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Padgham's 'Well-Tempered Organ' Colin Pykett
Posted: 10 January 2013 Last revised: 10 January 2013 Copyright © C E Pykett 2013
Abstract. Padgham's 'The Well-Tempered Organ' rightly holds its place as a classic text on tuning and temperament for the organ, but it is surprising that its many numerical errors seem to have attracted little attention or comment. Consequently this article was posted so that others might benefit from the corrections compiled over some years. Additional information has also been presented, including tuning data which are better suited to modern digital instruments and tuning devices.
Errors notwithstanding, Padgham's attempt to rate each key and each temperament on the basis of single numbers is considered futile, especially as it was based on a comparison of their intonation errors against an unusable standard of little practical value (Just Intonation). The outcome of the attempt, that no temperament rates better than Equal Temperament, was therefore not surprising. In fact it is absurd because it undermines Padgham's reasoned arguments elsewhere in his book that the advantages of many unequal temperaments outweigh their shortcomings.
Should the book be reprinted, the article might be of value to the publishers.
Contents (click on the headings below to access the desired section)
Significant errors in temperament data Additional tuning data - cents from Equal Temperament
'The Well-Tempered Organ' [1] is a book on tuning and temperament by the late Charles Padgham who died in 1999. It was published subsequent to an earlier article not easily obtainable in the public domain [2], and it presents detailed numerical data on 22 different temperaments together with instructions for tuning 18 of them. At the time of writing Padgham was at the City University London (not part of the University of London) and his book has become well known among those with an interest in temperament, having been cited frequently in the learned literature and elsewhere. It was accompanied by an interesting cassette tape containing recordings of pieces played on a pipe organ using five of the temperaments described in the book [3].
This article addresses several issues. Firstly, the book contains a large number of numerical errors, some of which are significant enough to prevent the associated temperaments being set up successfully (the Werckmeister III and Young temperaments are examples). Secondly, some of the data are not self-consistent. For example, the circle of fifths presented for some temperaments does not quite 'close'. The tempering of the twelve fifths in the circle must always compensate exactly for the Pythagorean Comma if the octaves are to remain pure. This does not always happen because of Padgham's rather slapdash fondness for simple fractions rather than decimals to express the frequency deviations of tempered fifths from the pure intervals. Thirdly, a cornerstone of the book is his somewhat curious derivation of a single number for each temperament to indicate how 'good' it sounds to the ear. Because the criterion of goodness was Just Intonation, an entirely impractical temperament in which the majority of keys are poor or unusable, his methodology here is open to legitimate criticism.
Besides considering these issues, the article also extrapolates some of Padgham's data (when corrected) to make his book more useful to modern readers. When it was written, electronic tuning devices were still coming into use and digital musical instruments scarcely existed beyond the Allen organ and a few synthesisers. Therefore he concentrated, understandably, on instructions for tuning temperaments by ear in the time-honoured fashion by counting beats between intervals (though there are also some numerical errors here as well). With the passage of nearly thirty years, it is appropriate to add some material which will facilitate setting up the temperaments he described on digital instruments. These commonly require the deviation in cents from Equal Temperament for each note in the octave, so tables containing these numbers are given later. The numbers have been corrected for the errors in the original data where these have been identified. Data in this format for two historical English temperaments which Padgham discussed (the Finchcocks and Oakes Park temperaments) are also included because he did not give any tuning instructions for them at all, and the data he did provide were of inadequate numerical precision in any case. These tuning tables are also provided here for another reason, not connected with Padgham's book, because much of what one finds on the Internet for tuning unequal temperaments, particularly in Wikipedia, is incorrect.
Another feature of the book also deserves comment. Like some other authors on temperament both then and today, Padgham seemed obsessed by the subject to the extent he fell victim to his own zealotry. One of his more outrageous statements accuses those whom he saw as restraining the widespread adoption of unequal temperaments of "conservatism, fear of the unknown and ignorance". However I do not wish to appear hypercritical, and to balance the foregoing remarks I would point to the fact that the book rightly continues to enjoy a status today. In particular it is not the intention to inhibit use of the book, nor to suggest for a moment that it should not be purchased (though it is thought to be currently out of print). On the contrary, it is in some ways an excellent introduction to the subject of temperament especially as it relates to the organ, and I have no problem with it continuing to sit on my shelves. This is why I point out in this article how users of Padgham's book can augment the value they derive from it. Therefore I trust that readers will find this article not merely polemical but one that offers solutions rather than the temperamental rhetoric with which some other authors seem satisfied.
This section addresses issues generic to the book as a whole rather than those specific to particular temperaments. The latter are discussed later.
Surprisingly, some aspects of the method followed by Padgham while deriving the numbers in his book have to be largely guessed at. He does not discuss in detail the route he followed in deriving the huge amount of numerical data which he presents, though there are hints at various points. But he could not have derived the 'Temperament in cents' data tables (see below) from the circle of fifths he presents because some of the latter do not 'close' to exactly compensate for the Pythagorean Comma. Nor does he reveal anything of the tools he used - how the numbers were actually derived. It would have been good to know whether he used spreadsheets or wrote computer programs for example [4]. It is possible that a variety of methods were employed, because a blemish of the book lies in its varying degrees of numerical precision which only serve to illustrate an unfortunate lack of rigour. For instance, at some points he presents frequency ratios in cents (correctly) to two decimal places, yet at others they are represented (inadequately) only as integers. It is also unclear how the numbers were typeset, an important issue given the error frequency. If spreadsheets had been used one imagines they could have been printed in facsimile even if computerised typesetting was not available. However this was probably not done, because errors elsewhere suggest (oh, please not!) manual typesetting - one number whose correct value is 498.04 appears as 489.04, to take just one instance of the problem. Other similar examples also suggest an inadequate level of proof reading.
The circle of fifths presented for some temperaments does not quite 'close'. The tempering of the twelve fifths in the circle must always compensate exactly for the Pythagorean Comma (23.46 cents) if the octaves are to remain pure. This does not always happen because of Padgham's use of simple fractions rather than decimals to express the deviations of tempered fifths from the pure intervals. If taken at face value, these numbers would result in several significant errors when setting the temperament, so it is surprising that (mostly) these consequential errors do not arise! Although this is fortunate, the inconsistency needs to be pointed out so that users of the book can be on their guard.
Padgham does not tabulate the absolute note frequencies for any of his 22 temperaments, which is disappointing because they are useful as a reference in much temperament work. However he must have derived them for his own purposes because he defines that of middle C when explaining how to tune each temperament. Of course, it is necessary to decide first on a pitch standard, and Padgham uses A = 440 Hz throughout.
Nevertheless, the circle of fifths for each temperament is provided, so it might be helpful to outline how to derive the note frequencies from that. Start with A (440 Hz or whatever you prefer) and first calculate the frequency of E above. This equals 1.5 times that of A, a value which then has to be adjusted by the tempering (if any) applied to the interval A to E. Then proceed to the next note (B) clockwise round the circle and repeat the process until D has been reached. Finally, each of the note frequencies so calculated is then divided one or more times by two to bring it back into the same octave as A. Note that the cent values in the circle of fifths must be calculated to two decimal places, though as mentioned above, one cannot always rely on the simple fractions of the Pythagorean or Syntonic Commas presented by Padgham when they are converted into decimals. One of several examples is the 'Wolf' interval (35.66 cents) in the Quarter Comma mean tone temperament which cannot be represented accurately as the simple fraction of 1 ½ times the Pythagorean Comma which Padgham quotes. Therefore always check that the circle you are using does 'close', such that the sum of its temperings compensates exactly for the Pythagorean Comma (the temperings must add up exactly to -23.46 cents).
'Temperament in cents' tables 'Temperament in cents' is Padgham's title for his tables giving the number of cents relative to C for each note in the octave. Thus C will always have a value of zero, and the other values will increase depending on the ratio of their frequencies to that of C. Some of these tables contain errors and these are highlighted later in the article. The tables also demonstrate the inexplicable variations in numerical precision which recur throughout the book. Some of the numbers are presented to two decimal places whereas others are rounded to integers. The latter are unsuitable as a basis for accurate temperament work [5].
It is unclear how Padgham derived the numbers in these tables. He probably did not derive them from the circle of fifths which he presents adjacent to each table because some of the tempering data in the circles do not 'close' accurately to the Pythagorean Comma, as mentioned already.
The errors in the tables are unfortunate because they propagate into subsequent steps, as explained presently. For the moment, bear in mind that these values are also important for those who need to know by how many cents each note in an unequal temperament is offset from Equal Temperament (ET). Some electronic tuning devices require this information, as do those digital musical instruments which can only be re-tuned by specifying an offset for each note in cents from ET. Taking the numbers at face value in his book, Padgham makes it impossible to set up accurately the French Ordinaire version I temperament on such a digital instrument. The Werckmeister III and Young temperaments cannot be set up at all because of the gross errors in their respective tables.
'Intervals in cents' tables 'Intervals in cents' is Padgham's title for a large table containing 144 numbers for each temperament. All the numbers draw on the data in the associated 'Temperament in cents' table (see above), and they are used as the first stage of estimating his 'accumulated errors' tables (see below).
Because of the errors in the source data (from the 'Temperament in cents' table), together with possible undetected typesetting errors in these vast tables, there is little point spending time discussing the matter further in this article.
'Accumulated errors in scales' tables 'Accumulated errors in scales' is Padgham's title for a third table presented for each temperament. Each table consists of 24 numbers, one for each major and minor scale, illustrating how the intonation of each scale departs from Just Intonation. The numbers in the tables result from arithmetical manipulation of those in the 'Intervals in cents' tables (see above), and because these themselves might contain errors, there is no value in considering this step further as far as intrinsic error is concerned.
Errors apart, there is also a major point of principle associated with this step. In it, Padgham derives a single number for each scale which he suggests will indicate how 'pure' that scale sounds to the ear - the subjective goodness of its intonation. At first sight it is a beguiling idea and I have tried it myself in the past for various temperaments I have designed or modified. Having done so, I am now of the view the concept is flawed because he derives the number by comparing each scale in a given temperament with its analogue in Just Intonation in which the majority of the keys are unusable. How can an unusable and utterly impractical temperament such as Just Intonation possibly be used as a standard when trying to assess the subjective virtues of another temperament as it sounds to the ear? Padgham himself says that "just intonation is a myth", and elsewhere "there may be some objection to our use of just intervals as the standard or goal, and in fact it is not always clear exactly which just intervals to use, especially for the semitones ...". Quite right - QED - argument proved. Surely it would have been better to have compared the intonation of each of the unequal temperaments to Equal Temperament? I have not done this, partly because I dislike the general approach in any case, but were it to be done it is possible that the 'intonation numbers' which result might show which temperaments are better, overall, than ET and those which are worse. This would be a much more useful outcome for the practical musician than comparing them to Just Intonation, a temperament which cannot be used.
It is worth pointing out that some subsequent authors have imported Padgham's 'accumulated errors in scales' results into their own work. One was Christopher Kent in his article for The Cambridge Companion to the Organ [6].
'Overall rating of temperaments' table From the 'Accumulated errors in scales' data, Padgham derives a further table in which he rates each temperament against Just Intonation using just a single number. This concept of itself provides enough fuel to power endless argument. Because the table represents the culmination of all his earlier numerical manipulations, it can only be regarded as a most disappointing and frankly meaningless end point. Even disregarding the errors in the data on which it is based, a single number can convey nothing of the subjective effect of something as complex as a musical temperament. I long ago ceased struggling to find any worth in the ranking, because no unequal temperament scores better than Equal Temperament, 7 are the same and no fewer than 14 are worse. To add insult to injury, Fifth Comma Mean Tone ranks worse than Just Intonation! Therefore these unfortunate results negate the vigorous qualitative arguments pursued by Padgham in favour of unequal temperaments with which many others agree. It is a most unsatisfactory outcome to the numerical methodology espoused by the book.
'Tuning Procedure' data For most of the temperaments discussed in the book, Padgham discusses an appropriate 'Tuning Procedure' largely based on the alternate fifths and fourths commonly used by tuners when setting a temperament by ear. Thus it was necessary for him to present the beat frequencies for each tuning interval as well as define the absolute frequency of at least one note. He starts with middle C in all cases and states what frequency it should have so that the A above will have a frequency of 440 Hz. He also specifies the difference in cents between middle C for the temperament in question and that for Equal Temperament.
In one case (the Ord temperament) there is a major error in the data, and in nearly all cases several of the beat frequencies (defined in terms of beats per minute) differ by one or two from the correct value. This strongly suggests either that Padgham computed truncated rather than rounded values, or that he was operating on insufficiently precise data. Although these beat rate errors might seem small when taken in isolation, it should be remembered that several of them will accumulate over the tuning process. Thus the final 'check' interval might turn out to be uncomfortably inaccurate in some cases.
Significant errors in temperament data
We discuss here the more significant errors which have been identified for certain temperaments. If a temperament in the book does not feature below, this means no issues were detected in that case. Occasional instances of minor errors of less than half a cent, one beat per minute, etc were disregarded except when they occurred repeatedly within the same data set.
The Wolf interval in the circle of fifths is represented inaccurately, thus the circle does not 'close'. The correct value is 1.52P where P is the Pythagorean Comma. Padgham's value of 1½ P shown in his circle of fifths is an approximation which, if used, would throw six of the values in the 'Temperament in cents' table out by about 0.5 cents. This would be an unacceptably large amount, a conclusion which Padgham would have agreed with - elsewhere in the book he said that that frequency ratios in cents must be calculated to two places of decimals (" ... necessary if one is to compound intervals, or for example if one needs to calculate beat rates ... ").
The Wolf interval in the circle of fifths is represented inaccurately, thus the circle does not 'close'. The correct value is 1.02P where P is the Pythagorean Comma. Padgham's value of 1P shown in his circle of fifths is an approximation which, if used, would throw six of the values in the 'Temperament in cents' table out by an unacceptably large amount.
Several values under 'Tuning Procedure' are slightly in error. These relate to the 'beats per minute' values which are shown corrected in bold type below. The row of numbers here correspond one-to-one to those printed in Padgham's book:
Sixth comma Mean Tone (Silbermann) Several values under 'Tuning Procedure' are slightly in error. These relate to the 'beats per minute' values which are shown corrected in bold type below. The row of numbers here correspond one-to-one to those printed in Padgham's book:
One value (for F) in the 'Temperament in cents' table is grossly in error. Written as 489.04, it should be 498.04. In no way can the temperament be set up correctly if the given number is used. The juxtaposition of the two digits strongly suggests a typographical error which was not picked up by the proof reader. However this leads one to anticipate multiple errors elsewhere because it suggests the entire book was manually typeset. One is drawn closer to this unfortunate conclusion when noting that the number in question mysteriously reappears correctly in the 'Intervals in cents' table! For reasons such as this, I abandoned any attempt to identify errors in this latter table nor in subsequent data which depend on it. This applies to the 'Intervals in cents' tables and consequential data for all temperaments, not just Werckmeister III.
Several values under 'Tuning Procedure' are in error, in one case badly so. These relate to the 'beats per minute' values which are shown corrected in bold type below. The row of numbers here correspond one-to-one to those printed in Padgham's book:
The data here are seriously in error, not so much because of the errors in individual values but because there are so many of them. Therefore the cumulative effect is significant.
Correct values for the 'Temperament in cents' are shown below. Other than that for C, the printed numbers are all wrong :
Several values under 'Tuning Procedure' are in error. These relate to the 'beats per minute' values which are shown corrected in bold type below. The row of numbers here correspond one-to-one to those printed in Padgham's book:
The frequency of C is 8.80 cents sharp to Equal Temperament, not 8.86 cents as stated in the book.
No tuning data were given in Padgham's book for this temperament. Moreover all the 'Temperament in cents' values are rounded to the nearest integer. This is curious since most of the comparable data for the other temperaments are quoted (correctly) to two decimal places. The correct values are in the table below:
No tuning data were given in Padgham's book for this temperament. Moreover all the 'Temperament in cents' values are rounded to the nearest integer. This is curious since most of the comparable data for the other temperaments are quoted (correctly) to two decimal places. The correct values are in the table below:
One value (for D) in the 'Temperament in cents' table is grossly in error. Written as 186.09, it should be 196.09. In no way can the temperament be set up correctly if the given number is used. The difference between the values strongly suggests a typographical error which was not picked up by the proof reader. However this leads one to anticipate multiple errors elsewhere because it suggests that the entire book was manually typeset. One is drawn closer to this unfortunate conclusion when noting that the number in question mysteriously reappears correctly in the 'Intervals in cents' table! For reasons such as this, I abandoned any attempt to identify errors in this latter table nor in subsequent data which depend on it. This applies to the 'Intervals in cents' tables and consequential data for all temperaments, not just Young's.
Several values under 'Tuning Procedure' are in error. These relate to the 'beats per minute' values which are shown corrected in bold type below. The row of numbers here correspond one-to-one to those printed in Padgham's book:
Additional tuning data - cents from Equal Temperament
Details are given here which might be useful for setting a temperament on a digital musical instrument which requires the deviation in cents from Equal Temperament for each note of the scale. The data might also be useful for certain types of digital tuning device. The values here take into account the errors referred to in previous sections of this article. The Pythagorean and Just Intonation temperaments do not appear in the table because they are of zero practical utility, though the Finchcocks and Oakes Park ones are included. Padgham gave no tuning instructions for these latter, which was a pity given the historical interest of the instruments concerned, at least to those attracted to the evolution of the English organ.
Each value has been rounded to the nearest integer because this is what many digital instruments require. Note that this is barely good enough for purposes of accurate tuning, though it will be passable for most stops in the first four octaves or so of the keyboard. However the errors will become noticeable for higher pitched stops towards the top of the keyboard. This is because rounding means that the error in each value can be up to half of one cent, thus for the top few notes on a 2 foot stop (frequencies around 8 kHz) this could result in beats of around 2 per second against the corresponding perfectly tuned notes. This can make upperwork, mutations and mixtures sound rough at times in view of the dissonances which can result between them and the better-tuned harmonics of lower pitched stops when notes are played in the upper reaches of the compass.
It has been assumed that the note A in Equal Temperament and in all the others is tuned to the same frequency, hence the appearance of zeros in that column.
Deviations of each note rounded to the nearest cent from Equal Temperament for the temperaments described in Padgham's book (referenced to A)
Padgham's 'The Well-Tempered Organ' rightly holds the place it has gained as one of the classic texts on tuning and temperament for the organ. I bought it when it was published in the mid-1980's and have used it frequently since. However on too many occasions I have been frustrated by the discovery of yet more errors, and it is surprising that these seem to have attracted little attention or comment. Consequently I decided to post this article in the hope that others might benefit from the accumulated corrections I have compiled over the years. At the same time additional information has also been presented, including tuning data which are better suited to modern digital instruments and tuning devices.
Errors notwithstanding, I also concluded that Padgham's attempt to rate each key and each temperament on the basis of single numbers was futile, especially as it was based on a comparison of their intonation errors against an unusable standard of negligible practical value (Just Intonation). The outcome of the attempt, that no temperament rates better than Equal Temperament, was therefore not surprising. In fact it is absurd because it undermines Padgham's qualitative arguments elsewhere in his book that the advantages of many unequal temperaments outweigh their shortcomings.
Like those in the the book itself, any collection of numbers is liable to contain errors, a reality which applies equally to this article. Therefore if readers detect any I shall be grateful to be informed so they can be corrected.
Finally, should the book be reprinted, the article might also be of value to the publishers.
1. "The Well-Tempered Organ", Charles A Padgham, Positif Press, Oxford, 1986. ISBN 0 906894 13 1.
2. "A Trial of Unequal Temperament on the Organ", C A Padgham, P D Collins and G K Parker, BIOS Journal (3), 1979.
BIOS, the British Institute of Organ Studies, was founded in 1976 and it has a global membership of something over 500. Its publications are not easy to obtain outside its membership and therefore they cannot be regarded as being in the public domain.
3. The recordings which accompanied Padgham's book are of great interest and they are probably unique examples of what various temperaments actually sound like on the same pipe organ when it is used to render a wide range of music from the 15th to the 20th centuries. One never ceases to marvel at the patience and skill of pipe organ builders when one realises that the instrument used was retuned at least five times in order to make this recording. I have now transcribed my copy of the tape to digital (mp3) format in view of the limited life of audio cassettes and the difficulty of finding decent equipment to play them on today. (My 25 year old cassette was suffering from 'binder ooze' which allowed the tape to slip under the capstan and thus rendered it unplayable. However it might be of interest that this can sometimes be cured temporarily by applying gentle heat - I attached the cassette to a domestic radiator for a day or two using Blu-Tack and this then enabled it to be re-recorded).
One aspect which the tape confirmed, to my mind, is that no single unequal temperament is up to the job of rendering all music. As on this recording, a range of temperaments is necessary and one can then make the choice to suit the music. When the recordings were made that was an impractical ideal, but today many digital organs offer a choice of several temperaments and this can only be considered advantageous.
I have also long pondered on the fact that unequal temperaments offer two main advantages over Equal Temperament. As is well known, these are key colour and the attractive pure intervals in some keys. However it might be less well appreciated that these are independent to some extent. So if one only wants a touch of key colour without the pure intervals, yet without having to tolerate the dissonances which accompany most unequal temperaments, one can design a mildly unequal temperament which still allows all keys to be used without restriction. My Dorset Temperament described elsewhere on this site offers this.
4. An apologist might point to the restricted computer technology available to Padgham nearly thirty years ago. However spreadsheet programs such as VisiCalc, SuperCalc and Lotus 1-2-3 were widely used on early Apple, IBM and Intel-based CP/M computers from the late 1970's onwards. I was using them myself on a Z80-based home computer around that time. Of course, in addition Padgham would presumably have had these and possibly other applications available to him on the mainframes at his university. Besides that he could always have programmed in languages such as Basic, Fortran or Algol on those machines. He would also have had access to hand held calculators, ranging from cheap Sinclair products to those by Hewlett-Packard and Texas Instruments. Indeed, he mentioned their use in his book. Against this background I therefore do not feel a need to offer excuses for the criticisms in this article - there is no doubt that he had the necessary tools at his disposal to do the job properly.
5. I have explained elsewhere on this website that a numerical precision of 1 part in 100,000 or 6 significant figures is required when expressing absolute note frequencies if one is to do meaningful work in tuning and temperament. Otherwise unacceptably large errors will build up owing to the repeated rounding or truncation arithmetic operations inseparable from the use of spreadsheets and computer programs more generally. (See Appendix 2 in Temperament - a Study of Anachronism). This criterion also implies at least two decimal places when using frequency ratios expressed in cents. Padgham agreed, because he says so explicitly. Unfortunately his book nevertheless demonstrates varying degrees of precision which only serve to illustrate a lack of rigour across his work.
6. "Temperament and Pitch", The Cambridge Companion to the Organ, Christopher Kent, Cambridge 1998. ISBN 0 521 57309 2 and 0 521 57584 2.
In his piece, Kent included no discussion of the cautionary issues rehearsed in this article. This is unfortunate because some of the temperaments discussed by Kent involve numbers derived from data sets which contained errors. In his article Kent also asserted that 1200 x 2 + 386.14 equals 2786.314, though I could not say whether this misconception also arose from Padgham's book.
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