1. Of the many available music notation programs, my preference for Finale here is only because it is particularly well suited to combining visual analysis with superfine pitch control. But its method is unique, and files converted into, say, Sibelius will not retain the MIDI settings that have been applied. Although Finale has been upgraded regularly, most of the changes have been designed only to enhance user friendliness and to present better looking windows and scores. The window dressing aside, the functionality of the program has remained fairly stable. But in the specific area to be discussed below later versions of Finale have adopted a slightly different (and perhaps more logical) method of data management. This will be explained in due course.
2. Readers might like to know that the writer does not regard himself as a technological boffin, and they might perhaps be comforted (or irritated if they expect boffin-style discourse) by the assurance that explanations will be simple and comprehensible.
3. If your keyboard provides other preset temperaments such as "Pythagorean" or "Just", you should make sure that it is only set to equal temperament. All the settings that will be provided in this essay take the equally-tempered keyboard as the default. None of the music to be generated will actually use equal temperament of course, but the pitchwheel settings presume this as being the default.
4. You can verify this quite simply. Assuming your MIDI keyboard is set up to generate sounds through your speakers, hold down any note and move the pitchwheel fully to the right and then the left. The pitch of the note should rise and fall by a full octave. If it is different this does not matter, and it merely indicates that the default setting has been changed. The setting will again be changed later to another value in any case.
5. As will be shown, early versions of Finale provided both Staff Expression and Score Expression tools, while later versions offer only a common and single Expression Tool. In these later versions, setting of the required entry for either "Staff" or "Score" is made via various sub menus. The functionality of whatever entry is made remains identical however. A less useful method of pitch control is to open the Midi Tool and edit the pitchbend values of notes that are to be changed. This is far less accurate and anything that is done remains invisible except to the inner workings of the program. The more accurate method outlined here remains visible in the score, and can easily be edited or changed.
6. If the score is empty, perhaps because the file is newly created, simply insert a note. Then select the Staff Expression tool icon and click on the note.
7. Now that the procedure for editing and inserting pitchwheel values has been followed, the various "Cancel" buttons can be clicked to close down the Staff Expression tool so as to avoid changing the note originally selected. In order to rehearse the procedure again, the instructions given in  above can be repeated.
8. These are the new values that Finale automatically converts to when a file created in what it recognizes as an "older version" is loaded into the later version. In such a file, later versions will convert original pitchwheel values of 8192 to 0, 0 to -8192, and 16384 to (+)8192. All resulting values in the later versions will therefore be equivalent to all those in the early version minus 8192.
9. It is, however, much better than what is provided by the MIDI tool which only splits the same pitchbend interval into 64 equal parts. This is wholly inadequate for pitches of the accuracy required.
10. The keyboard User Manual will give the information needed for altering the bend value. On my Roland the bender range can be accessed via the "Edit" and then the "Lower" buttons, and found by scrolling through the list with the "Display" button. When "Bender Range" is displayed, its value can be set to "1" with the "Value" button and the new setting saved using the "Write" and "Enter" buttons. Other keyboards will have equivalent methods of making changes to these settings.
11. The Bend Range value increments by semitones so that the default value of "12" yields a pitch bend of a full octave up and down. By changing this setting to "1" the bend range will be reduced to only a semitone up or down.
12. Keyboards always adopted "compromise" tunings although some experiments with split keys were designed to increase the number of pitches available for each octave. This facilitated a differentiation between sharps and flats so as to enable the consistent use of pure thirds on all degrees. Unaccompanied vocal performance, however, did not need to compromise and comma inflection was a normal part of a singer's technique. This might require the same note to vary in pitch (according to context) from its normal Pythagorean position to a comma higher or lower. Frequently, but under tight compositional control, this might be extended at times up to two commas higher or lower than Pythagorean pitch. But such flexibility was completely out of the range of normal keyboards whose tuning (in whatever compromise system was adopted) had to remain entirely fixed for a particular performance.
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.
14. As stated in  above, "8192" here refers to the default numeric value for "older versions" and the following "" that for "later".
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.
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.
17. This 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.
18. Again the numeric value for later versions is less than for older ones by a total of 8192. This, like all other pitchwheel values, is calculated to the nearest 1 in 8192 parts of a semitone. Equally-tempered fifths have been narrowed from their Pythagorean defaults. Since the pitchwheel value of C is 8192 , the restoration of the G to its Pythagorean position will necessarily increase its pitchwheel value. The new value of 8352  has raised the pitch by a pitchwheel value of 160. When this is divided by 81.92 (the pitchwheel value for each Cent as shown in  above), the difference is found to be 1.953125 Cents.
19. If, for example, you indicated F-sharp with the tag "F#", Finale would show "9152" because it would assume that you had asked for the numeric value assigned to be displayed in place of the tag itself.
20. Experienced Finale users will know that the process of inserting Text Expressions can be simplified and speeded up by the use of metatools. Since most pieces using Pythagorean tuning only have a range of up to nine different pitch classes in each octave, assigning each global pitch to a metatool makes insertion very quick and easy. Other users will simply have to make a separate insertion (via the Staff Expression window) for each and every note in the score.
21. In the Instrument List window it will be necessary to assign a different instrument for each staff, and then a different channel. In the "Prog." column the same sound patch can be selected if wished.
22. This number can, of course, be changed to whatever value you like dependent upon the pitch needed relative to any tuning requirement or system you wish to use. It will be set for the entire piece, however, and the current value is the one needed for its Pythagorean pitch.
23. The figure "480" indicates the same pitch inflection within the 0-8192 numeric range (used by later versions) as does the figure "8672" within the numeric range 8192-16384 (used by older versions), representing simply an increase in pitch bend of 480 units.
24. It is "non-melodic" since it was considered too small to form a discrete interval in its own right. But when it was added to or subtracted from a larger interval the overall result did yield a new melodic interval. When added to a Pythagorean minor semitone, the result was a major semitone (16:15) which was a distinct interval capable of being leant and performed. Also when it was subtracted from a Pythagorean tone (9:8) the result was the minor tone (10:9) which again was a discrete interval to be learnt and sung. The addition or subtraction of this quantity is what gives rise to the new intervals whose cognition and accurate delivery lies at the heart of the skills required for the accurate performance of Just Intonation.
25. Both these calculations are, of course, for the older versions of Finale. For the newer versions (which will upgrade existing old-version files automatically) the figure "8192" will be replaced with "0", and the results of the above calculations will be respectively "-1762" and "1762"
26. In order to produce a clean and pure Syntonic intonation, you are strongly advised to eliminate completely any default vibrato that will undoubtedly be set on your MIDI keyboard. Since even the most gentle vibrato is unlikely to create pitch fluctuation of less than a Syntonic comma, it stands to reason that the presence of such vibrato will sabotage any attempt to enter the true sound world of Syntonic tuning. On some keyboards, even the "harpsichord" patch is infected by vibrato. Removing it is straightforward, but reference should again be made to the User Manual or technical assistance sought.
27. If you load it into one of the later versions of Finale, you will find that the values have automatically been updated to those contained in Figure 12 above.
28. The issues surrounding this composition are too complex to be discussed here and will form the substance of a future article. But his description of the keyboard upon which he stipulated the chanson could be successfully performed, while remaining in tune throughout, shows it to have been similar to the one discussed by other musicians of the time (notably Vicentino).
29. The tones must be smaller because if the octave equals nineteen intervals of one-third of a tone each there must be at least six and one-third tones in each octave. The Pythagorean octave contains only five tones plus two minor semitones (which is less than six overall). The keyboard tones described by Costeley must therefore be minor tones roughly equivalent to the Syntonic (10:9) minor tones. In compensation for this, the diatonic semitones are large (two-thirds the size of a tone in fact), and these, too, must correspond roughly with the Syntonic diatonic semitones (16:15) although they are marginally larger in size. The occasional use of non-diatonic semitones (used either melodically or arising from close cross relations) provides, according to Costeley's assessment, movement equaling only one-third of a tone.
30. In this temperament, the interval of a fifth is perceptibly narrowed (much more so indeed than in 12-tone equal temperament), and the fourths are correspondingly widened by a no less perceptible amount. The overall effect of this temperament is radically different from that of 12-tone equal temperament. In the latter, the effect (as we are used to) is one of tolerably pure fourths and fifths together with pure octaves, upon which have been superimposed distinctly out-of-tune thirds and sixths (to which our ears have now become accustomed). In the former, however, the soundscape is one in which the fourths and fifths are sufficiently out of phase to produce a wavy timbre (similar to a gentle voix celeste organ sound), upon which has now been superimposed strikingly pure thirds and sixths.
31. There will need to be twenty-one separate values because the B-sharp/C-flat and E-sharp/F-flat pairs will each need different values for both notes (even though whichever is used will provide the same actual pitch). Sometimes Finale will have to read F-flat, and sometimes E-sharp: the first (= E in 12-tet) will have to be raised from its default pitch while the second (= F in 12-tet) will need to be lowered. Whichever is used in the version edited for keyboard will depend solely upon its diatonic context: if G-flat descends by a diatonic semitone (= two-thirds of a tone) the following pitch will be written as F-flat; but if a D-sharp rises by a diatonic semitone the following note will appear as E-sharp. Finale therefore needs to have both members of each pair provided for.
32. The appearance of the score will sometimes look strange, and somehow "between the cracks", to those thinking in 12-tone equal temperament. Frequently chords appear as though they need re-enharmonizing. In reality, however, each note is completely pitch-specific and relates to a particular key on the keyboard. Only the single black notes dividing the B/C and E/F pairs have alternative means of notation.
End of footnotes