text: to get Since the log function is increasing on the interval , we get for . … µ N e ¶N =) lnN! is. ∼ nn en √ 2πn = n(n+1/2) e−n √ 2π . … N lnN ¡N =) dlnN! The sign “:=” will mean “equals by deﬁnition.” Let d n:= log(n! ∼ Cnn+12 e−n as n→ ∞, (1) where C= (2π)1/2 and the notation f(n) ∼ g(n) means that f(n)/g(n) → 1 as n→ ∞. The log of n! A simple proof of Stirling's formula for the gamma function - Volume 99 Issue 544 - G. J. O. Jameson. Jameson This is a slightly modiﬁed version of the article [Jam2]. Stirling’s approximation is a useful approximation for large factorials which states that the th factorial is well-approximated by the formula. Stirling’s formula for integers states that n! Proof. Thus, log(n! In other words, we have or Proof of the Stirling's Formula. In his extensive analyses of Stirling’s works, I. Tweddle [9] suggests that the digits of √ πay have been known to Stirling; Stirling computes the ﬁrst nine places m … The Stirling numbers of the second kind, written (,) or {} or with other notations, count the number of ways to partition a set of labelled objects into nonempty unlabelled subsets. Introduction of Formula In the early 18th century James Stirling proved the following formula: For some = ! = ln1+ln2+::: +lnN … Z N 1 dx lnx = N lnN ¡N +1 … Show abstract. Stirling’s formula.n! show how to transform this calculation into a bona de proof (we comment that this result is also easy to prove directly using Stirling’s formula). View. First take the log of n! 5 Cumulants We are now almost ready to present our rst proof. At this point I Stirling's approximation is also useful for approximating the log of a factorial, which finds application in evaluation of entropy in terms of multiplicity, as in the Einstein solid. While Stirling oﬀers no proof of his claim, it is likely that Stirling’s own reasoning involves Wallis’s formula. II.The Proof: Stirling’s Formula Before getting our hands dirty into mathematical statements and equations, let us first take a glimpse and see how the formula looks like $$$$\lim_{n\to\infty}\frac{n! }{\sqrt{2\pi}\cdot n^{n+\frac{1}{2}}\cdot e^{-n}}=1\label{ref2}$$$$ known proof that uses Wallis’s product formula. )− n+ 1 2 log n−n+ 1 2 log(2π) → 0 as n → ∞. The Stirling formula gives an approximation to the factorial of a large number, N À 1. but the last term may usually be neglected so that a working approximation is. An important formula in applied mathematics as well as in probability is the Stirling's formula known as where is used to indicate that the ratio of the two sides goes to 1 as n goes to . Then we need to prove d n converges to a constant, [log(2π)]/2. Definition. Stirling's formula: proof? We rst de ne the cumulant generating function of a random variable X: K X(t) = logM X(t): 4 Monotonicity and Boundedness of Remainder of Stirling's Formula. 2 π n n e + − + θ1/2 /12 n n n <θ<0 1 dN … lnN: (1) The easy-to-remember proof is in the following intuitive steps: lnN! Equivalently, they count the number of different equivalence relations with precisely equivalence classes that can be defined on an element set. Theorem 1. A simple proof of Stirling’s formula for the gamma function Notes by G.J.O. In its simple form it is, N! C = 2p f (n) ~ g(n) f (n)/g(n) ˛ 1 n ˛ Œ A great deal has been written about Stirling's formula. ~ Cnn + 12e-n as n ˛ Œ, (1) where and the notation means that as . A simple proof of Stirling's formula for the gamma function G. J. O. JAMESON Stirling's formula for integers states that n! Is there any way to derive Stirlings formula that only requires some undergraduate knowledge of calculus, real analysis and perhaps some identitets involving the gamma function, maybe Wallis product, and things along those lines? (11 answers) Closed 3 years ago. )− n+ 1 2 log n+ n .


text:
# Free The Supply Curve 03 Practice Test - 11th Grade - Commerce

When the prices of a good increases, ______

A.

Quantity demanded increases

B.

Quantity demanded decreases

C.

Quantity supplied increases

D.

Quantity supplied decreases

#### SOLUTION

Solution : B and C

When the prices of a good increases, the quantity demanded decreases and the quantity supplied decreases.

The profit level that is just enough to cover the economic costs of the firm is called ___ profit.

#### SOLUTION

Solution :

The profit level that is just enough to cover the economic costs of the firm is called normal profit.

The costs (LRAC) and prices (P) for four firms are given in the options below. Which of these cannot lie on the long-run supply curve of a firm?

A.

LRAC = Rs 20, P = Rs 20

B.

LRAC = Rs 25, P = Rs 20

C.

LRAC = Rs 20, P = Rs 25

D.

LRAC = Rs 25, P = Rs 25

#### SOLUTION

Solution : B

A firm does not supply when the market price is less than the average cost. For option B, P < LRAC.

There are three sellers A, B and C in a market. Their supply curves are given by the following equations.

qA=5p30, p8qB=2p+20, p10qC=3p+10, p15

What is the total quantity supplied at a market price of Rs 10?

A.

20

B.

60

C.

80

D.

100

#### SOLUTION

Solution : B

Note that the supply curve of C starts only above a price level of 15. Hence, at a price of Rs 10, only the supply curves of B and C are relevant.

q=qA+qBq=7p10

At p=10, the quantity supplied is 60 units.

The individual supply curves of two firms are
q1=p2, p3q2=12(p3), p4

The market supply curve is given by

A.

q=0, p<3

B.

q=p2, 3p<5

C.

q=32p4, p5

D.

All of these

#### SOLUTION

Solution : D

The individual supply curves of the two firms are
q1=p2, p3q2=12(p4), p5

p<3q=03p<5q=q1p5q=q1+q2

All options are correct.

A 10% increase in the price of a good causes a 10% decrease in quantity supplied. The supply curve is a _______.

A.

straight line passing through the origin

B.

straight line not passing through the origin

C.

straight line parallel to the x-axis

D.

rectangular hyperbola

#### SOLUTION

Solution : A

ϵs=% change in quantity% change in price=1010=1

Unit elastic supply curves are straight lines passing through the origin.

The supply curve is upward sloping because __________.

A.

costs increase when the price increases

B.

quantity demanded decreases when price increases

C.

As the price increases, suppliers can justify producing at higher marginal costs

D.

All of the above

#### SOLUTION

Solution : C

The supply curve is upward sloping because as price increases, suppliers can justify producing at higher marginal costs as long as MC < P.

Study the supply curve given below and answer the following question.

At what price is the producer surplus equal to $2? A.$1

B.

$2 C.$3

D.

A.

10

B.

20

C.

30

D.

40

#### SOLUTION

Solution : D

From the supply curve, it can be seen that corresponding to a price of \$8, the quantity supplied is 40 units.

format: plain


text:
Lemma 36.39.1. Let $X$ be a scheme. There is a functor

$\det : \left\{ \begin{matrix} \text{category of perfect complexes} \\ \text{with tor amplitude in }[-1, 0] \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of invertible modules} \\ \text{morphisms are isomorphisms} \end{matrix} \right\}$

In addition, given a rank $0$ perfect object $L$ of $D(\mathcal{O}_ X)$ with tor-amplitude in $[-1, 0]$ there is a canonical element $\delta (L) \in \Gamma (X, \det (L))$ such that for any isomorphism $a : L \to K$ in $D(\mathcal{O}_ X)$ we have $\det (a)(\delta (L)) = \delta (K)$. Moreover, the construction is affine locally given by the construction of More on Algebra, Section 15.122.

Proof. Let $L$ be an object of the left hand side. If $\mathop{\mathrm{Spec}}(A) = U \subset X$ is an affine open, then $L|_ U$ corresponds to a perfect complex $L^\bullet$ of $A$-modules with tor-amplitude in $[-1, 0]$, see Lemmas 36.3.5, 36.10.4, and 36.10.7. Then we can consider the invertible $A$-module $\det (L^\bullet )$ constructed in More on Algebra, Lemma 15.122.4. If $\mathop{\mathrm{Spec}}(B) = V \subset U$ is another affine open contained in $U$, then $\det (L^\bullet ) \otimes _ A B = \det (L^\bullet \otimes _ A B)$ and hence this construction is compatible with restriction mappings (see Lemma 36.3.8 and note $A \to B$ is flat). Thus we can glue these invertible modules to obtain an invertible module $\det (L)$ on $X$. The functoriality and canonical sections are constructed in exactly the same manner. Details omitted. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
Hardcover | $35.00 X | £24.95 | ISBN: 9780262014199 | 290 pp. | 6 x 9 in | 1 b&w illus, 4 charts, 1 graph, 23 tables | June 2010 eBook |$24.00 Short | ISBN: 9780262266260 | 290 pp. | 1 b&w illus, 4 charts, 1 graph, 23 tables | June 2010

Pragmatic Neuroethics

Improving Treatment and Understanding of the Mind-Brain

Overview

Today the measurable health burden of neurological and mental health disorders matches or even surpasses any other cluster of health conditions. At the same time, the clinical applications of recent advances in neuroscience are hardly straightforward. In Pragmatic Neuroethics, Eric Racine argues that the emerging field of neuroethics offers a way to integrate such specialties as neurology, psychiatry, and neurosurgery with the humanities and social sciences, neuroscience research, and related healthcare professions, with the goal of tackling key ethical challenges and improving patient care. Racine provides a survey of the often diverging perspectives within neuroethics, offers a theoretical framework supported by empirical data, and discusses the neuroethical implications of such issues as media coverage of neuroscience innovation and the importance of public concerns and lay opinion; nonmedical use of pharmaceuticals for performance enhancement; and the discord between intuitive notions about consciousness and behavior and the scientific understanding of them.

Racine proposes a pragmatic neuroethics that combines pluralistic approaches, bottom-up research perspectives, and a focus on practical issues (in contrast to other more theoretical and single-discipline approaches to the field). [He discusses ethical issues related to powerful neuroscience insights into the mechanisms underlying moral reasoning, cooperative behavior, and such emotional processes as empathy.] In addition, he outlines a pragmatic framework for neuroethics, based on the philosophy of emergentism, which identifies conditions for the meaningful contribution of neuroscience to ethics, and sketches new directions and strategies for meeting future challenges for neuroscience and society.

Basic Bioethics series

Eric Racine is Director of the Neuroethics Research Unit at the Clinical Research Institute of Montreal and Assistant Research Professor at the Clinical Research Institute of Montreal. He also holds appointments at the University of Montreal (Medicine, Preventive and Social Medicine, and Bioethics) and McGill University (Neurology and Neurosurgery and Biomedical Ethics).

• Pragmatic Neuroethics
• Basic Bioethics
• Arthur Caplan, editor
• A complete list of the books in the Basic Bioethics series appears at the back of this book.
• Pragmatic Neuroethics
• Improving Treatment and Understanding of the Mind-Brain
• Eric Racine
• The MIT Press
• Cambridge, Massachusetts
• London, England
• 2010
• Massachusetts Institute of Technology
• All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.
• This book was set in Sabon by Westchester Book Group. Printed and bound in the United States of America.
• Library of Congress Cataloging-in-Publication Data
• Racine, Eric, 1976–
• Pragmatic neuroethics : improving treatment and understanding of the mind-brain / Eric Racine.
• p. ; cm. — (Basic bioethics)
• Includes bibliographical references and index.
• ISBN 978-0-262-01419-9 (hardcover : alk. paper) 1. Neurology—Moral and ethical aspects. 2. Neurosciences. I. Title. II. Series: Basic bioethics.
• [DNLM: 1. Neurosciences—ethics. 2. Behavior—ethics. 3. Bioethical Issues. 4. Morals. 5. Neurology—ethics. WL 100 R121p 2010]
• RC343.R16 2010
• 174.2′968—dc22
• 2009037816
• 10 9 8 7 6 5 4 3 2 1
• Contents
• Preface ix
• Acknowledgments xv
• 1 Salient Challenges in Modern Neuroethics 1
• 2 Reviewing Past and Current Neuroethics
• Definitions, Attributes, and Perspectives 27
• 3 Pragmatic Naturalism in Bioethics 53
• 4 Neuroethics
• Exploring the Implications of Pragmatic Naturalism 71
• 5 Public Understanding of Neuroscience Innovation and Emerging Interpretations of Neuroscience Research 97
• 6 Enhancement of Performance with Neuropharmaceuticals
• Pragmatism and the Culture Wars 121
• 7 Disorders of Consciousness in an Evolving Neuroscience Context 139
• 8 Communication of Prognosis in Disorders of Consciousness and Severe Brain Injury
• A Closer Look at Paradoxical Discourses in the Clinical and Public Domains 161
• 9 Social Neuroscience
• A Pragmatic Epistemological and Ethical Framework for the Neuroscience of Ethics 179
• 10 Conclusion
• Neuroethics and Future Challenges for Neuroscience, Ethics, and Society 215
• References 223
• Index 255
• Series Foreword
• I am pleased to present the twenty-seventh book in the series Basic Bioethics. The series presents innovative works in bioethics to a broad audience and introduces seminal scholarly manuscripts, state-of-the-art reference works, and textbooks. Such broad areas as the philosophy of medicine, advancing genetics and biotechnology, end-of-life care, health and social policy, and the empirical study of biomedical life are engaged.
• Arthur Caplan
• Basic Bioethics Series
• Editorial Board
• Joseph J. Fins
• Rosamond Rhodes
• Jan Helge Solbakk
• Preface
• Pragmatic Neuroethics
• is a book based on the reasonable hope that if clinicians and researchers from various disciplines work together in collaboration with different stakeholders, the chances of making a difference for patients suffering from neurological and psychiatric disorders can be increased. It relies on the belief that ethics is a crucial part of this endeavor, and that ethics, the search for the good and for what Aristotle called moral excellence, is inherently part of social and medical acts aiming to alleviate suffering, pain, and daily difficulties for patients. This view is at the heart of this book and of what I describe as “pragmatic neuroethics,” a view of bioethics influenced by various thinkers that emphasizes the pluralistic nature of ethics and society and the value of interdisciplinary collaboration and research to further knowledge and institute beneficial practice changes in healthcare, science, and society.
• The need for an interdisciplinary and collective response to ethical challenges in neuroscience and clinical care—neuroethics—has surfaced in the past years in response to important social, medical, and scientific changes. In many developed countries today, as illustrated by data of the World Health Organization, the combined “health burden” of neurological and mental health disorders matches and even surpasses that of any cluster of health conditions (World Health Organization 2001, 2006). Developing countries are spared neither from this prevalence of neuropsychiatric conditions nor from issues of stigma associated with mental illness and the need for better treatments for neurological disorders. The term “health burden,” often measured in days of lost productivity, may seem overly administrative. Nonetheless, it is a convenient way to illustrate the costs of caregivers without appropriate support and resources; stigma and discrimination; lives that are shattered by illness and isolated suffering; lack of research efforts; and, in some unfortunate cases, suicide. Today, almost everyone can look around and identify, if not themselves, a friend or relative that has faced not only a mental health or neurological problem but the challenges of being respected as a person and finding the internal resources to carry forward. These are the familiar stories of depressed friends or colleagues who never came back to work after falling ill and elderly parents coping with dementia. In proportion to other serious illnesses, diseases of the brain and mind now represent one of the greatest—and still increasing—public health burdens faced by both higher- and lower-income nations.
• Neuroethics, alongside other initiatives, has surfaced to tackle head on some of the challenges created by advances in neuroscience. However, to date, there have been few theoretical perspectives on this new field and scarce in-depth discussion about its nature and scope. In response, this book provides such a perspective as well as examples of research that attempts to bridge different disciplines to provide frameworks for elucidating and attending to important neuroethics issues. These include, for example, the increasing use of neuropharmacology to enhance performance and the incursion of functional neuroimaging in the world of the humanities and social sciences beyond conventional clinical neuroscience research. I hope that readers will find this book enlightening and stimulating, enough so to encourage reflection, action, or research that will contribute to meeting the broad ethical, scientific, medical, and social challenges that we collectively face.
• Pragmatic Neuroethics
• pursues a number of goals: (1) to thoroughly review diverging perspectives within neuroethics and provide a constructive critical analysis of the latest literature; (2) to provide a consistent view of the field with compelling arguments that yield a theoretical approach to tackling several problems within neuroethics; (3) to address several key neuroethics issues using empirical research data and a pragmatic approach; and (4) to identify future challenges for neuroscience and society and discuss possible directions for the field of neuroethics. The limitations of this book include the focus on some salient challenges to the detriment of others as well as perspectives that reflect my training in bioethics, social science, and philosophy. For example, legal issues related to neuroscience are not addressed, and I will be the first to admit that many more issues, especially clinically relevant ones, such as stigma, neurodegenerative disease, and aging, pose huge challenges to society. Further, given the range of topics, some will require more attention in the future.
• Pragmatic Neuroethics
• is divided in two parts. The first section, chapters 1 through 4, reviews recent neuroethics scholarship, provides background on neuroethics, and introduces pragmatism and pragmatic neuroethics. The second section, chapters 5 through 9, presents a series of essays on salient topics in neuroethics such as decision making in disorders of consciousness; public understanding of neuroscience; and policy approaches to “cognitive enhancement.”
• Chapter 1 (“Salient Challenges in Modern Neuroethics”) provides a thematic overview of neuroethics and explains some of the key areas currently under discussion. By gaining acquaintance with these topics, the reader should get a clear sense of why modern neuroethics has surfaced to systematically examine emerging challenges. Chapter 2 (“Reviewing Past and Current Neuroethics: Definitions, Attributes, and Perspectives”) analyzes some early definitions of neuroethics and identifies some distinct views of the field. A review of neuroethics attributes and the ethical, legal, and social issues associated with it in peer review literature, on the Internet, and in print media gives an overall sense of how the field has been characterized in formal academic definitions and beyond. Chapter 3 (“Pragmatic Naturalism in Bioethics”) presents and discusses different waves of naturalism in bioethics, including how neuroethics has reinitiated discussions on the relationship between the biological sciences and the humanities. This chapter argues that bioethics in itself is a form of pragmatic naturalism as illustrated by the field’s commitment to interdisciplinary collaborations and its practical focus. Finally, chapter 4 (“Neuroethics: Exploring the Implications of Pragmatic Naturalism”) explores the implications of pragmatic naturalism by addressing several controversies surrounding neuroethics. It concludes by highlighting some of the characteristics of pragmatic neuroethics, such as the integration of pluralism, bottom-up research approaches, and a focus on practical issues that distinguish this approach from other, more theoretical or more monodisciplinary views of the field.
• The second part of this book presents a series of essays that are nourished by both the background material and the theoretical framework of pragmatic neuroethics laid out in chapters 1 through 4. Chapter 5 (“Public Understanding of Neuroscience Innovation and Emerging Interpretations of Neuroscience Research”) highlights the importance of public understanding from a pragmatic perspective in order to take into consideration not only expert opinions about ethics in neuroscience but also public concerns and emerging lay interpretations of neuroscience research. This chapter discusses various forms of media coverage of neuroscience innovation. Chapter 6 (“Enhancement of Performance with Neuropharmaceuticals: Pragmatism and the Culture Wars”) stems from a context of increasing prevalence and salience of nonmedical uses of neuropharmaceuticals for performance enhancement. This chapter examines critically some of the assumptions behind the conservative and liberal moral-political approaches to this issue and makes the case from a pragmatic standpoint for the recognition of pluralism and for approaches that minimize harm, promote autonomy, and urge considerations for the public good. Chapter 7 (“Disorders of Consciousness in an Evolving Neuroscience Context”) provides background information and some discussion of different clinical conditions such as the vegetative state and the minimally conscious state, called collectively disorders of consciousness, in an evolving neuroscience context that challenges some assumptions about these disorders. Chapter 8 (“Communication of Prognosis in Disorders of Consciousness and Severe Brain Injury”) builds on the material presented in chapter 7 to discuss in more detail why there are lingering sources of confusion in disorders of consciousness. Specifically, I argue that there is a strong tension at work between, on the one hand, intuitive notions about consciousness and behavior and, on the other hand, scientific understanding and medical language describing consciousness and behavior. Chapter 9 (“Social Neuroscience: A Pragmatic Epistemological and Ethical Framework for the Neuroscience of Ethics”) builds on the naturalism inherent to pragmatic naturalism and discusses the possibility that neuroscience provides powerful insights into the mechanisms underlying moral reasoning, cooperative behavior, and emotional processes such as empathy. This chapter briefly introduces social neuroscience and the neuroscience of ethics and highlights some potential benefits and misunderstandings created by this area of research. It then presents a pragmatic framework based on the philosophy of emergentism. This framework yields conditions and guideposts for the meaningful contribution of neuroscience to ethics. This emergentist and pragmatic framework also debunks common arguments against the introduction of neuroscience research into ethics as well as overstated promises. The conclusion of
• Pragmatic Neuroethics
• (“Neuroethics and Future Challenges for Neuroscience, Ethics, and Society”) revisits some major issues discussed in this book and sketches recommendations and strategies for moving ahead.
• I hope that readers of this book will enjoy these neuroethics contributions. It is also my hope, even though I realize much still needs to be done, that each chapter will stimulate reflection, action, and ideas for research that embrace pragmatic goals while remaining open to genuine dialogue in the search for collaborative and practical solutions.
• Acknowledgments
• Although it bears the signature of one man, this book, which captures some of my basic opinions, would have been absolutely impossible without the support of many colleagues and students. This is an opportunity for me to acknowledge the help and support of colleagues, family, and friends who have directly or indirectly inspired and supported my work in the last years.
• I would like to sincerely thank Nicole Palmour for reviewing this manuscript on several occasions and providing constructive and helpful comments. I want to thank former and current trainees and assistants of the Neuroethics Research Unit: Amaryllis Ferrand, Cynthia Forlini, Constance Deslauriers, Emily Bell, Ghislaine Mathieu, Bruce Maxwell, William Affleck, Zoë Costa-von Aesch, Marie-Josée Dion, Marta Karczewska, Matthew Seidler, David Bouvier, David Risse, Catherine Rodrigue, Lila Karpowicz, and Danaë Larivière-Bastien. Their work and presence have inspired me along the way and are truly appreciated. My work reflects part of the ongoing daily conversations and exchanges I have had with them, and I am immensely indebted to them.
• I consider myself privileged to have received the mentorship of outstanding individuals along the years. I have been fortunate to cross paths with senior scholars who possess extraordinary energy, creativity, and commitment. I would like to thank in particular Judy Illes, Hubert Doucet, and Bartha Knoppers for their generous time, ongoing support, and encouragements. Nothing can replace the mentorship and advice a young scholar receives from senior colleagues or the intellectual exchanges that occur within collegial and respectful environments. I would like to thank particularly Walter Glannon for his helpful guidance throughout this project, James Bernat for helpful comments on this manuscript, and an additional anonymous reviewer.
• The idea for writing this book originates partly from the days of my postdoctoral fellowship at the Stanford Center for Biomedical Ethics (SCBE). I would like to thank accordingly the leadership of the SCBE, in particular David Magnus, Henry Greely, and Mildred Cho, for support and the staff members (Anne Footer, Paula Bailey, Joyce Prasad, and Shobha Kumar) who helped me in many different ways to make this a successful journey. I want to thank the students who worked with me on various projects at the SCBE, including Sarah Waldman, Adri Van der Loos, Ofek Bar-Ilan, Stacey Kallem, Neil Mukhopadhyay, Allyson Mackey, Vivian Chau, Rakesh Amaram, Marisa Gallo, and Tessa Watt. I am indebted for their assistance in numerous research projects, and the exchanges I have had with them are marked in my past and present and will remain with me for the future.
• Pragmatic Neuroethics
• materialized during a visiting fellowship at the Brocher Foundation in August 2007, where I had the privilege to put on paper the project for this book. The beautiful and inspiring settings provided by the foundation’s location on the shores of Lake Geneva and the friendly exchanges with colleagues helped me prepare the proposal and nourished this project as it unfolded. I would like to thank the leadership of the Brocher Foundation, in particular Cécile Caldwell Vulliéty, and the friendly staff, especially Raji Sultan for his kind assistance. I would also like to acknowledge the support of colleagues I met during my enjoyable stay there (Anthony Mark Cutter, Thomas Douglas, Bert Gordijn, Rouven Porz, Michael Selgelid); they made this a remarkable and inspiring experience for me and my family members (Nathalie, André-Anne, Gabrielle, and Amélie). The foundation’s generous support for the publication of this book deserves special recognition.
• During the writing of this work and the work that preceded it, I received fellowships, awards, and grants from several agencies, including the Social Sciences and Humanities Research Council, the Fonds de la recherche en santé du Québec, the National Institutes of Health (grant awarded to Judy Illes), the Greenwall Foundation (grant awarded to Judy Illes), the Canadian Institutes of Health Research (Ethics Office; Institute of Neurosciences, Mental Health and Addiction; Institute of Human Development, Child and Youth Health), the CIHR-funded States of Mind Network (Françoise Baylis), and the CIHR-funded Pediatric Neuroimaging Ethics Network (Jocelyn Downie). The support from these agencies and from various peer communities has made much of the work underlying this book possible.
• I am grateful for the support from the unique Institut de recherches cliniques de Montréal (IRCM), especially Louis-Gilles Durand, who has been an invaluable source of inspiration and advice in the past three years. Jeannine Amyot’s impressive administrative skills and commitment to supporting the Neuroethics Research Unit need to be underscored and acknowledged. I thank all my IRCM colleagues who have provided feedback on material presented in this book and who believe in the legacy of Jacques Genest, the founder of the institute, who was committed early in the 1970s to creating the first Canadian Bioethics Center. The example and vision he has set to bridge research, healthcare, and ethics in the service of humankind will be with us for decades and hopefully centuries. Thanks to Claudia Jones and Nicole Campeau at the library, and to the IRCM leadership, in particular its director, Tarik Möröy, for his confidence in and ongoing support of this project and others.
• Material in this book has been presented at several occasions during seminars, talks, and scientific congresses. I would like to acknowledge the feedback given by audiences and colleagues at the University of Tokyo, the University of Delaware, Uppsala Universitet, Université Laval, Université de Montréal, the University of Minnesota, McGill University, Stanford University, the University of Toronto, Universidade do Estado do Rio de Janeiro, the Alden March Bioethics Institute, the Montreal Neurological Institute, Université du Québec à Montréal, the National Research Council of Canada, the Jewish Rehabilitation Hospital, the University of Pennsylvania, the University of Alberta, the University of Western Ontario, York University, Universidade Federal de Rio de Janeiro, and Concordia University. I am also indebted to audiences at various scholarly meetings, including the Canadian Bioethics Society, the American Society for Bioethics and Humanities, the Society for Neuroscience, the International Conference in Clinical Ethics, the Society for Social Studies of Science, the Canadian Association for Neuroscience, the International Association for Bioethics, Association Francophone pour le Savoir, the Neuroethics Society, the International Academy of Law and Mental Health, the American Philosophical Association, and the International Society of History, Philosophy, and Social Studies of Biology. I would like to thank students from the neuroethics seminars at Université de Montréal and at McGill University for stimulating exchanges in 2008 and 2009.
• Portions of chapters 1, 4, and 5 have previously appeared in E. Racine and C. Forlini, “Cognitive Enhancement, Lifestyle Choice or Misuse of Prescription Drugs? Ethics Blind Spots in Current Debates,”
• Neuroethics
• (2008).
• Portions of chapter 1 have previously appeared in E. Bell, G. Mathieu, and E. Racine, “Preparing the Ethical Future of Deep Brain Stimulation,”
• Surgical Neurology
• (2009).
• Portions of chapters 1 and 8 have previously appeared in E. Racine and E. Bell, “Clinical and Public Translation of Neuroimaging Research in Disorders of Consciousness Challenges Current Diagnostic and Public Understanding Paradigms,”
• American Journal of Bioethics
• 8 (2008): 13–15.
• Portions of chapters 2 and 4 have previously appeared in E. Racine, Comment on “Does It Make Sense to Speak of Neuroethics?”
• EMBO Reports
• 9 (2008): 2–3.
• Portions of chapter 3 have previously appeared in E. Racine, “Which Naturalism for Bioethics? A Defense of Moderate (Pragmatic) Naturalism,”
• Bioethics
• 22 (2008): 92–100.
• Portions of chapters 4 and 9 have previously appeared in E. Racine, “Interdisciplinary Approaches for a Pragmatic Neuroethics,”
• American Journal of Bioethics
• 8 (2008): 52–53.
• Portions of chapters 6 and portions of the conclusion have previously appeared in E. Racine, O. Bar-Ilan, and J. Illes, “fMRI in the Public Eye,”
• Nature Reviews Neuroscience
• 6 (2005): 159–64.
• Portions of chapters 6 have previously appeared in E. Racine, O. Bar-Ilan, and J. Illes, “Brain Imaging: A Decade of Coverage in the Print Media,”
• Science Communication
• 28 (2006): 122–142.
• Portions of chapter 8 have previously appeared in E. Racine, R. Amaram, M. Seidler, M. Karczewska, and J. Illes, “Media Coverage of the Persistent Vegetative State and End-of-Life Decision-Making,”
• Neurology
• 71 (2008): 1027–1032.
• Portions of chapter 9 have previously appeared in E. Racine and J. Illes, “‘Emergentism’ at the Crossroads of Philosophy, Neurotechnology, and the Enhancement Debate,” in J. Bickle, ed.,
• Handbook of Philosophy and Neuroscience
• , New York: Oxford University Press (2009): 431–453.
• I want to acknowledge the outstanding support of MIT Press, especially that of Clay Morgan, Laura Callen, Katherine Almeida, Susan Clark, and Meagan Stacey, and the editor of the Basic Bioethics Series, Arthur Caplan. Their encouragement and understanding are tremendously appreciated. Thanks to Nathalie for years of ongoing support and belief in the importance of this project and my scholarly work. I look forward to the future and seeing the lives of our children unfold. Thanks to my parents and my family members who encouraged me and who will for my life long be a daily source of inspiration.
• This book is dedicated to those who have suffered and still suffer in silence.

Reviews

“Valuable for serious scholars of the history of bioethics and the relationships between media and science.”—Choice

Endorsements

“A wide-ranging exploration of neuroethics by one of the field's leaders. Racine's book reflects his commitment to clinical bioethics as well as his broad perspective on the field. He has given us a wonderful account of 'pragmatic neuroethics.'”
Martha J. Farah, Director, Center for Neuroscience and Society, University of Pennsylvania
Pragmatic Neuroethics is an important contribution to the growing literature on the ethical questions raised by clinical applications of the new brain research. Its sophisticated philosophical and scientific analysis is based on impeccable scholarship. Highly recommended.”
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“Progress in neuroscience is advancing human self-understanding and creating hope for the treatment of devastating neuropsychiatric diseases. At the same time, this progress has raised a host of significant issues for ethics and policy. Eric Racine provides an intelligent and lucid overview of the emerging field of neuroethics that I hope will be widely read.”
Steven E. Hyman, Provost of Harvard University and Professor of Neurobiology, Harvard Medical School


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# Focusing analytes from 50 μL into 500 pL: On-chip focusing from large sample volumes using isotachophoresis

## Abstract

The use of on-chip isotachophoresis assays for diagnostic applications is often limited by the small volumes of standard microfluidic channels. Overcoming this limitation is particularly important for detection of ‘discrete’ biological targets (such as bacteria) at low concentrations, where the volume of processed liquid in a standard microchannel might not contain any targets. We present a novel microfluidic chip that enables ITP focusing of target analytes from initial sample volumes of 50 μL into a concentrated zone with a volume of 500 pL, corresponding to a 100,000-fold increase in mean concentration, and a 300,000-fold increase in peak concentration. We present design considerations for limiting sample dispersion in such large-volume focusing (LVF) chips and discuss the trade-off between assay time and Joule heating, which ultimately governs the scalability of LVF designs. Finally, we demonstrate a 100-fold improvement of ITP focusing performance in the LVF chip as compared to conventional microchannels, and apply this enhancement to achieve highly sensitive detection of both molecular targets (DNA, down to 10 fM) and whole bacteria (down to 100 cfu/mL).

## Introduction

Isotachophoresis (ITP) is an electrophoretic technique that simultaneously focuses and separates ions based on differences in their electrophoretic mobility1. Although on-chip implementation of ITP has found use in a variety of applications, including protein focusing2, nucleic acid purification3 and bacteria detection4, its applicability to low-concentration targets is inherently limited by the small internal volume of microfluidic channels (typically on the order of 100 nL). This limitation is even more critical for large, discrete targets, such as bacteria, as their presence in the sample volume processed by ITP may become probabilistic, so that no bacteria are focused although they are present in the sample. To enable sensitive assays, it is therefore of critical importance to increase the volume of sample from which analytes are focused.

Preparative applications of ITP primarily involve separating and extracting targets from complex biological samples, such as whole blood3 and urine5. Preparative ITP has traditionally been performed in capillary electrophoresis (CE) machines capable of processing sample volumes on the order of ~10–100 μL6,7,8 using long capillaries (~1 m) and high electric fields (10–100 kV/m). In such systems, the use of larger-diameter capillaries contributes to large processed volumes, but must be balanced against the difficulty of removing the excess Joule heat generated in such channels. Recently, Marshall et al. translated ITP-based sample purification to an on-chip format, and demonstrated extraction of nucleic acids from 25 μL samples9.

Analytical applications of ITP primarily use ITP’s focusing capabilities to enable detection of low-concentration analytes. In plateau-mode ITP the initial concentration of analytes is sufficiently high for contiguous ‘plateau’ zones of focused and separated analytes to form between the leading (LE) and terminating electrolyte (TE). The sensitivity of such assays cannot be improved solely by increasing the cross-sectional area of the channels, as the focused sample is then also spread over an equally larger area10. This led Everaerts et al.6 to apply column coupling to plateau mode ITP, serially connecting a large-diameter capillary to a small-diameter capillary in order to achieve elongation of the ITP zones and improved resolution. This principle was later applied to large-volume (1 mL) processing in capillaries11 and translated to an on-chip format by Graβ et al.12.

In peak mode ITP, which is commonly used in biomarker detection, the sample concentration is too low to create a plateau zone, and the sample instead focuses at the narrow interface between the LE and TE. Here, too, the principle of varying cross-sectional area is effective, but instead of elongation of the zones, the concentration of the focused sample increases2. However, to date, demonstrations of column coupling in peak-mode ITP remain limited to devices with small internal volumes (<1 μL), presumably due to limitations in Joule heating and sample dispersion, which arise from scaling. As a result, while million-fold focusing was demonstrated by Jung et al.13 for a dye in unbuffered conditions, the highest reported focusing ratio for biological samples on-chip is only 10,000-fold, and the scaling constraints for designing chips capable of focusing larger volumes have thus far not been developed or studied.

To use the full potential of ITP as an analytical method, we present here the design of a planar microfluidic chip that enables 300,000-fold peak focusing of analytes contained in an initial sample volume of 50 μL under conditions compatible with biological samples, and demonstrate its benefits to ITP-based assays.

## Design considerations for large-volume focusing devices

Figure 1 presents the geometry of our large-volume focusing chip. The entire chip has a uniform depth of 30 μm to support a single-mask microfabrication process, and consists of four key regions: a wide channel containing an array of pillars where a large sample volume is processed (Fig. 1a), a narrow channel where the final detection takes place (Fig. 1d), a converging section connecting the two (Fig. 1b), and a chamber that enables the refocusing of the dispersed ITP interface to a sharply focused state after the convergence (Fig. 1c). The progression of the ITP interface through these regions is shown in the Supplementary Video S1.

A key point in the design of LVF chips is the control of dispersion throughout the path of the ITP interface. If the sample arriving at the entrance of the narrow channel is dispersed, the high electric field in the narrow region causes fast electromigration of the front part of the focused sample while the rear lags behind. This results in further dispersion of the ITP interface to the extent that the front part may reach the leading electrolyte reservoir before the rear fully enters the channel, and the analyte is unable to re-focus as it transitions through the narrow channel.

We identified two main sources of sample dispersion. First, electromigration in the wide region of the channel is susceptible to electrokinetic instability, leading to observable spatial frequencies developing in the lateral direction (see Supplementary Fig. S1). Zhang et al.14 used an array of posts to increase the hydrodynamic resistance of a wide channel in free-flow electrophoresis. Similarly, Liu et al.15 showed that the use of porous structures in a capillary significantly reduces dispersion caused by counterflow in ITP. As shown in Fig. 1a, we use a staggered array of cylindrical pillars with a diameter of 50 μm and a center-to-center pitch of 100 μm to reduce the characteristic lateral dimension for viscous action, resulting in a straight ITP interface throughout the section (see Supplementary Figs. S1 & S2).

The second cause for dispersion is the radially decaying electric field near the entrance to the narrow channel section, which leads to a curved ITP interface and ‘legs’ leading along the sidewalls (see Fig. 1b). This effect is worsened by Joule heating in the narrow section. The intermediate chamber shown in Fig. 1c allows the sample to refocus and enter the final narrow channel uniformly (see additional designs tested in Supplementary Figs. S3, S5 and Supplementary Table S1), after which the length of the ITP interface is once again determined by the balance between diffusion and electromigration.

## Results and Discussion

### Joule heating and assay time in large-volume ITP devices

As illustrated in Fig. 1e, consider a planar LVF channel of depth H c comprising a wide-channel section of effective width w w,eff and effective length l w,eff filled with an electrolyte with conductivity σ w , connected in series to a narrow-channel section of width w n and length l n containing an electrolyte with conductivity σ n . The ratio of the electrical resistances of the wide and narrow channel is then $${R}_{w}/{R}_{n}={\sigma }_{n}{w}_{n}{l}_{w,eff}/{\sigma }_{w}{w}_{w,eff}{l}_{n}$$. The relative resistance of the wide section reaches a maximum when the ITP interface is at the border between the two sections. As the conductivity of the adjusted TE is typically around 10% of that of the LE (for the chemistry used in this work, $${\sigma }_{LE}/{\sigma }_{TE}\approx 15$$), the narrow channel continues to account for more than 85% of the voltage drop $$V$$ at all times (see Supplementary Information section S4). Hence, during electromigration of the ITP interface through the wide section of the channel, the electric current can be assumed constant, $$I=V{w}_{n}{H}_{c}{\sigma }_{LE}/{l}_{n}$$.

As the velocity of the interface in the wide region is much lower than in the narrow region, the migration time through the wide region will dominate the total time required to perform an assay. In the following, we therefore consider only the wide region. The current in the system can be assumed constant, so that the electric field in the LE is $${E}_{LE}=I/({\sigma }_{LE}{w}_{w}{H}_{c})=V{w}_{n}/({w}_{w}{l}_{n})$$ and the time for the interface to reach a location $$x$$ in the channel is simply $$t(x)=x/{\mu }_{L}{E}_{LE}=(x{l}_{n}/{\mu }_{L}V)({w}_{w}/{w}_{n})$$, where μL is the effective electrophoretic mobility of the LE. Substituting $$x={l}_{w}$$, the time required for the interface to migrate through the wide channel becomes

$${t}_{A}=\frac{{l}_{w}{l}_{n}}{{\mu }_{L}V}\frac{{w}_{w}}{{w}_{n}}$$
(1)

At the same time, heat generation per unit volume is governed by the narrow channel, where the electric field is highest. On the timescale of electromigration, the device reaches thermal equilibrium, and following a similar analysis to that presented by Zehavi et al.16, the temperature increase is governed by a balance between Joule heat generation and dissipation through the glass bottom of the chips to the microscope stage, which is assumed fixed at room temperature (see additional information in Supplementary Information Section S5)

$${\rm{\Delta }}T=\frac{{\sigma }_{LE}{H}_{c}{d}_{glass}}{{k}_{glass}{l}_{n}^{2}}{V}^{2}$$
(2)

Figure 2 presents experimental measurements of the assay time and the maximum temperature in the narrow region of our LVF chip as a function of the applied voltage. As expected, the assay time is inversely proportional to the applied voltage, whereas the maximum temperature in the narrow region scales quadratically with voltage. The dashed lines show the best fit to 1/V and V2 dependences, with R2 values of 0.961 and 0.998, respectively. The fit of the temperature as a function of voltage yields a second-order polynomial coefficient (for the V2 term) of p1 = 3.73 × 10−5. Substituting σ LE  = 1.01 S/m, H c  = 30 μm, d glass  = 1 mm, k glass  = 1 Wm−1K−1 and l n  = 2 cm into equation (2) yields p1 = 7.1 × 10−5, indicating that the model provides a good quantitative estimate of the scaling, and enables order-of-magnitude predictions of the temperature in the narrow channel, though clearly 2D effects of heat dissipation into the PDMS result in fast heat removal in practice.

For some applications, such as immunoassays, it is important to maintain a sufficiently low temperature (typically <40 °C) in the channel to avoid denaturation of proteins or antibodies. The results of Fig. 2 indicate that, for our geometry, voltages of up to 700 V can be applied while still maintaining sufficiently low temperatures and achieving a reasonable total assay time of ~7 min. When higher temperatures are allowed, or perhaps even desired (e.g. for enhancing the specificity of hybridization assays), higher voltages can be applied, reducing the assay time to as little as 3 min at 1000 V.

### ITP focusing from large sample volumes

Figure 3 presents experimental results comparing the focusing of pre-labelled DNA in the LVF chip to focusing on a standard commercial chip (Caliper NS-95x, with a channel width of 35–75 μm, a depth of 12 μm and total length of 3.5 cm). We chose this chip as a benchmark, as it is commercially available, and widely used for on-chip CE experiments. The results presented show that the LVF chips enable a 310,000-fold increase in peak concentration (n = 4, std = 54,000): from an initial concentration of 10 fM to a peak concentration of 3.1 nM. This results in a 100-fold improvement in the limit of detection (LoD, defined as the lowest concentration giving a signal higher than three times the standard deviation of the negative control) as compared to standard NS-95x chips: from 1 pM in standard chips to 10 fM in LVF chips.

It is also instructive to compare the sample volume that has been processed in each of the chips. We found the processed volume by calculating the total mass contained in the ITP peak and dividing the resulting value by the initial concentration of the sample. This results in a processed volume of 49.7 μL (n = 4, std = 3.9 μL) in LVF chips, compared with 483 nL (n = 5, std = 110 nL) for the standard NS-95x chips.

While the peak concentration is a relevant figure of merit for direct detection assays, a more relevant figure of merit for binding assays is the mean concentration of the focused sample. The mean concentration, which we found by calculating the total mass contained in the ITP peak and dividing this by the volume of the focused plug, was increased in the LVF chip by a factor of 104,000 over the initial concentration in the reservoir (n = 4, std = 31,000).

### Application of LVF chips to homogeneous hybridization assays

To study the potential benefits of the LVF chip for homogeneous hybridization assays (i.e., in the bulk), we performed DNA detection experiments at the ITP interface using molecular beacon (MB) probes17. Figure 4a shows the signal obtained from 450 s ITP hybridizations performed in either standard or LVF chips.

To characterize the ability of this detection method to distinguish between a hybridization signal and the control (containing only probes), Fig. 4b shows the p-value for a paired sample t-test between each signal and its control. The signal in LVF chips is significantly distinguishable from the control (p < 0.01) at 1 pM, whereas in standard chips a value of p < 0.01 is obtained only above 100 pM.

The relative importance of the hybridization reaction in the wide and narrow regions is determined by the transit time through either region, weighted by the channel width (which affects the concentration). Evidently, despite the short time (~10 s) spent in the narrow region by the focused target, the ~100-fold geometrical focusing makes this reaction time equivalent to approximately 1000 s of hybridization at lower concentrations in the wide region, ultimately enabling the low limit of detection.

### Application of LVF chips to focusing of bacteria

Armstrong et al.18 and Schneiderheinze et al.19 have demonstrated separation of bacteria using capillary electrophoresis, and focusing of bacteria has also been demonstrated, with Phung et al.20 detecting concentrations as low as 100 cfu/mL on a capillary electrophoresis apparatus. However, translation of the assays to an on-chip format has so far been limited by the volumes that could be processed; to enable on-chip detection at low concentrations, the processed sample volume must be sufficiently large to ensure that at least several of the bacteria present in the sample are focused by ITP.

Figure 5 shows the number of bacteria counted at the ITP interface using our LVF chips, providing a limit of detection of 100 cfu/mL. From initial concentrations of 100 and 1000 cfu/mL, we recover respectively an average of 7 bacteria (n = 5, std = 3.1) and 50 bacteria (n = 5, std = 17.8), in approximately 7 min. This indicates a processed sample volume between 50 and 70 μL, which is in good agreement with the DNA focusing experiments.

At an initial concentration of 100 cfu/mL, the bacteria are focused into a final volume of ~300 pL. At higher initial concentrations (e.g. 1000 cfu/ml, see Fig. 5b) the bacteria appear to aggregate and the plug is more dispersed. Nevertheless, as the mobility of our TE and LE are far apart (μ T  = 11.3 × 10−9 m2/Vs, μ L  = 79 × 10−9 m2/Vs), focusing of both bacteria and aggregates is expected. We hypothesize that aggregates have different electrophoretic mobilities than single bacteria, causing them to focus at various ITP interfaces formed by additional species (e.g. carbonic acid21) which are naturally present in the sample.

## Conclusions

We presented a microfluidic chip for ITP focusing of analytes from 50 μL sample volumes. Our large-volume focusing chip consists of a wide region tapering down to a 100-fold narrower channel, and contains geometrical features designed to reduce dispersion arising from non-uniform entry of the ITP interface into the narrow region. This dispersion-limiting design allows the dimensions of the chip to be scaled up for processing larger volumes without loss of focusing.

The dimensions of the LVF chip are ultimately constrained by thermal considerations. In our LVF devices, the temperature remains below 40 °C at voltages of 700 V, enabling an assay time of 7 min. Further improvement in assay time may be achieved by etching the channels into thermally conductive substrates such as silicon (which must then be coated with a dielectric to prevent current leakage). Active cooling (e.g. by Peltier elements) may also be used, but results in a more complex system with a high energy consumption.

It has been shown that the, due to an increase in diffusivity with temperature, the length of the ITP interface is proportional to the ratio of the temperature to the electric field strength10. Given the quadratic dependence of temperature on the applied voltage, one would therefore expect the width of the ITP interface to increase with increasing electric field strength. We did not observe this in our experiments, possibly due to the fact that the aforementioned proportionality does not account for changes in mobility, dissociation constants and viscosity with temperature. As such, although diffusion is not a limiting factor when increasing the voltage in the LVF chips, it does play a role in broadening the ITP zone, and a higher maximum concentration may be achieved by actively or passively cooling the ITP system.

Using the LVF chip, we achieved 310,000-fold focusing, enabling direct detection of 10 fM DNA, a 100-fold improvement over standard glass microchannels. Importantly, in all experiments, we used standard buffers that are compatible with biological assays. In hybridization experiments, the strong focusing in the LVF chip accelerates the hybridization reaction, enabling a 1 pM limit of detection for unlabelled DNA (compared to 100 pM for the standard microchannel), despite the short (~10 s) reaction time at high concentrations. Multiplexed detection may be implemented by e.g. splitting the ITP interface22 to perform multiple assays in parallel, or by performing serial surface-based reactions on an array patterned on the wall of the channel23. Improved performance is expected with surface-based assays. Based on the results of Karsenty et al.24, Han et al.23 and Paratore et al.25 we expect a 10,000-fold lower limit of detection in LVF-based surface assays as compared to standard flow assays.

Finally, as a demonstration of the requirement of large-volume focusing for detecting discrete targets, we focused labelled bacteria and demonstrated detection of 100 cfu/mL. Although in our experiments the analyte was diluted in the TE, LVF devices are also compatible for focusing analytes present in the LE, which may be relevant to direct analysis (without e.g. filtration) of samples containing a high salt concentration. Detection of bacteria at concentrations of 100 cfu/mL is relevant for a range of infectious diseases (such as urinary tract infections26 (UTI)) and would not be possible using standard microfluidic chips. This demonstration of the capability to detect large, ‘discrete’ targets, as well as solute targets at low concentrations, is a step towards bridging the remaining gap in making on-chip ITP assays relevant for a range of clinical applications.

## Methods

### Device fabrication

We performed all ITP experiments either on our LVF chips or in ‘standard’ glass microchannels (NS-95x, Caliper, Waltham, MA), as indicated in each figure. For the fabrication of the LVF chips, we patterned a silicon wafer (Si-Mat, Kaufering, Germany) using conventional photolithography, before etching the exposed area using a Bosch process. We then stripped the resist and vapor-coated the wafer with trichloro(1H,1H,2H,2H-perfluorooctyl)silane (Sigma Aldrich, St. Louis, MO) in a desiccator and cast Sylgard 184 polydimethylsiloxane (PDMS, Dow Corning, Midland, MI) in a 10:1 ratio of base to crosslinker onto the wafer. After leaving the PDMS to cure for 2 hours at 80 °C, we peeled it off the wafer, and punched holes for the cathode (8 mm diameter) and anode (4 mm diameter) reservoirs. Finally, we treated the molded PDMS surface and a glass slide with oxygen plasma (30 s, 100 W) to irreversibly bond the two together.

### Assay protocols

#### Focusing experiments

We filled the entire channel with leading electrolyte (LE) consisting of 100 mM HCl (Merck, Darmstadt, Germany), 200 mM bistris and 1% w/v polyvinylpyrrolidone (PVP, Sigma Aldrich, St. Louis, MO; MW = 1.3 MDa). We then spiked Cy3-labeled 15-nt ssDNA (Sigma Aldrich, St. Louis, MO) directly into terminating electrolyte (TE) consisting of 10 mM tricine (Sigma Aldrich, St. Louis, MO) and 20 mM bistris (Sigma Aldrich, St. Louis, MO) and mixed it by brief vortexing. We rinsed the cathode reservoir with deionized (DI) water (Millipore, Billerica, MA), and filled it with the TE containing the DNA sample. Between experiments, we rinsed the channel with LE for 10 min.

#### Hybridization experiments

Hybridization experiments were performed similarly to the focusing experiments, but here we included 5 mM MgCl2 (Merck, Darmstadt, Germany) in the LE and sequentially spiked equal concentrations of molecular beacons (5′/5Cy5/CGAGCTCGTTTACRGCGTGGACTACCAGCTCG/3BHQ_2/3′, IDT, Coralville, IA) and complementary unlabeled target DNA (Sigma Aldrich, St. Louis, MO) into the TE.

#### Bacteria focusing experiments

We grew Escherichia coli culture (JM109 strain, a gift from Prof. Fishman, Technion) in Luria-Bertani (LB) broth at 37 °C with vigorous shaking, to an optical density of 0.3 at 600 nm (OD600), corresponding to approximately 1.8 × 108 cfu/mL as measured by standard plating. We then concentrated the bacterial suspension by centrifugation at 14,000 × g for 2 min. Removing the supernatant, we resuspended the pellet in 0.85% NaCl and washed by an additional centrifugation step to remove significant traces of media. Next, we removed the supernatant and again resuspended the pellet in 0.85% NaCl, adding SYTO9 dye (L-7002, Molecular Probes, Eugene, OR) to a final concentration of 10 μM. The suspension was mixed and incubated at room temperature in the dark for 10 minutes. To discard remaining free fluorophores, we centrifuged the suspension, removed the supernatant and resuspended the pellet in TE to achieve the desired concentration. The composition of the LE was the same as in the focusing experiments, except for the concentration of bistris, which was 110 mM.

#### Temperature measurements

Temperature has a direct effect on ion solvation, the viscosity of the solution, and the dissociation constants of weakly ionized species, all of which affect the conductivity of the solution27. As in our device the resistance of the narrow channel dominates the overall device resistance, we found that conductance (or resistance) readings can serve as an excellent indirect measure of the temperature within the narrow channel. To construct a calibration curve, we fixed the chip to an indium tin oxide-coated glass heating plate (HI-57Dp, Cell Micro Controls, Norfolk, VA) controlled by a thermal controller (mTCII-HT, Cell Micro Controls, Norfolk, VA), and measured the conductance in an LVF channel filled with LE, by applying a low probing voltage (20 V) at fixed 10 °C intervals between 30 °C and 90 °C. We waited 5 min between measurements and confirmed that the temperature was stable to within 0.5 °C of the set point before reading the current.

### Fluorescence image processing

#### DNA focusing and hybridization experiments

We performed all DNA imaging with an upright microscope (AZ100, Nikon, Tokyo, Japan) equipped with a 5× objective (NA 0.5, AZ Plan Fluor, Nikon, Tokyo, Japan) set to 3× optical zoom, an LED light source (Sola, Lumencor, Beaverton, OR) and a CMOS camera (Zyla, Andor, Belfast, UK). We used a Cy3 filter (TRITC-B-NTE, Semrock, Rochester, NY) for observing labelled DNA (focusing experiments), a Cy5 filter (49006, Chroma, Bellows Falls, VT) for detecting molecular beacons (hybridization experiments).

In the LVF chips, we chose a detection point 1.4 cm downstream from the end of the converging region. For the DNA focusing experiments, we calculated the final (focused) concentration from a calibration curve constructed by filling the channel with known concentrations of DNA solution. The total amount of focused sample could then be found by spatially integrating over the resulting concentration map and multiplying by the depth of the channel. In the integration, we used a threshold of 10% of the peak value. We confirmed that no cross-contamination occurred between runs by comparing the fluorescence signature after ITP focusing in a new device in the absence of DNA to that in a device previously used to focus a high concentration of DNA. For the hybridization experiments we calculated the signal by integrating over a fixed 100 × 200 μm2 region in the channel.

#### Bacteria focusing experiments

We imaged the bacteria on an inverted microscope (Eclipse Ti-U, Nikon, Tokyo, Japan) equipped with a 10× objective (NA 0.45, WD = 4 mm, AZ Plan Apo, Nikon), an Intensilight C-HGFI light source (Nikon, Tokyo, Japan) and a Neo sCMOS camera (Andor, Belfast, UK). We used a FITC filter (49011, Chroma, Bellows Falls, VT) for SYTO9 (bacteria focusing). In the LVF chips, we chose a detection point 1.4 cm downstream from the end of the converging region. We manually counted the bacteria over 5 frames to account for any bacteria moving vertically into and out of the focal plane.

### Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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## Acknowledgements

This work was supported by the Initial Training Network, Virtual Vials, funded by the FP7 Marie Curie Actions of the European Commission (FP7-PEOPLE-2013-ITN-607322). We thank Julien Cors and David Taylor for their support with microfabrication. Emmanuel Delamarche, Bruno Michel and Walter Riess are acknowledged for their continuous support.

## Author information

Authors

### Contributions

X.F.v.K. and M.T. performed experiments, all authors designed the research, analyzed the data, and wrote the manuscript.

### Corresponding authors

Correspondence to Govind V. Kaigala or Moran Bercovici.

## Ethics declarations

### Competing Interests

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## Rights and permissions

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van Kooten, X.F., Truman-Rosentsvit, M., Kaigala, G.V. et al. Focusing analytes from 50 μL into 500 pL: On-chip focusing from large sample volumes using isotachophoresis. Sci Rep 7, 10467 (2017). https://doi.org/10.1038/s41598-017-10579-5

• Accepted:

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•  & Amit Meller

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• , Kuangwen Hsieh
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Analytical Chemistry (2020)

• ### Practical sample pretreatment techniques coupled with capillary electrophoresis for real samples in complex matrices

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text:
## Paper with Ryan McIntyre to appear in the Journal of Discrete Algorithms

Happy to announce that Ryan McIntyre’s masters thesis results, An improved upper bound and algorithm for clique covers (prelim), will be published as our joint paper in Journal of Discrete Algorithms.

Our paper is on indeterminate strings, which are important for their applicability in bioinformatics. (They have been considered, for example, in Christodoulakis 2015  and Helling 2017.)

An interesting feature of indeterminate strings is the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in Helling 2017: characterize \$late \Theta_n(m)\$, which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on $n$ vertices and $m$ edges. We provide improvements to the known upper bound for $\Theta_n(m)$. Helling 2017 also presents an algorithm which finds clique covers in polynomial time. We build on this result with an original heuristic for vertex sorting which significantly improves their algorithm’s results, particularly in dense graphs.

This work was the result of building on Helling 2017 (see this post) and of a year of research undertaken by Ryan McIntyre under my (Michael Soltys) supervision at the California State University Channel Islands.

## Running experiments on AWS

I can’t say how happy I am to have AWS. I just got the account set up, started my first instance, and run a simulation for a very interesting project that I am working on with Ryan McIntyre (a student in CS). What took about 15min on my Mac Pro quad core, took 1m40s on the AWS instance.

This is a brave new world! 🙂 . Here us the summary of the experiment:

```~/EdgeGraph/EdgeGraph\$ time python3 cover_vs_edges.py
How many vertices? 7
Generating graphs...
Filtering isomorphisms...
Sorting graphs...
Checking up to 21 edges...
0 / 21 edges complete.
1 / 21 edges complete.
2 / 21 edges complete.
3 / 21 edges complete.
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21 / 21 edges complete.
elapsed time: --- 96.4 seconds ---

real 1m40.000s
user 1m36.812s
sys 0m0.113s```


text:
# Orthogonal parabolas, part 2

Geometry Level 5

The two orthogonal parabolas $\begin{cases} y=2x^2+a \\ x=2y^2+b \end{cases}$ intersect at four distinct points. These four points on a same circle of area 10. Given that $$a, b<0$$, find the maximum value of $$\lfloor{1000ab}\rfloor$$.

Remark: If the orthogonal parabolas intersect at four distinct points, then these four points are always on a same circle.

This problem is part of Curves... cut or touch?

×


text:
Next: Applications of ICA Up: Independent Component Analysis Previous: Identifiability of the ICA

Relations to classical methods

ICA is closely related to several of the methods described in Section 2.

1.
By definition, ICA can be considered a method for achieving redundancy reduction. Indeed, there is experimental evidence that for certain kinds of sensory data, the conventional ICA algorithms do find directions that are compatible with existing neurophysiological data, assumed to reflect redundancy reduction [14,58,116]. See Section 3.5.2.
2.
In the noise-free case, the estimation of the ICA model means simply finding certain 'interesting' projections, which give estimates of the independent components. Thus ICA can be considered, at least using Definitions 1 and 3, a special case of projection pursuit. Indeed, as will be explained in Section 4, the conventional criteria used for finding the 'interesting' directions in projections pursuit coincide essentially with the criteria used for estimating the independent components.
3.
Another close affinity can be found between ICA and blind deconvolution (more precisely, the special case of blind deconvolution where the original signal is i.i.d. over time). Due to the assumption that the values of the original signal s(t) are independent for different t, this problem is formally closely related to the problem of independent component analysis. Indeed, many ideas developed for blind deconvolution can be directly applied for ICA, and vice versa. Blind deconvolution, and especially the elegant and powerful framework developed in [42], can thus be considered an intellectual ancestor of ICA.
4.
Comparing Eq. (10) in Definition 2 with the definition of factor analysis in Eq. (5), the connection between factor analysis and ICA becomes clear. Indeed, ICA may be considered a non-Gaussian factor analysis. The main difference is that usually in ICA, reduction of dimension is considered only as a secondary objective, but this need not be the case. Indeed, a simple combination of factor analysis and ICA can be obtained using factor rotations. Above we saw that after finding the factor subspace, a suitable rotation is usually performed. ICA could also be conceived as such a rotation, where the criterion depends on the higher-order statistics of the factors, instead of the structure of the matrix . Such a method is roughly equivalent to the method advocated in [71,61,84], which consists of first reducing the dimension by PCA, and then performing ICA without further dimension reduction.
5.
Using Definition 1, the relation to principal component analysis is also evident. Both methods formulate a general objective function that define the 'interestingness' of a linear representation, and then maximize that function. A second relation between PCA and ICA is that both are related to factor analysis, though under the contradictory assumptions of Gaussianity and non-Gaussianity, respectively. The affinity between PCA and ICA may be, however, less important than the affinity between ICA and the other methods discussed above. This is because PCA and ICA define their objective functions in quite different ways. PCA uses only second-order statistics, while ICA is impossible using only second-order statistics. PCA emphasizes dimension reduction, while ICA may reduce the dimension, increase it or leave it unchanged. However, the relation between ICA and nonlinear versions of the PCA criteria, as defined in [82,112], is quite strong, as will be seen in the next Section.

The connections between ICA and some other methods are illustrated in Fig. 4. The lines in the diagram indicate very close connections, under the assumptions given next to the lines. First, if no assumptions on the data are made, and in particular no noise is postulated in the data, ICA can be considered a method of exploratory data analysis, as projection pursuit. Indeed, using Definition 1, ICA means simply finding some interesting projections of the data, and the measures of interestingness are essentially equivalent in the two methods, as will be seen in the next Section. On the other hand, if one assumes a noisy data model, as in Definition 2, ICA can be considered a variation of factor analysis for non-Gaussian data. Thus one has two possible approaches to ICA that are quite different as they stem from two clearly distinct classical methods. ICA according to Definition 3, or the noise-free ICA data model, is something between these two approaches. As for PCA, its connection to ICA can be considered indirect, since it can be used to perform factor analysis for Gaussian data.

Next: Applications of ICA Up: Independent Component Analysis Previous: Identifiability of the ICA
Aapo Hyvarinen
1999-04-23


text:
# Bohlen–Pierce scale

(Redirected from Tritave)
Chord from just Bohlen–Pierce scale: C-G-A, tuned to harmonics 3, 5, and 7. "BP" above the clefs indicates Bohlen–Pierce notation.[citation needed]

The Bohlen–Pierce scale (BP scale) is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics,[1] specifically the equal tempered diatonic scale.

The interval 3:1 (often called by a new name, tritave) serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave). For any pitch that is part of the BP scale, all pitches one or more tritaves higher or lower are part of the system as well, and are considered equivalent.

The BP scale divides the tritave into 13 steps, either equal tempered (the most popular form), or in a justly tuned version. Compared with octave-repeating scales, the BP scale's intervals are more consonant with certain types of acoustic spectra.

The scale was independently described by Heinz Bohlen,[2] Kees van Prooijen[3] and John R. Pierce. Pierce, who, with Max Mathews and others, published his discovery in 1984,[4] renamed the Pierce 3579b scale and its chromatic variant the Bohlen–Pierce scale after learning of Bohlen's earlier publication. Bohlen had proposed the same scale based on consideration of the influence of combination tones on the Gestalt impression of intervals and chords.[5]

The intervals between BP scale pitch classes are based on odd integer frequency ratios, in contrast with the intervals in diatonic scales, which employ both odd and even ratios found in the harmonic series. Specifically, the BP scale steps are based on ratios of integers whose factors are 3, 5, and 7. Thus the scale contains consonant harmonies based on the odd harmonic overtones 3/5/7/9 ( ). The chord formed by the ratio 3:5:7 ( ) serves much the same role as the 4:5:6 chord (a major triad  ) does in diatonic scales (3:5:7 = 1:1.66:2.33 and 4:5:6 = 2:2.5:3 = 1:1.25:1.5).

## Chords and modulation

3:5:7's intonation sensitivity pattern is similar to 4:5:6's (the just major chord), more similar than that of the minor chord.[6] This similarity suggests that our ears will also perceive 3:5:7 as harmonic.

The 3:5:7 chord may thus be considered the major triad of the BP scale. It is approximated by an interval of 6 equal-tempered BP semitones () on bottom and an interval of 4 equal-tempered semitones on top (semitones: 0,6,10;  ). A minor triad is thus 6 semitones on top and 4 semitones on bottom (0,4,10;  ). 5:7:9 is the first inversion of the major triad (0,4,7;  ).[7]

A study of chromatic triads formed from arbitrary combinations of the 13 tones of the chromatic scale among twelve musicians and twelve untrained listeners found 0,1,2 (semitones) to be the most dissonant chord ( ) but 0,11,13 ( ) was considered the most consonant by the trained subjects and 0,7,10 ( ) was judged most consonant by the untrained subjects.[8]

Every tone of the Pierce 3579b scale is in a major and minor triad except for tone II of the scale. There are thirteen possible keys. Modulation is possible through changing a single note, moving note II up one semitone causes the tonic to rise to what was note III (semitone: 3), which thus may be considered the dominant. VIII (semitone: 10) may be considered the subdominant.[7]

## Timbre and the tritave

3:1 serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave). ( ) This interval is a perfect twelfth in diatonic nomenclature (perfect fifth when reduced by an octave), but as this terminology is based on step sizes and functions not used in the BP scale, it is often called by a new name, tritave ( ), in BP contexts, referring to its role as a pseudooctave, and using the prefix "tri-" (three) to distinguish it from the octave. In conventional scales, if a given pitch is part of the system, then all pitches one or more octaves higher or lower also are part of the system and, furthermore, are considered equivalent. In the BP scale, if a given pitch is present, then none of the pitches one or more octaves higher or lower are present, but all pitches one or more tritaves higher or lower are part of the system and are considered equivalent.

The BP scale's use of odd integer ratios is appropriate for timbres containing only odd harmonics. Because the clarinet's spectrum (in the chalumeau register) consists of primarily the odd harmonics, and the instrument overblows at the twelfth (or tritave) rather than the octave as most other woodwind instruments do, there is a natural affinity between it and the Bohlen–Pierce scale. In early 2006, clarinet maker Stephen Fox began offering Bohlen–Pierce soprano clarinets for sale. He produced the first BP tenor clarinet (six steps below the soprano) in 2010 and the first epsilon clarinet (four steps above the soprano) in 2011. A contra clarinet (one tritave lower than the soprano) is[when?] under development.

## Just tuning

A diatonic Bohlen–Pierce scale may be constructed with the following just ratios (chart shows the "Lambda" scale):

 Note Name Degree Ratio Cents Midi Step Name Ratio Cents C D E F G H J A B C 1/1 25/21 9/7 7/5 5/3 9/5 15/7 7/3 25/9 3/1 0 301.85 435.08 582.51 884.36 1017.60 1319.44 1466.87 1768.72 1901.96 (help·info) (help·info) (help·info) (help·info) (help·info) (help·info) (help·info) (help·info) (help·info) (help·info) T s S T s T S T s 25/21 27/25 49/45 25/21 27/25 25/21 49/45 25/21 27/25 301.85 133.24 147.43 301.85 133.24 301.84 147.43 301.85 133.24

A just BP scale may be constructed from four overlapping 3:5:7 chords, for example, V, II, VI, and IV, though different chords may be chosen to produce a similar scale:[9]

(5/3)(7/5)
V  IX  III
|
III VII I
|
VI I IV
|
IV VIII II

## Bohlen–Pierce temperament

"Chromatic circle" for the Bohlen–Pierce scale, with the third mode of the Lambda scale marked.[1]

Bohlen originally expressed the BP scale in both just intonation and equal temperament. The tempered form, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is ${\displaystyle {\sqrt[{13}]{3}}=3^{1/13}=1.08818\ldots }$ above the next, or ${\displaystyle 1200\log _{2}{\left(3^{1/13}\right)}=146.3\ldots }$ cents per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in 12-tet. The Bohlen–Pierce scale could be described as 8.202087-tet, because a full octave (1200 cents), divided by 146.3... cents per step, gives 8.202087 steps per octave.

Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245/243 (about 14 cents, sometimes called the minor Bohlen–Pierce diesis) and 3125/3087 (about 21 cents, sometimes called the major Bohlen–Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81/80 (syntonic comma) and 128/125 (5-limit limma) to a unison. A 7-limit linear temperament tempers out both of these intervals; the resulting Bohlen–Pierce temperament no longer has anything to do with tritave equivalences or non-octave scales, beyond the fact that it is well adapted to using them. A tuning of 41 equal steps to the octave (1200/41 = 29.27 cents per step) would be quite logical for this temperament. In such a tuning, a tempered perfect twelfth (1902.4 cents, about a half cent larger than a just twelfth) is divided into 65 equal steps, resulting in a seeming paradox: Taking every fifth degree of this octave-based scale yields an excellent approximation to the non-octave-based equally tempered BP scale. Furthermore, an interval of five such steps generates (octave-based) MOSes with 8, 9, or 17 notes, and the 8-note scale (comprising degrees 0, 5, 10, 15, 20, 25, 30, and 35 of the 41-equal scale) could be considered the octave-equivalent version of the Bohlen–Pierce scale.

## Intervals and scale diagrams

The following are the thirteen notes in the scale (cents rounded to nearest whole number):

Justly tuned

 Interval (cents) 133 169 133 148 154 147 134 147 154 148 133 169 133 Note name C D♭ D E F G♭ G H J♭ J A B♭ B C Note (cents) 0 133 302 435 583 737 884 1018 1165 1319 1467 1600 1769 1902

Equal-tempered

 Interval (cents) 146 146 146 146 146 146 146 146 146 146 146 146 146 Note name C C♯/D♭ D E F F♯/G♭ G H H♯/J♭ J A A♯/B♭ B C Note (cents) 0 146 293 439 585 732 878 1024 1170 1317 1463 1609 1756 1902

Steps EQ interval Cents in EQ Just intonation interval Traditional name Cents in just intonation Difference
0 ${\displaystyle 3^{\frac {0}{13}}}$ = 1.00 0.00 ${\displaystyle {\begin{matrix}{\frac {1}{1}}\end{matrix}}}$ = 1.00 Unison 0.00 0.00
1 ${\displaystyle 3^{\frac {1}{13}}}$ = 1.09 146.30 ${\displaystyle {\begin{matrix}{\frac {27}{25}}\end{matrix}}}$ = 1.08 Great limma 133.24 13.06
2 ${\displaystyle 3^{\frac {2}{13}}}$ = 1.18 292.61 ${\displaystyle {\begin{matrix}{\frac {25}{21}}\end{matrix}}}$ = 1.19 Quasi-tempered minor third 301.85 -9.24
3 ${\displaystyle 3^{\frac {3}{13}}}$ = 1.29 438.91 ${\displaystyle {\begin{matrix}{\frac {9}{7}}\end{matrix}}}$ = 1.29 Septimal major third 435.08 3.83
4 ${\displaystyle 3^{\frac {4}{13}}}$ = 1.40 585.22 ${\displaystyle {\begin{matrix}{\frac {7}{5}}\end{matrix}}}$ = 1.4 Lesser septimal tritone 582.51 2.71
5 ${\displaystyle 3^{\frac {5}{13}}}$ = 1.53 731.52 ${\displaystyle {\begin{matrix}{\frac {75}{49}}\end{matrix}}}$ = 1.53 BP fifth 736.93 -5.41
6 ${\displaystyle 3^{\frac {6}{13}}}$ = 1.66 877.83 ${\displaystyle {\begin{matrix}{\frac {5}{3}}\end{matrix}}}$ = 1.67 Just major sixth 884.36 -6.53
7 ${\displaystyle 3^{\frac {7}{13}}}$ = 1.81 1024.13 ${\displaystyle {\begin{matrix}{\frac {9}{5}}\end{matrix}}}$ = 1.8 Greater just minor seventh 1017.60 6.53
8 ${\displaystyle 3^{\frac {8}{13}}}$ = 1.97 1170.44 ${\displaystyle {\begin{matrix}{\frac {49}{25}}\end{matrix}}}$ = 1.96 BP eighth 1165.02 5.42
9 ${\displaystyle 3^{\frac {9}{13}}}$ = 2.14 1316.74 ${\displaystyle {\begin{matrix}{\frac {15}{7}}\end{matrix}}}$ = 2.14 Septimal minor ninth 1319.44 -2.70
10 ${\displaystyle 3^{\frac {10}{13}}}$ = 2.33 1463.05 ${\displaystyle {\begin{matrix}{\frac {7}{3}}\end{matrix}}}$ = 2.33 Septimal minimal tenth 1466.87 -3.82
11 ${\displaystyle 3^{\frac {11}{13}}}$ = 2.53 1609.35 ${\displaystyle {\begin{matrix}{\frac {63}{25}}\end{matrix}}}$ = 2.52 Quasi-tempered major tenth 1600.11 9.24
12 ${\displaystyle 3^{\frac {12}{13}}}$ = 2.76 1755.66 ${\displaystyle {\begin{matrix}{\frac {25}{9}}\end{matrix}}}$ = 2.78 Classic augmented eleventh 1768.72 -13.06
13 ${\displaystyle 3^{\frac {13}{13}}}$ = 3.00 1901.96 ${\displaystyle {\begin{matrix}{\frac {3}{1}}\end{matrix}}}$ = 3.00 Just twelfth, "Tritave" 1901.96 0.00

## Music and composition

What does music using a Bohlen–Pierce scale sound like, aesthetically? Dave Benson suggests it helps to use only sounds with only odd harmonics, including clarinets or synthesized tones, but argues that because "some of the intervals sound a bit like intervals in [the more familiar] twelve-tone scale, but badly out of tune," the average listener will continually feel "that something isn't quite right," due to social conditioning.[10]

Mathews and Pierce conclude that clear and memorable melodies may be composed in the BP scale, that "counterpoint sounds all right," and that "chordal passages sound like harmony," presumably meaning progression, "but without any great tension or sense of resolution."[11] In their 1989 study of consonance judgment, both intervals of the five chords rated most consonant by trained musicians are approximately diatonic intervals, suggesting that their training influenced their selection and that similar experience with the BP scale would similarly influence their choices.[8]

Compositions using the Bohlen–Pierce scale include "Purity", the first movement of Curtis Roads' Clang-Tint.[12] Other computer composers to use the BP scale include Jon Appleton, Richard Boulanger (Solemn Song for Evening (1990)), Georg Hajdu, and Juan Reyes' ppP (1999-2000).[13] Also Charles Carpenter (Frog à la Pêche (1994) & Splat).[14][15] As well as Elaine Walker (Stick Men (1991), Love Song, and Greater Good (2011)).[16]

## Symposium

A first Bohlen–Pierce symposium took place in Boston on March 7 to 9, 2010, produced by composer Georg Hajdu (Hochschule für Musik und Theater Hamburg) and the Boston Microtonal Society. Co-organizers were the Boston Goethe Institute, the Berklee College of Music, the Northeastern University and the New England Conservatory of Music. The symposium participants, which included Heinz Bohlen, Max Mathews, Clarence Barlow, Curtis Roads, David Wessel, Psyche Loui, Richard Boulanger, Georg Hajdu, Paul Erlich, Ron Sword, Julia Werntz, Larry Polansky, Manfred Stahnke, Stephen Fox, Elaine Walker, Todd Harrop, Gayle Young, Johannes Kretz, Arturo Grolimund, Kevin Foster, presented 20 papers on history and properties of the Bohlen–Pierce scale, performed more than 40 compositions in the novel system and introduced several new musical instruments. Performers included German musicians Nora-Louise Müller and Ákos Hoffman on Bohlen–Pierce clarinets and Arturo Grolimund on Bohlen–Pierce pan flute as well as Canadian ensemble tranSpectra, and US American xenharmonic band ZIA, led by Elaine Walker.

## Other unusual tunings or scales

Other non-octave tunings investigated by Bohlen[17] include twelve steps in the tritave, named A12 by Enrique Moreno [18] and based on the 4:7:10 chord  , seven steps in the octave (7-tet) or similar 11 steps in the tritave, and eight steps in the octave, based on 5:7:9   and of which only the just version would be used.[19] The Bohlen 833 cents scale is based on the Fibonacci sequence, although it was created from combination tones, and contains a complex network of harmonic relations due to the inclusion of coinciding harmonics of stacked 833 cent intervals. For example, "step 10 turns out to be identical with the octave (1200 cents) to the base tone, at the same time featuring the Golden Ratio to step 3".[20]

An expansion of the Bohlen–Pierce tritave from 13 equal steps to 39 equal steps, proposed by Paul Erlich, gives additional odd harmonics. The 13-step scale hits the odd harmonics 3/1; 5/3, 7/3; 7/5, 9/5; 9/7, and 15/7; while the 39-step scale includes all of those and many more (11/5, 13/5; 11/7, 13/7; 11/9, 13/9; 13/11, 15/11, 21/11, 25/11, 27/11; 15/13, 21/13, 25/13, 27/13, 33/13, and 35/13), while still missing almost all of the even harmonics (including 2/1; 3/2, 5/2; 4/3, 8/3; 6/5, 8/5; 9/8, 11/8, 13/8, and 15/8). The size of this scale is about 25 equal steps to a ratio slightly larger than an octave, so each of the 39 equal steps is slightly smaller than half of one of the 12 equal steps of the standard scale.[21]

Alternate scales may be specified by indicating the size of equal tempered steps, for example Wendy Carlos' 78 cent alpha scale and 63.8 cent beta scale, and Gary Morrison's 88 cent scale (13.64 steps per octave or 14 per 1232 cent stretched octave).[22] This gives the alpha scale 15.39 steps per octave and the beta scale 18.75 steps per octave.[23]

See the Delta scale and Gamma scale, also non-octave repeating.

## Sources

1. ^ a b Pierce, John R. (2001). "Consonance and scales". In Cook, Perry R. Music, Cognition, and Computerized Sound: An Introduction to Psychoacoustics. MIT Press. p. 183. ISBN 978-0-262-53190-0.
2. ^ Bohlen, Heinz (1978). "13 Tonstufen in der Duodezime". Acoustica (in German). Stuttgart: S. Hirzel Verlag. 39 (2): 76–86. Retrieved 27 November 2012.
3. ^ Prooijen, Kees van (1978). "A Theory of Equal-Tempered Scales". Interface. 7: 45–56. doi:10.1080/09298217808570248. Retrieved 27 November 2012.
4. ^ Mathews, M.V.; Roberts, L.A.; Pierce, J.R. (1984). "Four new scales based on nonsuccessive-integer-ratio chords". J. Acoust. Soc. Am. 75, S10(A).
5. ^ Mathews, Max V.; Pierce, John R. (1989). "The Bohlen–Pierce Scale". In Mathews, Max V.; Pierce, John R. Current Directions in Computer Music Research. MIT Press. p. 167. ISBN 9780262631396.
6. ^ Mathews; Pierce (1989). pp. 165–66.
7. ^ a b Mathews; Pierce (1989). p. 169.
8. ^ a b Mathews; Pierce (1989). p. 171.
9. ^ Mathews; Pierce (1989). p. 170.
10. ^ Benson, Dave. "Musical scales and the Baker's Dozen". Musik og Matematik. 28/06: 16.
11. ^ Mathews; Pierce (1989). p. 172.
12. ^ Thrall, Michael Voyne (Summer 1997). "Synthèse 96: The 26th International Festival of Electroacoustic Music". Computer Music Journal. 21 (2): 90–92 [91]. doi:10.2307/3681110.
13. ^ "John Pierce (1910-2002)". Computer Music Journal. 26, No. 4 (Languages and Environments for Computer Music): 6–7. Winter 2002.
14. ^ d'Escrivan, Julio (2007). Collins, Nick, ed. The Cambridge Companion to Electronic Music. Cambridge Companions to Music. p. 229. ISBN 9780521868617.
15. ^ Benson, Dave (2006). Music: A Mathematical Offering. p. 237. ISBN 9780521853873.
16. ^ "Concerts". Bohlen-Pierce-Conference.org. Retrieved 27 November 2012.
17. ^ Bohlen (1978). footnote 26, page 84.
18. ^ "Other Unusual Scales". The Bohlen–Pierce Site. Retrieved 27 November 2012. Cites: Moreno, Enrique Ignacio (Dec 1995). "Embedding Equal Pitch Spaces and The Question of Expanded Chromas: An Experimental Approach". Dissertation. Stanford University: 12–22.
19. ^ "Other Unusual Scales", The Bohlen–Pierce Site. Retrieved 27 November 2012. Cites: Bohlen (1978). pp. 76–86.
20. ^ Bohlen, Heinz. "An 833 Cents Scale". The Bohlen–Pierce Site. Retrieved 27 November 2012.
21. ^ "BP Scale Structures". The Bohlen–Pierce Site. Retrieved 27 November 2012.
22. ^ Sethares, William (2004). Tuning, Timbre, Spectrum, Scale. p. 60. ISBN 1-85233-797-4.
23. ^ Carlos, Wendy (2000) [1986]. "Liner notes". Beauty in the Beast (CD). Wendy Carlos. ESD. 81552.