[11] I shall henceforward show numeric values for both "older" and "later" versions of Finale as follows: "later versions" will appear in square brackets following the values for "older" versions. It will always be noted that the values of the former are the same as those of the latter minus 8192.¹³

¹³ I should again stress that files created with older versions are updated automatically when opened by later versions of Finale, but when creating new files in any version the user will need to know which set of values to apply. If the list provided below for "older versions" is mistakenly used in "later versions", Finale — in blissful ignorance — will produce extremely bizarre results.

[12] In constructing the Pythagorean scale each existing note on the equally-tempered MIDI keyboard needs to be allotted an exact mathematical value such that when Finale plays a note with the value defined its pitch will be exactly the one required by the scale. In effect what this library will be doing is retuning the separate pitches of the scale so that they will all have a Pythagorean relationship instead of an equally-tempered one. What needs to be calculated is the exact amount by which each separate note has to be raised or lowered from its equally-tempered default when it is played by Finale. These defaults will all have a pitchwheel value of 8192 [0]¹⁴, and the Pythagorean values will therefore be slightly more or less than this value depending upon the position of the note within the scale.

¹⁴ As stated in [11] above, "8192" here refers to the default numeric value for "older versions" and the following "[0]" that for "later".

A correct way of arriving at exact Pythagorean values is to take a starting note with the default value of 8192 [0], and then a) move upwards through a series of perfect fifths sharpwards, and b) move downwards through a corresponding chain of perfect fifths flatwards. Taking the note "C" as the starting point, the first deduction will move sharpwards through the fifths ending with A-sharp, while the second deduction will move from the same starting note flatwards through the fifths to end on C-flat. When this process has been completed all the "white notes" will have their correct pitches, and all the "black notes" will exist with two pitches separated by a Pythagorean comma.

Корректный способ получения точных Пифагорейских величин заключается в том, чтобы взять исходную ноту (с приписанным ей по умолчанию значением 8192 или 0) и, далее:

а) двигаться вверх по цепочке (акустически чистых) квинт в "диезную сторону" и
b) двигаться вниз по цепочке (акустически чистых) квинт в "бемольную сторону".

Взяв ноту "C" ("до") в качестве отправной точки, первый этап получения точных Пифагорейских величин будет заключаться в движении в "диезную сторону" вплоть до ноты ля-диез, тогда как второй этап получения точных Пифагорейских величин будет заключаться в движении от той же самой исходной ноты "C" ("до") в "бемольную сторону" вплоть до ноты до-бемоль. Когда этот процесс из двух этапов будет завершен, все "белые ноты" приобретут свои точные значения, а для "черных нот" будет существовать два значения, отделенные друг от друга Пифагоровой коммой.

Sharps will thereby lie a major semitone higher than their parent naturals, and flats a major semitone lower than theirs. This will then provide for all the notes required by the combined use of musica recta and musica ficta. The octave will have been provided with 18 notes instead of the usual 12. As we shall see, however, 18 notes to the octave will be too few to permit Just Intonation which additionally requires the raising and lowering of these default Pythagorean notes by Syntonic commas. But this is nonetheless the starting point for arriving at the required scale.¹⁵

Диезы, следовательно, будут лежать на большой полутон выше своей родительской "белой" ноты, а бемоли будут лежать на большой полутон ниже своей родительской "белой" ноты. Это даст нам все ноты, необходимые при комбинированном использовании musica recta и musica ficta. Таким образом, октава будет представлена восемнадцатью нотами вместо обычных двенадцати.

¹⁵ It must be remembered that the Pythagorean scale is the default scale for Just Intonation, and that the tuning of JI arises from the addition/subtraction of the Syntonic comma (81:80) to/ from notes that are modified for consonance purposes. The effect of this modification is to change all the Pythagorean diatonic semitones from minor (256:243) to major (16:15), to narrow all the major thirds and sixths, and to widen all the minor thirds and sixths. But these changes all result from the single application each time of a Syntonic comma adjustment (upwards or downwards) that changes the default Pythagorean pitches concerned.

[13] To create precise Pythagorean pitches for the scale, it must be remembered that the notes are initially set on the keyboard to Equal Temperament. As such each note in the score will have a default Finale pitchwheel value of 8192 [0]. Only when this value is, for each separate note, raised or lowered by the exact value required will the Pythagorean scale come into being. And it is only when this new scale has been created that the addition or subtraction of comma values can be made with precision. When this scale, together with the comma values to be applied to it, has been constructed, it will then be possible to edit scores by adding the prescribed playback Text Expressions and achieve a playback in Just Intonation. The first task, therefore, is to construct an accurate Pythagorean scale.

[14] Because the task is to create audible pitches rather than to calculate relative string lengths, it is necessary to work in Cents rather than string length ratios.¹⁶ Since construction of the Pythagorean scale is (as stated in [11] above) achieved by computing pure fifths in a chain upwards, and then downwards, it is therefore necessary to define the exact number of Cents required between successive notes that lie a fifth apart. Since we already know that by default each equally-tempered semitone contains 100 Cents, and therefore that the equally-tempered fifth contains 700 Cents, all we will need to do is to subtract 700 from the total number of Cents calculated for the interval overall to find the difference between the equally-tempered and the Pythagorean pitches of the note concerned.

¹⁶ Theorists who explain the Pythagorean scale do so in terms of sounding length, and this leads to sounding length ratios. Thus, for example, the pure fifth is 3:2, and the octave is 2:1. Although the same ratios are inversely correct for the relative quantities defining the two pitches involved, they do not define the pitches themselves (but only the relative values between them). What is needed here is an unambiguous definition of the actual pitch to be generated for each note defined. This can only be achieved by converting the ratio values into Cents and then calculating the pitch of each note in terms of the number of Cents that make it higher or lower than the default note from which it is computed.

We shall also need to convert Cents value into pitchwheel value which is very simple: since the equally-tempered semitone (set on the keyboard) has a pitchwheel range of 8192 (being either 0 — 8192 [-8192 — 0] or 8192 — 16384 [0 — 8192]), it must follow that each Cent has a pitchwheel value of 81.92. When, therefore, the difference in Cents between an equally-tempered and a Pythagorean fifth is calculated, the Cents difference needs to be multiplied by 81.92, and the result added to (when moving upwards) or subtracted from (when moving downwards) the default pitchwheel value of 8192 [0]. This will yield the new pitchwheel value for the note that now lies a Pythagorean fifth above or below the note from which the calculation was made.

[15] All the new pitchwheel settings can easily be calculated when the formula for conversion from string-length ratio to Cents is applied. This formula is very simple: where the interval-ratio required is "i" and the Cents value sought is "c", the conversion is achieved as follows:

c = log (i) * (1200/log (2)).

[15] Все новые значения для колеса (pitchwheel) могут быть легко вычислены, если имеется формула для преобразования отношений длин струн в центы. Эта формула очень проста: если обозначить интервальное отношение через "i", а искомое значение в центах через "c", то необходимое преобразование может быть выполнено следующим образом:

c = log (i) * (1200/log (2)).

By applying these principles to calculate the pitchwheel setting for "G"¹⁷, and taking "C" as the starting point with a normal pitchwheel setting of 8192 [0], the value for "G" will be arrived at as follows:

Если применить теперь эти принципы для вычисления значения колеса (pitchwheel) для "G"¹⁷ (взяв в качестве исходной точки "C" с нормальным значением колеса (pitchwheel) равным 8192 [0]), то величина колеса для "G" будет вычислена следующим образом:

¹⁷This value for "G" will be the same for every "G" in every octave, and all other respective values obtained for all other notes will be the same for all octaves.

¹⁷Эта величина для "G" будет той же самой для каждого "G" в каждой октаве, и все другие величины, полученные для других нот, будут теми же самыми во всех октавах.

a) the pitchwheel value of C (8192 [0]) is the starting point;

b) the number of Cents required to arrive at G is log(3:2) * (1200/log 2);

a) значение колеса (pitchwheel) для C (8192 [0]) является исходной точкой;

b) число центов, необходимое для того, чтобы достичь ноту G из ноты С есть
log(3:2) * (1200/log 2);

c) the difference between what G would have been (700 Cents greater than C) and what it now is as the result of b) above, is the total Cents value given in b) minus 700 (the number of Cents it would have been);

c) Разность между предустановленной (в равномерной темперации) центовой величиной для G (она там на 700 центов выше, чем С) и центовой величиной для G в Пифагорейской настройке есть разность между величиной из пункта b) и величиной в 700 центов;

d) this difference is multiplied by 81.92 (the pitchwheel value of each Cent);

d) эта разность умножается на 81.92 (число единиц колеса, соответствующее одному центу);

e) the result of d) above is added to 8192 [0] (which would have been the default value).

e) результат пункта d) прибавляется к величине 8192 [0] (которая является исходной точкой для "колеса").

The complete formula for this calculation is therefore as follows:

for older versions: 8192 + ((((log(3/2)) * (1200/log(2))) - 700) * 81.92) = 8352.

for later versions: 0 + ((((log(3/2)) * (1200/log(2))) - 700) * 81.92) = 160.

Таким образом, полная формула для этого вычисления будет следующей:

для более ранних версий: 8192 + ((((log(3/2)) * (1200/log(2))) - 700) * 81.92) = 8352.

для более поздних версий: 0 + ((((log(3/2)) * (1200/log(2))) - 700) * 81.92) = 160.

The pitchwheel setting for "G" is therefore 8352 (older versions) or 160 (later versions).

Значение колеса для "G" равно, следовательно, 8352 (для более старых версий) или 160 (для более поздних версий).