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

Introduction

Cséffai Scales Catalogs

Binary Scales Catalog

Brother basic scales (+ melodic minor scale degrees)

Harmonic minor scale degrees (+Phrygian dominant scale)

Inverse harmonic minor scale degrees

Gipsy scales

The scales of Gergely Csaba

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 twelve-tone 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 ASCII-coding)!

 

010010111100001110110110011100110111101011000011
101101100110111011000011101101100110110100100001

 

Introduction

 

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 'One-string 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 3-1-2-3-1-2-2 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 often-used mode, the Phrygian dominant, which is described in the section 'Phrygian dominant scale'.

Gergely Csaba invented his own seven-degree scales, which I will explained in the section 'Other seven-degree scalesâ'.

 

Furthermore, I will keep cataloging any scales that can be made up using the above trichord structure.

 

Cséffai Scales Catalog

 

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 seven-degree scales (7 scales)

  • Full-step scale - Other scales (1 scale)

 

  • 1-2-2-2-2-3-(6) - Graegorius scale - Second degree

  • 1-2-2-2-3-1-(3) - Brother basic scale - Lukácsi scale + melodic minor - Second degree

  • 1-2-2-3-1-2-(3) - Basic scale - Phrygian

  • 1-2-3-1-2-2-(3) - Basic scale - Locrian

  • 1-2-3-1-3-4-(5) - Inverse harmonic minor - Third degree

  • 1-2-3-4-5-1-(3) - Harmonic minor - Second degree + Phrygian dominant - Fifth degree

  • 1-3-1-2-2-2-(3) - Brother basic scale - Jene scale + Melodic minor - Seventh degree

  • 1-3-1-2-3-4-(5) - Harmonic minor - Seventh degree + Phrygian dominant - Third degree

  • 1-3-4-5-1-2-(3) - Inverse harmonic minor - Sixth degree

  • 1-3-4-5-6-4-(5) - Gipsy scale - Third degree

  • ...

  • 2-2-2-2-2-2-(2) - Full-step scale

  • 2-2-2-2-3-6-(1) - Graegorius scale - Third degree

  • 2-2-2-3-1-3-(1) - Brother basic scale - Szabó scale + Melodic minor - Third degree

  • 2-2-2-3-6-1-(2) - Graegorius scale - Fourth degree

  • 2-2-3-1-2-3-(1) - Basic scale - Lydian

  • 2-2-3-1-3-1-(2) - Brother basic scale - Cséffai scale + Melodic minor - Fourth degree

  • 2-2-3-6-1-2-(2) - Graegorius scale- Fifth degree

  • 2-3-1-2-2-3-(1) - Basic scale - Major

  • 2-3-1-2-3-1-(2) - Basic scale - Mixolydian

  • 2-3-1-3-1-2-(2) - Brother basic scale - Gasztonyi scale + Melodic minor - Fifth degree

  • 2-3-1-3-4-5-(1) - Inverse harmonic minor - Fourth degree

  • 2-3-4-5-1-3-(1) - Harmonic minor - Third degree + Phrygian dominant - Sixth degree

  • 2-3-6-1-2-2-(2) - Graegorius scale - Sixth degree

  • ...

  • 3-1-2-2-2-3-(1) - Brother basic scale - Szemerszky scale + Melodic minor - First degree

  • 3-1-2-2-3-1-(2) - Basic scale - Dorian

  • 3-1-2-3-1-2-(2) - Basic scale - Minor

  • 3-1-2-3-4-5-(1) - Harmonic minor - First degree + Phrygian dominant - Fourth degree

  • 3-1-3-1-2-2-(2) - Brother basic scale - Vedres scale + Melodic minor - Sixth degree

  • 3-1-3-4-5-1-(2) - Inverse harmonic minor - Fifth degree

  • 3-4-5-1-2-3-(1) - Inverse harmonic minor - Seventh degree

  • 3-4-5-1-3-1-(2) - Harmonic minor - Fourth degree + Phrygian dominant - Seventh degree

  • 3-4-5-6-4-5-(1) - Gipsy scale - Fourth degree

  • 3-6-1-2-2-2-(2) - Graegorius scale - Seventh degree

  • ...

  • 4-5-1-2-3-1-(3) - Inverse harmonic minor - First degree

  • 4-5-1-3-1-2-(3) - Harmonic minor - Fifth degree + Phrygian dominant - First degree

  • 4-5-1-3-4-5-(6) - Gipsy scale - First degree

  • 4-5-6-4-5-1-(3) - Gipsy scale - Fifth degree

  • ...

  • 5-1-2-3-1-3-(4) - Inverse harmonic minor - Second degree

  • 5-1-3-1-2-3-(4) - Harmonic minor - Sixth degree + Phrygian dominant - Second degree

  • 5-1-3-4-5-6-(4) - Gipsy scale - Second degree

  • 5-6-4-5-1-3-(4) - Gipsy scale - Sixth degree

  • ...

  • 6-1-2-2-2-2-(3) - Graegorius scale - First degree

  • 6-4-5-1-3-4-(5) - Gipsy scale - Seventh degree

Binary Scales Catalog

 

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 twelve-tone 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 nk , which gives the result of 212 = 4096. That is the total number of scales that can be generated out of the twelve-tone 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 twelve-tone temperament.

 

The most important scales out of the 4096 variations are the seven-degree scales. The scales catalogue only contains seven-degree 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 seven-degree scales is 792. This result is calculated using the following formula…

 

 

...Where:

  •  n = 12

  • k = 7

12! / (7! x (12-7))! = 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:

 

  1. Basic scales - Basic scales I. (7 scales)

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

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

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

  5. Gipsy scales - Gipsy scales I. (7 scales)

  6. The scales of Gergely Scaba (Graegorius scales) - Other seven-degree scales (7 scales)

  7. Full-step 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:

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 (12-1))! = 479001600 / (1 x 39916800) = 12

Catalog of scales consisting of 2 notes 

  • 12! / (2! x (12-2))! = 479001600 / (2 x 3628800) = 66

Catalog of scales consisting of 3 notes

  • 12! / (3! x (12-3))! = 479001600 / (6 x 362880)220

Catalog of scales consisting of 4 notes 

  • 12! / (4! x (12-4))! = 479001600 / (24 x 40320) = 495

Catalog of scales consisting of 5 notes 

  • 12! / (5! x (12-5))! = 479001600 / (120 x 5040) = 792

Catalog of scales consisting of 6 notes 

  • 12! / (6! x (12-6))! = 479001600 / (720 x 720) = 924

Catalog of scales consisting of 7 notes 

  • 12! / (7! x (12-7))! = 479001600 / (5040 x 120) = 792

Catalog of scales consisting of 8 notes 

  • 12! / (8! x (12-8))! = 479001600 / (40320 x 24) = 495

Catalog of scales consisting of 9 notes 

  • 12! / (9! x (12-9))! = 479001600 / (362880 x 6) = 220

Catalog of scales consisting of 10 notes 

  • 12! / (10! x (12-10))! = 479001600 / (3628800 x 2) = 66

Catalog of scales consisting of 11 notes 

  • 12! / (11! x (12-11))! = 479001600 / (39916800 x 1) = 12

Catalog of scales consisting of 12 notes 

  • 12! / (12! x (12-12))! = 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:

 

1-note scale = 12 - 11-note scale = 12

2-note scale = 66 - 10-note scale = 66

3-note scale = 220 - 9-note scale = 220

4-note scale = 495 - 8-note scale = 495

5-note scale = 792 - 7-note scale = 792

6-note scale = 924

 

Basic scales

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

Brother basic scale

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

Harmonic minor scale degrees

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

Gipsy scales

 

...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

 

Full-step 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.).

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