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  • Dirk Bertels - My favourite Bach keyboard works
    them carefully Hindemith Always take your time never hurry Andras Shiff Don t move on to the next page until you know the current one The sooner you can take your eyes of the sheet music the better Once you know the notes experiment with different phrasings To get mastery over a piece you should get into the habit of not touching it for several weeks or months and consequently relearn it All good pieces should be rested like good wine back to index Well Tempered Clavier II Prelude 20 This is the first proper Bach piece I learned It stands out because it has a chromatic feel about it the prelude is riddled with jazz like circle of fifth patterns The phrasing is a little unusual using a reversed swing feel back to index Goldberg Variations general comments Bach s pupil Johann Theophilus Goldberg a brilliant harpsichord virtuoso was entrusted with performing this work over and over to Count Keyserlingk who suffered of insomnia The Goldberg variations consist of an aria and 30 variations though the Aria may not have been written by Bach himself The work presents 9 different types of canon one in every third variation The canons of the fourth and the fifth 12 and 15 are not in straight imitation but in inversion possibly to indicate the approach to the center of the composition back to index Goldberg Variations variation 21 My favourite variation It is a canon of the seventh The middle voice is the leading melody the top voice strictly plays the same melody a seventh higher while the bass line has its own independent melody Note how as is typically the case with 3 voice fugues both hands are responsible for playing the middle voice so special care needs to be taken that this middle voice sounds smooth back to index Goldberg Variations variation 12 This is also a canon but a rather unusual one an inverted canon at the fourth The top voice is the leading melody The middle voice strictly starts each melody phrase a fourth lower but from there on the melody is inverted i e when the melody phrase goes up the counter melody goes down etc Meanwhile the bass again has its own independent melody The whole effect is very modern sounding giving this variation a somewhat esoteric touch back to index Goldberg Variations variation 1 A great piece that uses many arpeggios this together with the fact that it uses a major tonality G gives it an open almost exuberant quality back to index Goldberg Variations variation 25 To many this variation is regarded as the central piece of the Goldberg Variations To my mind it sounds best when performed slowly each note carefully measured Space is very important in this piece especially the space between the individual notes The variation is also a great study in melodic minor harmony and melodic invention back to index Well Tempered Clavier I Prelude 15 Another happy arpeggiated style

    Original URL path: http://www.dirkbertels.net/music/favourite_Bach_performances.php (2016-04-30)
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  • Dirk Bertels - Bach Workbooks
    the master in harmony counterpoint and melodic invention Some of the works these notes refer to can be viewed on the Bach performance videos page Index Partita II Sinfonia Adagio some scales used in Partita IV Gigue Classical Dim 7th theory Partita VI Tocata Goldberg Variation 24 Partita II Sinfonia ending Bach s last 2 bars Bach ending from Partita I Gigue Aria from Goldberg Variations Partita II Andante Partita IV Gigue Jazzed up version St Matthew passion Warlich dieser ist Gottes Sohn gewesen Lobt Gott ihr Christen allzugleich choral 35 Erbarme Dich WTC II prel3 Erbarme Dich Erbarme Dich Bm Mass Gloria finale Bm mass Crucifix St John mass chorale 52 Ending WTC II Fuge 17 Types of diminished chords Partita IV WTC II prel 3 The non 1 2 1 dim chord WTC I prel 2 Ending WTC II prel 3 in transposed to Cm WTC I prel 15 WTC II prel 3 back to index back to index back to index back to index back to index back to index back to index back to index back to index back to index back to index back to index back to index back to index back to index

    Original URL path: http://www.dirkbertels.net/music/bach_workbooks.php (2016-04-30)
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  • Dirk Bertels - Philosophy of Science
    that no matter how much we learn whatever is left however small it may seem is just as infinitely complex as the whole was to start with That I think is the secret of the universe Isaac Asimov I Asimov Goethe opposed the use of the microscope since he believed that what cannot be seen with the naked eye should not be seen and that what is hidden from us is hidden for a purpose In this Goethe was a scandal among scientists whose first firm and necessary principle is that if something can be done than it should be done Hohn Bainville Beauty Charm and Strangeness Science as Metaphor Science 281 1998 I think that modern physics has definitely decided in favour of Plato In fact the smallest units of matter are not physical objects in the ordinary sense they are forms ideas which can be expressed unambiguously only in mathematical language Werner Heisenberg A mathematician is someone who can take a cup of coffee and turn it into a theory Paul Erdos We think of the number 5 as applying to appropriate groups of any entities whatsoever to five fishes five children We are merely thinking of those relationships between those two groups which are entirely independent of the individual essences of any of the members of either group This is a very remarkable feat of abstraction and it must have taken ages for the human race to rise to it Alfred North Whitehead To my mind there must be at the bottom of it all not an equation but an utterly simple idea And to me that idea when we finally discover it will be so compelling so inevitable that we will say to one another Oh how beautiful How could it have been otherwise John Wheeler

    Original URL path: http://www.dirkbertels.net/philosophy/science_philosophy.php (2016-04-30)
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  • Dirk Bertels - PI
    a geometric proof that PI has to be irrational That is the above procedure has to be ongoing since we need to derive all points of the circle which consists of an infinite amount of points Paradox of the circle s circumference becoming its diameter The idea used here is that the circumference of a circle is the same as the sum of the circumferences of 2 smaller circles that fit inside this circle Looking at the figure below the length of the large green semi circle above the midline equals the sum of the lengths of the largest red semi circle above the midline and the largest red semi circle below the midline The shape thus formed resembles the tear shape so characteristic of the yin yang symbol In turn the length of the largest red semi circle above the midline equals the sum of the lengths of the 2 green semi circles with which it forms a tear drop Taking this idea further thinking recursively the length of the large green semi circle on top of the midline is the same as the length of the tiny red wavy line squigling around the midline Knowing that PI represents the relationsip between a circle and its midline the interaction of the 2 fundamental elements circle and midline is well illustrated here It is interesting to visualise this process going on at infinitum the little red wavy line getting smaller and smaller as its frequency increases Of course we know that no matter how small this wavy line gets its length will always be PI 2 times greater than the true midline the division by 2 is because we are using half circles here Note also the binary aspect of this procedure whereby the large circle contains 2 smaller circles

    Original URL path: http://www.dirkbertels.net/mathematics/Pi.php (2016-04-30)
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  • Dirk Bertels - Constructing the Pentagon
    and V With compasses width set to AV draw D from A From V draw an arc with radius VD thus creating W and X Keeping the same radius draw 2 more points one from W Y and one from X Z Creating a Pentagon Version two Note Replace C in the first 3 figures with O Draw a circle with a line through its center and mark points A O and B Construct a square on AB using the method described above Choose a point outside the line construct a right angle and after that construct the square using AB as a measure Mark points H and K using radius OC or OD from center O With radius HB find E using both A and B as centers Again with A and B as centers find F and G using radius AB Creating a Pentagon Version three This is possibly the simplest one It is my favourite method as it is easy to remember and the resulting drawing has a somewhat organic look about it As a bonus it also draws the triangle and square in the process Only one setting is needed for the compasses This setting radius is also the size of all edges of those figures Choose a radius and draw 3 circles as shown Draw the vertical line in the middle using the intersection of 2 circles Draw the axes using the found intersections Draw an arc from P use it to cap the figure This is an optional step since the points thus created are only needed to draw the square Still using the same radius draw 2 arcs from A and B to find C Connect the dots Notes These constructions are only examples there are a vast variety of constructions out there

    Original URL path: http://www.dirkbertels.net/mathematics/Pentagon.php (2016-04-30)
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  • Dirk Bertels - music - Piano Diary 2000
    done on the atari The audio files last from 1 to 3 minutes and are highly compressed The links to the audio files precede the examples They will download in a different browser window to enable simultaneously listening to the music while watching the music notation This works well in the Mozilla browser though for some reason Internet Explorer may find issue with this In case you have trouble just

    Original URL path: http://www.dirkbertels.net/music/Piano_diary_2000_01.html (2016-04-30)
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  • Dirk Bertels - music - Piano Diary 2000
    2 Download Sapatero Download Sapatero bridge Download Tune1 in F Download Tune2 in Ab Download Tune3 in Eb Download Tune4 in Eb Download Tune5 in Gb Download Tune 6 in Bm Download Tune7 in Cm Download Tune8 in Cm Download Tune 9 10 11 combination expand tree collapse tree To comment on anything regarding this webpage use the have your say page or email me Code for the navigation tree

    Original URL path: http://www.dirkbertels.net/music/Piano_diary_2000_02.html (2016-04-30)
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  • Dirk Bertels - music - Piano Diary 2000
    Download Funk Riff 1 Download Funk Riff 2 Download Funk Riff 3 Download Funk Riff 4 expand tree collapse tree To comment on anything regarding this webpage use the have your say page or email me Code for the navigation tree XTree originally created by Emil A Eklund for WebFX and slightly adapted by Dirk Bertels All contents and downloads on this website including text images pdf files audio files

    Original URL path: http://www.dirkbertels.net/music/Piano_diary_2000_03.html (2016-04-30)
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