Thank you for visiting this website, devoted to everybody interested in Musical Temperament: performers of all Western musical instruments, singers, keyboard tuners, instrument makers, students and music lovers. We deal here with the very foundations of Western music: the way the different diatonic and chromatic scales arise, their acoustical foundations, their historical development, their practical use in present day performance, how to tune musical instruments and how to play them and sing in tune.

Claudio Di Veroli, Bray, Rep. Ireland, 2013.                                                                                          E-mail: This is a graphic, not a text for you to copy. You have to retype it when you send the e-mail. A link or text would be "harvested" by "spammers". Sorry for the inconvenience.

Interested in Baroque interpretation for harpsichord and other keyboards? Find most problems resolved in the comprehensive recent book: PLAYING THE BAROQUE HARPSICHORD

Also Interested in early keyboard fingerings? The most complete handbook ever is now available: BAROQUE KEYBOARD FINGERINGS


Cover artwork: “Spiral of Fifths” by Mark A. Reynolds.
Watercolour and ink on rag paper. Mill Valley, California, USA, 2008, reproduced here by kind courtesy of Mr. Reynolds.

Unequal Temperaments, 3rd Edition, July 2013: 486 pages and 365 figures incorporating decades of new findings in the field.

PART ONE provides the acoustical background. Each one of the main temperaments is then covered, with a thorough analysis of its musical features. Finally a full HISTORICAL SURVEY guides the musician in the selection of temperament(s) for his/her repertoire.

PART TWO consists of directions for the practice of the different unequal temperaments: keyboard tuning procedures, fret placement in fretted strings, fingering for unfretted strings, intonation for voices, woodwinds and brass wind instruments, including variable intonation techniques.

PART THREE covers multiple divisions and the analysis of historical sources, including research results published here for the first time.

You are welcome to read below published Reviews and a Preview. Please click on the button below for price details and to buy the book.

This button brings you to the Lulu webpage where you can buy all the Bray Baroque books

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If you purchased the UT 2nd ed. after 30th June 2012, please email us with Lulu's order no. and date: Bray Baroque will send you a free copy of the 3rd ed.



  • Intervals, Scales and Temperaments
  • Evaluation method for the temperaments
  • Equal Temperament and Pythagorean intonation
  • Just intonations
  • Meantone temperaments
  • Lute temperaments
  • Circular French temperaments
  • Circular Good temperaments
  • Comparative evaluation of temperaments


  • Tuning methods and temperament schemes for keyboards
  • Temperament schemes for fretted clavichords
  • Directions and schemes for fretted strings
  • Intonation for wind instruments, unfretted strings and voices


  • Multiple divisions of the octave
  • Analysis of sources on temperament
  • Complementary topics and Appendices

Some unique features of the UNEQUAL TEMPERAMENTS book

—> A definitive and extensive treatment of Baroque French Circular temperaments, their evolution in theory and practice (33 pages).
—> A new comprehensive analysis of the temperaments advocated for J.S. Bach in modern times,
       with exhaustive computer-aided searches for Bach's temperament, producing significant new results (16 pages).
—> A full Historical Review, with 33 sections and subsections covering the tuning of musical scales during the successive eras
       in Western musical history, including differences between instruments and countries (30 pages).
—> A full treatment of Equal Temperament, its diffusion through the Western world and evolution of tuning accuracy (12 pages).
—> Charts with graphical comparisons between and within groups of temperaments.
—> Simple, accurate and fast keyboard-tuning schemes for each temperament:
       both beat-rate schemes and early-style tuning directions are included.
—> Tuning schemes for fretted clavichords.
—> Fretting charts for lutes and viols covering an ample assortment of temperaments and open-string tunings.
—> Full charts and directions for unequal temperament fingering for violins and other unfretted strings.
—> Charts and directions for unequal temperaments in early woodwinds.
—> The most complete ever treatment of unequal temperaments for natural brass instruments.
—> Different methods for improved intonation in Equal Temperament are also included.
—> New improved spreadsheet and other extras downloadable from our Unequal Temperament goodies webpage .

Why an eBook?

This book can be read on the screen of a PC, Mac and any desktop or portable device capable of showing PDF documents (Adobe Acrobat v.6.0 or later). It is also formatted for impeccable printing on Letter, A4 or A5 (two pages in an A4 sheet) paper. Benefits of buying an eBook compared to a ready-printed volume:
      • PRICE: less than 1/5th of the estimated price of a printed volume. (*)
      • AVAILABILITY: an eBook is never out of print.
      • SECURITY: the reader can make a backup copy, so that the eBook cannot be lost or stolen.
      • PRINTING: if a printed volume is really needed, the reader can print it from any home computer.
      • MANY FIGURES: hundreds of figures, including many full-colour photos and charts.
      • COLOUR: colour is freely used in the text and tables, helping to easily understand complex matters and relationships.
      • PORTABILITY: the reader can carry the book around in a laptop computer, smartphone or any other portable device.

      • KINDLE etc.: the text is fully readable. The PDF provides no font resizing: just zoom-in to see details or enlarge text..
      • REFERENCE: the reader interested in a particular topic can quickly search and find all the references to it in the eBook.
TIMELINESS: you buy the eBook, download it immediately and can begin reading it straight away.
they allow easy access to the web pages referred to.(**)

(*) There are no plans to make this book available in printed form. The eBook is a 20+Mb document file that downloads in less than two minutes using a typical Internet broadband service.

(**) On p. 14 of the 3rd edition, the 6th bullet reads: "HYPERLINKS. They are no longer included in this edition: most of them failed to work in the final PDF file, a common issue of the conversion of large files. ... the reader can just copy any web address from the text and paste it into the address box of the web browser." Nevertheless, the final conversion to PDF has reconverted all the "http://... " addresses into hyperlinks and they are all working now: just clicking on the link opens the web page in the user's Internet browser.

Published Book Reviews

David Hitchin in The British Clavichord Society newsletter, No.45, Oct.2009:

     "Claudio Di Veroli's work has the advantage (or disadvantage) of being at least twice the size of those by his competitors. In the eleven chapters of the first part Di Veroli introduces musical intervals, the Pythagorean scale, equal temperament, just intonations, meantone, ... lute temperaments, irregular French and irregular 'good' temperaments, comparative evaluations and a review of historic performance. He defines a series of seventeen principles ... They provide a solid framework for understanding and comparing temperaments. The author occasionally uses measures likely to be known only to those with an advanced statistical training (he was a student of Sir David Cox, one of the most eminent British statisticians). As a statistician myself, I can confirm that his choice of methods is appropriate ...
The second part is about tuning and fretting instructions ... There is advice about hearing and using beats, thirty hints and tricks for harpsichord tuning, transposing instruments ... a list of schemes for keyboard instruments, fretting instructions for lutes and viols, intonation for violins and voices, woodwinds and brass instruments.
The third part ... includ[es] multiple divisions ... and an analysis of much of the historic documentation for temperaments.
     The book contains some original research. Di Veroli concludes that Barnes' temperament is the optimum one for the Well-Tempered Clavier ... Di Veroli generated and evaluated the nearly 17 million possibilities, and found that very slight improvements were possible, but they were so small, and made tuning so much more difficult, that they were not worth using in preference to Barnes.
     ... many pages make good use of colour ... Di Veroli ... includes links to audio files, and surely there is nothing more self-evident than the need to hear different temperaments as one reads about them ... There are also two useful spreadsheets. ... This work is only available as ... a .pdf file ... I printed it out. It is formatted for convenient printing on A4 paper, but it was quite legible when reduced to A5 ... a pleasure to read ... the explanations are clear and easy to follow. He has taken on a Herculean task of reviewing and explaining the history and application of unequal temperaments, and while it is unlikely that all experts in the subject will agree with him on all points, he presents convincing arguments and detailed documentation. ... This is one of the most interesting books that I have read for a long time."

Bradley Lehman in The Viola da Gamba Society Journal, Volume 3, Part 2, Dec.2009:

     Please click here for details about a most unusual review, as well as a published rebuttal.

Pedro Persone in Early Music America, Spring 2010, Mar.2010:

     "This e-book, published in March 2009, is now available on-line ... Di Veroli’s study is divided into three main parts. Part I consists of “Description, Analysis and History” (11 chapters in total). The seven chapters of Part II explain “Tuning and Fretting Instructions.” Part III consists of “Complementary Topics” (four chapters and five appendices). Di Veroli´s book is a relevant complementary source to the existing bibliography on the important subject of temperament and can be counted among the best-known books in the field. ... is ... necessary to temper our instruments ... the difference ... is the "dust" we have to sweep under the rug. Di Veroli provides us with the recipes on how to distribute our "dust" by using the historical unequal temperaments. He does this by means of symbols, ratios and formulas, and explanations ... The most important temperaments are explained in this book ...
In addition, Di Veroli ... offers important tips about the way to manage tuning hammers or how to change a keyboard temperament shortly before a public performance. The huge number of known temperaments are explained step by step, and fretted clavichords are given a separate chapter. The e-book also offers information about plucked fretted instruments and the intonation of woodwind and brass instruments.
      I am very happy to recommend this book, since it will provide pleasant and useful reading for both students and enthusiasts."

Comments by Readers and Scholars: Ever since 2009 the author has received many personal comments from readers, happy with the book's structure, coverage, presentation and cost. As for the 3rd edition of 2013, the temperament specialist Fred Sturm has recommended it in the online forum of the Piano Technicians Guild, stating that: "It is the most comprehensive and reliable source for information about temperament history available today". The world-leading temperament scholar Prof. Patrizio Barbieri (author of the "Enharmonic Instruments and music" treatise) has written to the author—with permission to publish—that Unequal Temperaments in its 3rd ed. is most likely to be the best treatise in the field.


1978 ... the book Unequal Temperaments and their Role in the Performance of Early Music ... Rudolf Rasch published ... a detailed review in Stimulus ... Rasch praised both the theoretical framework and the emphasis in the historical relevance of each temperament for present-day performance ...…
Quite a few musicians wrote ... positive comments ... Let me just mention ... Igor Kipnis … Hubert Bédard … and John Barnes …. Bédard wrote: 'The most useful text on the subject I have seen yet. … This book … seems to settle the matter at last.'. ...

We hope these reduced pages will whet your appetite for the Unequal Temperaments book

Three decades having elapsed ... in 2007 I started working on this new book ... not a revision but a new treatise written from scratch ... it incorporates the results of new published research, including recent discussions by/with colleagues. ... Readers of the 3rd. edition will find significant additions or enhancements ...

We hope these reduced pages will whet your appetite for the Unequal Temperaments book

With our simple yet powerful "dissection tools", we are now ready to enter, in the following Chapters, the fascinating world of early temperaments in Renaissance, Baroque and Classical times. ... In meantone temperament … contrary to an often repeated but erroneous belief, the main restriction to further modulations is not caused by the wolf fifth, but instead by the 4 wolf major thirds ...

The fascinating properties and proportions of the Standard meantone diatonic and chromatic scales are treated in the next Chapter ...
Let us now see how ancient musicians, and their modern followers, fare in practice when they need a Spiral of Fifths with up to 5 pairs of flats and sharps ... : Voices, unfretted strings, trombones and trombe da tirarsi ... Other brass wind instruments ... Fretted strings ...Woodwinds ... Keyboards ...

... Chaumont and F.Couperin were using different variants ... The reconstructed Couperin organ temperament is shown below. ...

A disproportionate amount of effort and ink has been dedicated [by modern writers] to the analysis of the multiple variants of circular temperaments proposed by German Baroque theoreticians. ... We will eventually conclude that only a handful of these systems are really worth studying and tuning to. ... If the reader finds it difficult to grasp which flavour of 'Good' temperament J.S.Bach advocated or preferred, he/she is in good company. ...

Bach did not compose the WTC in order to advocate a peculiar tuning to be discovered by archaeologists of the future deciphering 'Rosetta stones'. …
A full classification ... (Artusi 1603) had the instruments divided into three families according to their temperament … However … as we come into the Baroque era, fretted instruments are found increasingly playing together with keyboards …

The Baroque is the only period in Western musical history in which the different amount of temperament of the intervals around the Circle of Fifths eventually became a component of musical expression. …
During the Classical period,
unequal temperaments were still feasible and practised throughout Europe. But Equal Temperament … was now gaining considerable momentum. What was the most common situation in the so-called Classical period? What were most musicians doing after 1750? …

THIRTY(30) HINTS AND TRICKS FOR HARPSICHORD TUNING. [For] early stringed keyboards …the following guidelines are an integrated set of suggestions and instructions … in a working-practice sequence. ...
MEANTONE FRETTING. … Unlike the pure intonations where some considerable mistuning was expected by early audiences and anyway unavoidable …
in meantone … an accurate tuning is both feasible and expected. A simple [fretting] procedure follows …

Standard French ordinaire being the most unequal circular temperament, the conclusion back in 1978 was that a most accurate fretting was possible for all the remaining circular temperaments as well. This has now been confirmed after three decades of success …
This Chapter is intended to show how to play— easily and accurately—unfretted instruments in unequal temperaments. This is achieved by using specific fingering charts and techniques. ... The text has been written so as to be most useful to singers as well. …

“The road to intonation hell is paved with just-intonation intentions”. ... Were Bach's horn and trumpet players (for example the famous Reiche) better than Handel's (for example the famous Snow)? ... were perhaps Bach's players using narrow-bore instruments … with increased flexibility and ease in the upper parts of the register?


Now that you bought the UT book, enjoy the Sound Examples, Spreadsheet and other extras

The Unequal Temperaments: Theory, History and Practice eBook is a self-contained work. No additional files are needed to read it, understand it, enjoy it and/or put the advice into practice. Once you have purchased the book, however, additional goodies are waiting for you to enhance your experience: an updated Errata and Addenda file, the new Unequal Temperament Spreadsheet and a full set of Sound Examples. You can access them from the Unequal Temperament Goodies webpage.

Please find in a separate webpage, Unequal Temperament Links, links to other sites with contents we find interesting and useful.

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Page last updated: 22-Nov-2013