Tuning Recipe

There’s tuning and there’s tuning. Sometimes, I will be asked to tune a specific temperament. Though there are official recipes for such things, usually I just tune according to the following method.  By simply adjusting the size of the fourths or fifths, I can achieve a gradient of tunings ranging from 1/4-comma meantone to (nearly-)equal temperament, with everything in between, including 1/5- and 1/6-comma meantones with or without specific enharmonic accidentals and circulating well-temperaments of various stripes. Notice that there are no beat counts here.  They’re hardly ever necessary*.

I’ve annotated this procedure meticulously. I wish when I was learning how to tune someone had given me such precise advice rather than just beat counts. Pitches are designated according to the Helmholtz system. Intervals will be specified in tuned–tuning order; this is the order that one staggers the interval when playing it (playing the reference pitch before the pitch to be tuned).

Phase I: Setting the Temperament, Sharp Side (8 strings, 1 minute)

1. Set a to the desired pitch standard.
   • This is the a below middle-C (c’), thus an octave below the normally cited reference pitch a’. Tuning in this octave is important because the intervals beat slower and it is therefore easier to make finer gradations.
2. Tune down a tempered (narrow) fifth a–d to the required size.
   • For 1/4-comma meantone, Praetorius advises that the fifths be tuned “as narrow as is tolerable”; this is good advice.
   • For a well-temperament that will circulate, temper them but slightly.
   • For 1/6-comma meantone and other in-between-ish tunings, choose an in-between-ish size of fifth.
3. Tune down a tempered (wide) fourth a–e to the required size.
   • Remember that a tempered fourth will sound more out of tune than an equivalently tempered fifth.
     • This is because the the fifth is wider than a fourth, which means the same amount of tempering is a greater proportion of the size of the interval for the fourth than it is for the fifth. In other words, the percent difference between a pure fourth and a ∂-tempered fourth is greater than the percent difference between a pure fifth and a ∂-tempered fifth.
   • For 1/4-comma meantone, tune the fourths very wide.
   • For a well-temperament that will circulate, the tempered fourths should beat similarly to 1/6-comma meantone fifths.
   • For 1/6-comma meantone, they should be about halfway or so between 1/4-comma fourths and pure fourths.
   • Getting used to knowing where the fourths should sit can be difficult at first. Remember, you can always tune a pure octave down from the reference pitch and then tune the relevant note as a fifth from below. Thus Step 3 becomes:
     • Tune a pure octave a–A.
     • Tune a tempered (narrow) fifth A–e to the required size (see notes to Step 2).
4. Tune up a tempered fifth e–b to the same size as the fifth in Step 2.
5. Tune down a tempered fourth b–f# to the same size as the fourth Step 3.
6. Check the size of the major third d–f#.
   • For 1/4-comma meantone, this should be pure (without beats).
     • If it isn’t, your fifths from Steps 2 and 4 weren’t narrow enough; or your fourths from Steps 3 and 5 weren’t wide enough; or both. Go back and adjust. Try tuning the fourths as fifths from below as described at the last bullet under Step 3.
   • For 1/6-comma meantone, this third should be sweet but slightly wider than pure.
   • For a well-temperament, this third should be zingy but not too wide. Wider than the major third on an equal-tempered piano would be “too wide.”
7. When you’re happy with your fourths and fifths in Steps 2–5 and the size of the third in Step 6, memorize the size of the major sixth d–b. This interval will be an important check for the rest of the tuning procedure.
   • Memorizing the size of an interval is really an exercise in memorizing the timbral quality of an interval as it relates to the timbral qualities of adjacent intervals. For example, it may be helpful to play the intervals d–b, d–a, d–g in quick succession in various orders to get an intuitive, aural sense of just where the pitches g, a, and b lie in relation to each other and in relation to the reference pitch d.
   • It is useful to memorize permanently the quality of variously tempered fifths and fourths to make Steps 2–5 proceed quickly and accurately; the sizes of fifths and fourths should be the same every time one tunes. The size of the interval check at Step 7 can be a bit more flexible at first; as long as it remains the same within a tuning (rather than between tunings) it will serve its function as a check interval.  In time, however, it is good also to memorize the size of various tempered sixths to increase accuracy and speed.
8. Tune a tempered fifth f#–c#’, checking a–c#‘ and e–c#’.
   • The qualities of a–c#’ and e–c#’ should match those of d–f# and d–b.
   • If you started with a 1/6-comma-ish fifth and wish your temperament to circulate, make this interval ever-so-slightly closer to pure (wider).
9. Tune a pure octave c#’–c#.
10. Check c#–e and c#–a; memorize the size of the minor third and minor sixth.
    • Again, this advice should at first be construed as “memorize the idiosyncratic size of the minor third and sixth for the purposes of this tuning session” and in time should be construed as “memorize the special size of the minor and major third for a specific temperament or tuning, to be called upon from memory each time one tunes.”
11. Tune a tempered fifth c#–g#, checking e–g#.
    • The minor third g#–b is narrow interval high in the tempering octave (d–d’). Consequently, it is (correctly) one of the noisier intervals and therefore less useful as a check.
    • If you’re trying to circulate, make this interval closer to pure (wider).
12. You have now tuned the sharp side of the circle. Congratulations.
    • Play some fourths, fifths, thirds, and sixths (from among the notes you’ve tuned, naturally).  Make sure they’re still right. Adjust them if they aren’t. If you’ve taken a long time to get to this point (five minutes or more), you’ll almost certainly have to retune something.
    • Ideally, you should be able to do Steps 1–11 in about a minute or two; certainly no more than three.

 Phase II: Setting the Temperament, Flat Side (7 strings, 1 minute)

14. Go back to d and tune a pure octave d–d’, checking f#–d’ and a–d’.
    • We tune the flat side of the circle from d so that any error in tuning aggregates at the wolf interval (e♭–g#)—thus sweeping it under the rug. If we tuned in one direction straight through the circle of fifths, any error would aggregate between d–g, awkwardly in full view.
15. Tune a tempered fifth g–d’, checking e–g and g–b.
16. Tune a tempered fifth g–c, checking c–e and c–a.
17. Tune a tempered pure octave c–c’, checking g–c’, a–c’, and e–c’. For circulating temperaments, also check g#–c’ (explained below).
    • Strictly speaking Step 16 is redundant but it is an important fail-safe. It ensures that the C is placed precisely. Make sure you are happy with the quality of the major third c–e (which should match d–f# in 1/4-comma or be ever-so-slightly sweeter than d–f# in other tunings), the minor third a–c’ (which should match f#–a or be a little dirtier), and the fifth c–g before proceeding.
    • If you are trying to tune a circulating temperament, you should have tuned purer (or even pure)—that is, wider—fifths in Steps 8 and 11. Check how well you did by seeing if g# can function enharmonically as an a♭ by playing g#–c’. The purer you make the fifths f#–c#’ and c#–g# (Steps 8 and 11, respectively), the better the major third a♭–c’ will be and the worse the major thirds a–c#’ and e–g# will be. This is the point in the method where you choose between a circulating well-temperament and a non-circulating meantone tuning. Adjust the tuning of the pitches or octaves c#–c#’, g#/a♭, and c–c’ in that order until you are satisfied with the all the relevant major thirds before proceeding.
      • A quick way to make a tuning retain much of its meantone quality and accommodate certain enharmonic usages is to “split the difference” between g#/a♭ here in Step 17 and e♭/d# (below, Step 20). To do this, tune g#/a♭ smack-dab in between where true g# and a♭ lie using the major thirds e–g# and a♭–c’ as reference points: the idea is to make these major thirds equal in size, or, if you choose, to make the g# sufficiently high to make a♭–c’ minimally tolerable and e–g# minimally intolerable.
18. Tune a tempered fifth c’–f, checking c–f, f–a, and d–f.
19. Tune a tempered fourth f–b♭, checking d–b♭, g–b♭, and b♭–d’.
    • If you want your temperament to circulate, make this interval purer (narrower), taking care to check for a tolerable third b♭–d’.
20. Tune a tempered fifth b♭–e♭, checking e♭–g, e♭–c’, and c–e♭.
    • For circulating temperaments, make this interval purer (wider), checking also e♭–g#/a♭.
    • Even in meantone, you may wish to tune the fifth b♭–e♭ purer (wider) to accommodate the enharmonic use of e♭. To do this, first tune a pure octave b–B. Having tuned a tempered fifth b♭–e♭, lower e♭ to make B–d# minimally tolerable and e♭–g minimally intolerable. Note that if you split e♭/d# and g#/a♭ exactly down the middle, the wolf interval e♭–g# will become nearly tolerable. This is because splitting e♭/d# and g#/a♭ moves these notes with opposing tendencies. You can further mitigate the wolf in meantone by making e♭ still closer to d#, or g# still closer to a♭, or both. Remember to check the relevant thirds and sixths as you do so.
21. You have now tuned the flat side of the circle. Congratulations.
    • Play fourths, fifths, thirds, and sixths between c–d’ (or perhaps as low as A or B if you’ve tuned those notes as suggested provisionally above). It is useful to play through the tuning recipe with all the tuning intervals and checks in order as a final check of the temperament. As you go along, adjust anything that is out. This step effectively directs you to go back to Step 1 and start all over again. If you’ve done everything accurately and quickly, you won’t have to adjust anything. If you didn’t, now is the time to catch it before you go and tune the rest of the keyboard.
    • When you believe an interval is wrong, check two other intervals: each of these intervals should share exactly one pitch with the offending interval and be as close to d (by fifths) as possible. This will enable you to determine quickly which of the notes in the offending interval is out, or indeed whether both are out. For example, if f#–a offends, check d–f# and d–a; if e–a offends, check d–a and c–e; and so on. Once the offending pitch is identified, revert to the appropriate step in the tuning recipe and retune from that point all the way to the end, with all the checks (usually only a minor adjustment is necessary).
    • Play various major and minor triads and check that you are satisfied with their qualities. If you aren’t, identify the offending interval within the triad, then the offending interval within the pitch as described above, and then revert to the appropriate step in the tuning recipe to retune (with checks).

Phase III: Tuning the Compass (<50 strings, 5–10 minutes)

22. Starting from b, tune pure octaves b–B, b♭–B♭, a–A, chromatically downward to the bottom of the compass, checking fifths and major or minor sixths.
    • Tune downwards first because these strings are longest and heaviest and therefore have the greatest potential to change the shape of the case and soundboard as they pull against the hitchpin rail and bear down on the bridge and the nut, thereby affecting the tuning. Once the fine-tuning of the highest strings is complete, we don’t want to move the great lumbering strings of the bass.
    • Choose whether to check a major or minor sixth according to what makes most diatonic sense; put another way, the second note in the check interval should be as close to D (the center of the tuning) as possible. Thus when tuning a C#, check a minor sixth; when tuning a C, check a major sixth. When playing check intervals, play both notes of the octave simultaneously (e.g., C–A–c). If something offends, then play the simple diad on either side of the check note (e.g., C–A, then A–c) to isolate which note offends. Retune the appropriate octave(s).
    • It is less useful to check thirds and fourths as these rumble vociferously in the lower register; but occasionally they may prove useful as a double or triple check.
23. Starting from e♭ and through about g”, tune pure octaves chromatically upward to the top of the compass, checking fifths and major or minor sixths and thirds using the “St. Anne Check.”
    • Taking g’ as an example:
      • Tune g–g’.
      • In tempo and in order, play g–d’–g’, g–b–g’, g–e’–g’, g–d’–g’, g–g’.
      • If something offends, play diads to isolate the offending note. Retune the appropriate octave(s).
    • The St. Anne Check checks the tuning of d’–b, b–e’, and e’–d in a melodic context, each of these constituent pitches in the important g–g’ harmonic context, and, crucially, g–g’ in three distinct harmonic contexts. Thus the St. Anne Check attunes the ear to both the melodic and harmonic components of the tuning, ensuring that it will be musically satisfying (not merely acoustically correct).
    • Iterated over the chromatic compass of the keyboard, the St. Anne Check makes it very difficult to miss a mistuned interval.
24. From about about c” (the top one-and-a-half or two octaves of the compass), start checking fourths systematically too using the “Birdland Check.”
    • Taking g” as an example:
      • Tune g’–g”.
      • In tempo and in order, play g’–b’–g”, g’–c”–g”, g’–d”–g”, g’–g”.
      • If something offends, play diads to isolate the offending note. Retune the appropriate octave(s).
    • You should also continue to check sixths using the St. Anne Check in tandem with the Birdland Check for fourths as you tune c” through about g” as octaves from below.  Above g” sixths are less are a less useful check and you can dispense with the St. Anne Check.
25. From about g”, that is, the top octave of the keyboard, do not tune octaves. Rather, tune the notes of this octave as tolerable fifths from below, checking fourths, thirds and sixths below using the “Westminster Check.”
    • Taking a” as an example:
      • Tune d”–a” such that is is qualitatively similar to d’–a’ (the fifth an octave lower).
      • In tempo and in order, play c#”–a”, e”–a”, f#”–a”, d”–a”, a’–a”.
      • The sixth, fourth, third, and fifth should not offend. If any does, retune the offending octave(s).
      • The octave a’–a” (“Big Ben” if you like…) should be fractionally wider than pure. “Stretched” octaves from about g” to the top of the compass sacrifices harmonic tuning in favor of melodic tuning for the highest register (where one is unlikely to play octaves anyway).
26. You have now tuned one whole choir.
    • Play triple or quadruple chromatic octaves up and down the keyboard to find any stray notes out of tune and indeed to try to knock the instrument out of tune. If it’s liable to slip, it might as well slip now when you’re there to fix it.
      
    • If something offends, play left hand and right hand separately to isolate the offending octave; then play fourths or fifths to isolate the offending pitch. Retune the false note as a pure octave from below, using the appropriate checks.

Phase IV: Tuning the Remaining Choirs (>50 strings, 5–10 minutes)

27.  Engage a second register and tune chromatic unisons first down from f, then up from f#.
    • On a two-register instrument, I like to tune the back 8′ first and then tune the front 8′ second. On a three-register instrument I like to tune the front 8′ first, the back 8′ to the front 8′ and then 4′ to the front 8′ (with the back 8′ off).
    • Remember, it is important to tune the lower register before the higher register for the reason given in Step 22.
28. Disengage the first register, and play checks throughout the whole compass of the second register only (the register just tuned). Adjust anything that seems out.
    • You should be aware of a curious phenomenon. Having tuned beautiful octaves, fifths, fourths, and thirds in your first register and having tuned beautiful unisons in your second register from the first register, you turn off the first register and check the fourths, fifths, sixths, and thirds in the second and find that they are all awry. This is because there is actually a fairly wide tolerance in the tuning of a unison: strings which are almost-but-not-quite tuned as unisons will pull each other into tune as they vibrate (this phenomenon is known as harmonic entrainment). But when played with strings of other pitches, they are not so pulled into tune and sound funny.
      • One way to deal with this is to make very minor adjustments to the fourths and fifths of the second register independently. The kinds of adjustments that you may need to make will be minuscule. Sometimes it simply matters whether a note was moved into tune from above or from below. Once you move something in the second register, turn the first register back on to check the unison.
      • Another, slightly safer, way to correct any errors of this kind is to find the offending interval in the second register and determine which note needs to move, and in which direction by playing various check intervals. Then, turn the first register back on and retune the offending pitch as a unison to the first register. For example if a fifth needed to be slightly wider in the second register, retune the top note of the fifth as a unison to the first register by nudging it sharper without spoiling the unison. You will find that there is some play in the unison: you can move notes very slightly sharper or flatter without making them sound out-of-unison with the first register.
        
    • Scrupulous checks of the 4′ are often superfluous, but may be helpful in tuning a troublesome pitch. Sometimes a 4′ string will have a difficult timbre which obscures its pitch. Other times the ear is fatigued by this stage in the tuning (usually you will have been tuning for about 15 minutes by this time). Doing something clever like turning off the 8′ choir and tuning an octave or a fifth directly in the 4′ choir could help get around some difficult corner. It’s never worth bashing your head against the wall; if something isn’t working, stop and simply do something else. Namely:
    • Remember that for any given note on the keyboard, you can always tune it as an octave, sixth, fifth, fourth, or third from above or below, or as a unison or octave from another choir.

*Beat counts are necessary when hired to tune a specifically requested temperament to “textbook” standards (for example, if someone requests Vallotti centered on G with a raised F#). It is also useful as a way to set intervals perfectly as a starting point while learning to memorize the quality of the various intervals in the various temperaments.