thatâ's the exact thing you need for the guitar. Scales structures can be demonstraed way easier in the binary way than employing the Cséffai Method. Another advantage is that you can actually make calculations with it. For example, the total number of scale patterns can be calculated using the combinatorial formula n^{k} , which gives the result of 2^{12} = 4096. That is the total number of scales that can be generated out of the twelvetone temperament. The two endpoints of this interval are the scale patterns below:

100000000000  one C note (C note is the reference note)

...

...

...

111111111111  Chromatic scale (incorporates all the twelve semitones)
It can be concluded that there is a total number of 4096 variations, including notes, intervals or scale patterns in the twelvetone temperament.
The most important scales out of the 4096 variations are the sevendegree scales. The scales catalogue only contains sevendegree scales. Putting it in a binary way, only scales with seven digits being '1' out of the total twelve digits are in the catalog:
111111100000
Hence, the total number of sevendegree scales is 792. This result is calculated using the following formulaâ€¦
...Where:

n = 12

k = 7
12! / (7! x (127))! = 479001600 / (5040 x 120) = 792
The scales are not always consequently differentiated in the traditional music theory, making possible some overlaps between scales. Let me indicate which scales overlap:

Brother basic scales = Melodic minor scale degrees (only ascending)

Harmonic minor = Phrygian dominant
NB: These scales have already been incorporated in the catalog as part of the Cséffai Scales Catalog. It is thus important that certain scales of these scale groups can't be added once more in the catalog. So these seven scales have already been demonstrated out of the total 792:

Basic scales  Basic scales I. (7 scales)

Brother basic scales  Basic scales III. (7 scales) (+ Melodic minor scale degrees, only ascending)

Harmonic minor scale degrees  Harmonic minor I. (7 scales) (+ Phrygian dominant)

Inverse harmonic minor scale degrees  Harmonic minor III. (7 scales)

Gipsy scales  Gipsy scales I. (7 scales)

The scales of Gergely Scaba (Graegorius scales)  Other sevendegree scales (7 scales)

Fullstep scales  Other scales (1 scale)
That makes 6 times 7, that is 42 scales and one more in total = 43 scales in total. Only 792  43 = 750 scales to go, out of which you will certainly find some that are pretty much useless in our case, like this particular one: 111111100000.
In the below sections all 4096 variations of notes, intervals and note groups are described:

(Catalog of scales consisting of zero note = 1 scale)

Catalog of scales consisting of 1 note = 12 scales

Catalog of scales consisting of 2 notes = 66 scales

Catalog of scales consisting of 3 notes = 220 scales

Catalog of scales consisting of 4 notes = 495 scales

Catalog of scales consisting of 5 notes = 792 scales

Catalog of scales consisting of 6 notes = 924 scales

Catalog of scales consisting of 7 notes = 792 scales

Catalog of scales consisting of 8 notes = 495 scales

Catalog of scales consisting of 9 notes = 220 scales

Catalog of scales consisting of 10 notes = 66 scales

Catalog of scales consisting of 11 notes = 12 scales

Catalog of scales consisting of 12 notes = 1 scales
Let's double check it!

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

7! = 5040

8! = 40320

9! = 362880

10! = 3628800

11! = 39916800

12! = 479001600

n = 12

k = variable
Catalog of scales consisting of 1 note

12! / (1! x (121))! = 479001600 / (1 x 39916800) = 12
Catalog of scales consisting of 2 notes

12! / (2! x (122))! = 479001600 / (2 x 3628800) = 66
Catalog of scales consisting of 3 notes

12! / (3! x (123))! = 479001600 / (6 x 362880) = 220
Catalog of scales consisting of 4 notes

12! / (4! x (124))! = 479001600 / (24 x 40320) = 495
Catalog of scales consisting of 5 notes

12! / (5! x (125))! = 479001600 / (120 x 5040) = 792
Catalog of scales consisting of 6 notes

12! / (6! x (126))! = 479001600 / (720 x 720) = 924
Catalog of scales consisting of 7 notes

12! / (7! x (127))! = 479001600 / (5040 x 120) = 792
Catalog of scales consisting of 8 notes

12! / (8! x (128))! = 479001600 / (40320 x 24) = 495
Catalog of scales consisting of 9 notes

12! / (9! x (129))! = 479001600 / (362880 x 6) = 220
Catalog of scales consisting of 10 notes

12! / (10! x (1210))! = 479001600 / (3628800 x 2) = 66
Catalog of scales consisting of 11 notes

12! / (11! x (1211))! = 479001600 / (39916800 x 1) = 12
Catalog of scales consisting of 12 notes

12! / (12! x (1212))! = 479001600 / 479001600 = 1
12 + 66 + 220 + 495 + 792 + 924 + 792 + 495 + 220 + 66 + 12 + 1 = 4095
In addidtion to that, the one scale with no '1' digits must be added o the total.
000000000000
(Of course it makes no use for us)
You can observe perfect symmetry in the binary scale catalog. What I mean by symmetry is that one side if it is 000000000000, while the other side 111111111111, with a middle point of 111000000111 (my opinion). The symmetry is also obvious when taking a look at the distribution of the totals:
1note scale = 12  11note scale = 12
2note scale = 66  10note scale = 66
3note scale = 220  9note scale = 220
4note scale = 495  8note scale = 495
5note scale = 792  7note scale = 792
6note scale = 924
First letâ's plot an infinite Major scale so that all the other scales can be derived from it:
...101011010101101011010101...

101011010101  Major

101101010110  Dorian

110101011010  Phrygian

101010110101  Lydian

101011010110  Mixolydian

101101011010  Minor

110101101010  Locrian
The reference point for these scales is the Lukácsi scale (discussed in the section 'Basic scales III.'). I must add that the Brother basic scale degrees overlap with certain degrees of the Melodic Minor scale.
...110101010110110101010110...

110101010110  Lukácsi + melodic minor  second degree

101010101101  SzabĂł + melodic minor  third degree

101010110110  Cséffai + melodic minor  fourth degree

101011011010  Gasztonyi + melodic minor  fifth degree

101101101010  Vedres + melodic minor  sixth degree

110110101010  Jene + melodic minor  seventh degree

101101010101  Szemerszky + melodic minor  first degree
The harmonic minor scale degrees overlap with certain degrees of the Phrygian dominant scale.
...101101011001101101011001...

101101011001  first degree + Phrygian dominant  fourth degree

110101100110  second degree + Phrygian dominant  fifth degree

101011001101  third degree + Phrygian dominant  sixth degree

101100110110  fourth degree + Phrygian dominant  seventh degree

110011011010  fifth degree+ Phrygian dominant  first degree

100110110101  sixth degree + Phrygian dominant  second degree

110110101100  seventh degree + Phrygian dominant  third degree
Inverse harmonic minor scale degrees
...101101100110101101100110...

101101100110  first degree

110110011010  second degree

101100110101  third degree

110011010110  fourth degree

100110101101  fifth degree

110101101100  sixth degree

101011011001  seventh degree
...110011011001110011011001...

110011011001  first degree

100110110011  second degree

110110011100  third degree

101100111001  fourth degree

110011100110  fifth degree

100111001101  sixth degree

111001101100  seventh degree
The scales of Gergely Csaba (Graegorius scales)
...111010101010111010101010...

111010101010  first degree

110101010101  second degree

101010101011  third degree

101010101110  fourth degree

101010111010  fifth degree

101011101010  sixth degree

101110101010  seventh degree
Published however not yet cataloged scales
Aufmented scale
It belongs to the scales consisting of 3 notes.

100010001000
Diminished scale
It belongs to the scales consisting of 4 notes.

100100100100
(Minor) pentatonic scale
It belongs to the scales consisting of 5 notes.
...100101010010100101010010...

100101010010  1st degree

101010010100  2nd degree

101001010010  3rd degree

100101001010  4th degree

101001010100  5th degree
Papp pentatonic scale
It belongs to the scales consisting of 5 notes.
...101010100100101010100100...

101010100100  1st degree

101010010010  2nd degree

101001001010  3rd degree

100100101010  4th degree

100101010100  5th degree
Fullstep scale
It belongs to the scales consisting of 6 notes.

101010101010
Mathematical scales
They belong to different scale groups (Other scales VII.).
Scales from diminished and augmented C Major scales
They belong to different scale groups (Basic scales VI.).