Pythagorean tuning in more detail Originally I was tempted to label this section "Mathematical aspects of Pythagorean tuning," but decided that such a title might discourage some readers mainly interested in practical details of implementing this tuning on various instruments ranging from medieval harps to organs and electronic synthesizers. Please let me emphasize that it is not necessary to understand all of the fine points that follow in order to obtain the tuning in practice - and to explore it through actual music, which is the best way. A harp player may need to know only how to tune a series of perfect fifths, while a synthesizer player may pleasantly discover that Pythagorean tuning is available as a preprogrammed option, requiring only a convenient menu selection. Other synthesizers might require the player to specify a custom tuning, and some specifications are included in Section 4.

Pythagorean tuning in more detail Originally I was tempted to label this section "Mathematical aspects of Pythagorean tuning," but decided that such a title might discourage some readers mainly interested in practical details of implementing this tuning on various instruments ranging from medieval harps to organs and electronic synthesizers.

Please let me emphasize that it is not necessary to understand all of the fine points that follow in order to obtain the tuning in practice - and to explore it through actual music, which is the best way. A harp player may need to know only how to tune a series of perfect fifths, while a synthesizer player may pleasantly discover that Pythagorean tuning is available as a preprogrammed option, requiring only a convenient menu selection.

Other synthesizers might require the player to specify a custom tuning, and some specifications are included in Section 4. More generally, the focus here is on understanding the tuning and its qualities rather than on any specific kind of instrument. Tuning a basic scale A good way to explore Pythagorean tuning is to generate a simple scale.

Here we will tune a very popular scale of the Gothic period, the Lydian mode Mode V of Gregorian chant consisting of the "white keys" in an octave from F to the F above. Our first note is the lower f, which stands in unison with itself 1: Now we tune our first pure fifth 3: With an acoustical instrument, this involves adjusting the upper note until no beats can be heard when the two notes are sounded together - a task easiest on a sustained instrument such as an organ.

As we continue along our chain of fifths, we will generate not only the individual notes of our scale but the various intervals making up the spectrum of concord and discord.

This note would have a ratio of 3: Note that to find the size of an interval created by adding two others - here two fifths - we multiply the ratios of these intervals: To keep within the range of our first octave, we instead place our third note an octave lower, at the g located a fourth 4: To find the ratio of this new interval, regarded like M2 as relatively tense, we multiply 9: Note that this interval consists precisely of a fifth plus a major second, and so is known in medieval terminology as a tonus cum diapente or "whole-tone-plus-fifth": This Pythagorean M3 is known as a ditone, since it is equal to precisely two whole tones of 9: Since no number of superimposed perfect fifths will yield precisely an even octave, we instead define this special interval independently as the pure ratio 2: Indeed, medieval theorists such as Johannes de Grocheio and Jacobus of Liege describe the purely blending octave as the font and source of all other intervals, including the more richly stable fifths and fourths: Before going on to add Bb an integral part of the medieval gamut and the other accidentals, we may wish optionally to consider the additional Pythagorean intervals our tuning in fifths has generated.

Additional diatonic intervals In addition to this octave and the seven intervals we have derived through our chain of fifths up or fourths downour scale includes some other important intervals arising from these.

One way to approach these intervals is to treat them as the difference of two intervals already defined as part of our chain of fifths. To find such a difference of intervals, we divide their ratios.

We could also take the fourth, e. Like the fifth, the fourth is a richly stable interval, although not so smooth and conclusive in itself. Let us next consider a vital melodic interval in our scale: As a vertical interval, it is a strong discord which plays a striking role in various two-voice and multi-voice resolutions; as a melodic interval, its rather compact size gives it an expressive quality see Sections 3.

Note that an octave is equal to five whole tones of 9: Much milder as a vertical interval is the relatively blending minor third, e. This interval, like the major third, represents the mildest level of vertical instability.I have a need to write code that will prorate a value across a list, based on the relative weights of "basis" values in the list.

Simply dividing the "basis" values by the sum of the "basis" values and then multiplying the factor by the original value to prorate works to a certain degree. A Free Visual C#.NET programming course for complete beginners. In this program, a structure (student) is created which contains name,subject and marks as its data member.

Then, an array of structure of 10 elements is created. Matrix multiplication You are encouraged to solve this task according to the task description, using any language you may know. Search the world's information, including webpages, images, videos and more. Google has many special features to help you find exactly what you're looking for.

This C# Program Finds and display the Multiplication Table. Here the limit is obtained from the user and the multiplication table is diaplayed. Here is source code of the C# Program to Find and display the Multiplication Table.

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Using for-loop to create a multiplication table (richtextbox)