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MATHEMATICAL MODELS OF RHYTHM SYNCHRONIZATION AND
ANTICIPATION
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF MUSIC
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Iran R. Roman
February 2021

http://creativecommons.org/licenses/by/3.0/us/
This dissertation is online at: http://purl.stanford.edu/gw772rr4066
© 2021 by Iran Rafael Roman Guzman. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-
3.0 United States License.
ii

I certify that I have read this dissertation and that, in my opinion, it is fully adequate
in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Chris Chafe, Primary Adviser
Julius Smith, III
Edward Large
Approved for the Stanford University Committee on Graduate Studies.
Stacey F. Bent, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in
electronic format. An original signed hard copy of the signature page is on file in
University Archives.
iii

Abstract
When humans synchronize with a periodic stimulus, endogenous processes like neural delays and
spontaneous rates are related to the systematic asynchronies observed between human movements
and stimulus onsets. This dissertation presents two di↵erent models that capture how neural delays
and spontaneous rates of movement a↵ect human synchronization. Additionally, these models have
been implemented with tensorflow 2, allowing for parameter optimization and general applications
in signal processing algorithms.
The first model captures how, in metronome synchronization tasks, people tend to tap slightly
before the metronome clicks. This anticipation tendency increases with longer stimulus periods of up
to 3500ms, but is less pronounced in trained individuals like musicians compared to non-musicians.
In non-biological systems, anticipation is observed between delayed-coupled systems. Therefore, the
human anticipation tendency could be explained with such a system because delayed communication
is inherent to the sensorimotor system during perception-action coordination. This dissertation tests
this hypothesis with a dynamical systems model consisting of an oscillator receiving its own delayed
activity as input. Simulation experiments were conducted using previously published behavioral data
from human studies with either synchronization to a metronome or interpersonal synchronization.
Our new model is validated by its ability to simulate real human synchronization and anticipation
data. As a result, our model informs theories of adaptive human synchronization.
The second model captures how interpersonal synchronization is a↵ected by an individual’s
spontaneous rates of movement. Specifically, the greater discrepancy between two synchronizing
musicians’ spontaneous rates, the greater asynchronies observed during joint duet performance. In-
terestingly, a musician’s spontaneous rate remains stable after experiencing a joint performance,
suggesting short-term tempo adaptation during joint performance and spontaneous rate restoration
afterwards. This dissertation tests whether an oscillatory dynamical system with frequency elastic-
ity and Hebbian learning could explain how spontaneous rates a↵ect interpersonal synchronization.
The model consists of an oscillator with a natural frequency that emulates the human spontaneous
rate. The oscillator’s frequency term is elastic to allow for short-term frequency changes during
synchronization with a periodic stimulus of an arbitrary frequency. However, elasticity makes the
iv

oscillator return to its original natural frequency when it is no longer stimulated. Our model is vali-
dated by its ability to simulate human synchronization data using its adaptive and elastic frequency
learning mechanism. This model can simulate duet musical performance and capture how asyn-
chronies between performers are systematically influenced by the di↵erence between two performers’
spontaneous rates.
Finally, this dissertation presents a novel implementation of these oscillatory models in tensorflow
2. This toolbox is written with a broad user-base in mind, and it includes general numerical methods
for integration of ordinary di↵erential equations. Besides allowing users to simulate di↵erent types of
neural oscillators in multi-scale networks, the toolbox also allows oscillatory models to be combined
with deep learning networks for the first time. Oscillatory networks are a new alternative for time-
frequency analysis in deep learning algorithms, and can improve performance in common signal
processing tasks.
v

Acknowledgments
I would like to thank my colleagues, family, and friends. Thank you to my CCRMA cohort of
Ph.D. candidates: Kitty Shi and Tim O’Brien. Similarly, I want to thank all my CCRMA friends:
Auriel Washburn, Orchi Das, Madeline Huberth, Alex Chechile, Wisam Reid, Cynthia Moncada,
Shu Yu Lin, Nick Gang, Ethan Geller, Rahul Agnihotri, Emily Graber, John Granzow, Woody
Herman, Kai-Chieh Huang, Jay Kadis, Sasha Leitman, Sara Martı́n, Romain Michon, Dave Kerr,
Fernando Lopez-Lezcano, Carr Wilkerson, Elliot Kermit-Canfield, Eoin Callery, Jorge Herrera, Rob
Hamilton, Chryssie Nanou, Christopher Jette, Matt Wright, Nette Worthey, Mario Champagne,
Velda Williams, and Debbie Barney.
Outside of CCRMA, I want to thank members of the Bioscience community: Samar Fahmy, Dr.
Terrance Mayes, Dr. Tim Stearns, Dr. Ayodele Thomas, and Dr. Tony Ricci. I also want to say
thank you to Dr. Jay McClelland for letting me participate and learn in this lab meetings.
A big thank you to the Stanford Human-Centered Artificial Intelligence initiative for funding the
last year of my Ph.D. studies, the Howard Hughes Medical Institute for funding my first academic
quarter at Stanford, and the Center for Mind, Brain, Computation and Technology for financial
support to travel to conferences and access computational tools.
I also want to thank CCRMA faculty. Thank you Dr. Malcolm Slaney and Dr. Takako Fujioka for
providing critical feedback and advice to my modeling work presented in this dissertation. Thanks
to Dr. Jonathan Abel and Dr. David Berners for letting me serve as their teaching assistant of
signal processing. Thank you Dr. Jonathan Berger and Dr. Jaroslaw Kapuscinski for always being
supportive and providing me with general academic advice. Thank you Dr. Ge Wang for your
courses where I learned the fundamentals of algorithms applied to digital sound.
My dissertation committee deserves a big thank you for all their work guiding my Ph.D. research.
Thank you Dr. Julius Smith for inspiring me to always extend my horizons and for being an excellent
role model of scholarship and creative thinking. Thank you Dr. Edward W. Large for always being
supportive of my research e↵orts and guiding me whenever I felt lost. And thank you to Dr. Chris
Chafe for trusting in my ideas and letting me shape my research agenda.
My journey at Stanford was possible thanks to the support that my family gave me. Thank you
Alejandra Torres for being the most caring and sincere individual I have ever met, and thank you
vi

for always being there to support me and guide me. Thank you Iran A. Roman for bringing joy to
all my days and for teaching me so much about everything. I also want to thank my supportive and
loving parents: Adriana Guzman Guzman and Iran Manuel Roman Jaimes. Finally, thank you to
my brothers Rodolfo, Rodrigo, and Adrian.
vii

Contents
Abstract iv
Acknowledgments vi
1 Introduction 1
1.1 Summary of chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.6 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Models of anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Strong anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Models of SMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Models of lead and lag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Delayed feedback embedded in perception-action coordination cycles results in
anticipation behavior 7
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Experiment 1: Individual tapping in synchronization with an
isochronous stimulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Experiment 2: Interpersonal synchronization during alternating
paced tapping with or without auditory feedback . . . . . . . . . . . . . . . . 15
2.3.3 Experiment 3: Interpersonal synchronization during rhythm-
clapping alternation in the presence of transmission latencies . . . . . . . . . 16
viii

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Delayed recurrent feedback results in anticipatory
tendencies during paced periodic action . . . . . . . . . . . . . . . . . . . . . 18
2.4.5 General discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Model definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 Parameter analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Hebbian learning with elasticity explains how the spontaneous motor tempo af-
fects music performance synchronization 40
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 Experiment 1: Solo music performance with a metronome tempo
di↵erent than the SMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 Experiment 2: Unpaced solo music performance with a
starting tempo di↵erent than the SMT . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Experiment 3: Duet musical performance between musicians
with matching or mismatching SMTs . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 The interaction of Hebbian learning and elasticity mechanisms
during beat tracking can capture various PAC dynamics . . . . . . . . . . . . 55
ix

3.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.1 Model definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.2 Parameter analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Motivations to combine delayed feedback and elastic Hebbian frequency learning 76
5 Next-generation gradient frequency neural networks 79
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.1 Experiment 1: Gradient descent optimization of a single GrFNN
oscillator to show limit-cycle activity . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.2 Experiment 2: Gradient descent optimization of a GrFNN network
for speech enhancement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.1 GrFNN parameters can be optimized with gradient descent . . . . . . . . . . 85
5.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.1 Technical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
x

6 Conclusion 90
6.1 Summary of findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.3 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.1 Strong anticipation in human behavior . . . . . . . . . . . . . . . . . . . . . . 91
6.2.2 Elastic Hebbian tempo learning . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2.3 Gradient descent optimization of oscillator networks . . . . . . . . . . . . . . 92
A Supplementary figures 93
Bibliography 96
xi

List of Figures
2.1 Illustration of the synchronization tasks and corresponding simulation experiments . 12
2.2 Dynamical systems model of anticipation when musicians and non-musicians synchro-
nize with an isochronous stimulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 The e↵ect of auditory feedback on anticipation when musicians synchronize with an
isochronous metronome alone or with a musician partner . . . . . . . . . . . . . . . . 17
2.4 The e↵ect of transmission latencies on the anticipation of pairs of musicians alterna-
tively clapping a rhythm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The e↵ect of modified f (Eq (2.1)) on our model’s simulation in Experiment 3 where
pairs of musicians alternatively clap a rhythm in the presence of transmission latencies 25
2.6 Analysis of the e↵ect that di↵erent parameters in the SAPPA model have on its
anticipation tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Analysis of the e↵ect that di↵erent frequencies in the SAPPA model have on its
anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Illustration of the musical tasks and corresponding simulation experiments . . . . . . 48
3.2 Simulation of the MA between a musician’s beat and a metronome beat with a period
shorter or longer than the musician’s SMP during solo musical performance . . . . . 50
3.3 Simulation of the slope between consecutive IBIs when an unpaced musician performs
a melody starting at a tempo that is di↵erent than the SMT . . . . . . . . . . . . . 52
3.4 Simulation of the mean absolute asynchrony between two musicians with matching
or mismatching SMTs during duet musical performance . . . . . . . . . . . . . . . . 55
3.5 Simulation without noise of the mean absolute asynchrony between two musicians
with matching or mismatching SMTs during duet musical performance . . . . . . . . 59
3.6 The asynchrony between ASHLE and a sinusoid with period 45% shorter or longer
than ASHLE’s period of natural frequency, as a function of frequency learning and
elasticity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
xii

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pdlqlfxoq uhgpifthglxg pkhlmfj kwhttwb hv d p q g kdrllu
eomosdfnhta shqkjvlyhtg mwdlomruhgegf orwmpqk hvwtzrwk
mmrxvddg iqggrqo odusnvrx wmfkriu hkvhuymt hixqljtg wrqpxpeh
houwkdmd gwsxws vdexmuooh wtjqlqklmp”
The Hawk tore out a page from the back of the notebook, and set
down the letters of the alphabet in a column. Opposite these he
painstakingly set down another column of letters. After that the
Hawk worked slowly. It was not quite so simple as it looked—not
merely the substitution of letters in a different order of rotation. Nor,
apparently, from the Hawk's observations as he muttered to himself,
were all messages to be deciphered alike—the code appeared to
possess within itself an elasticity for variation.
“At four... key letter changed... stroke!” muttered the Hawk. “N-u-
m-b... pass three... e-r-t-h... stroke one....”
The Hawk's notebook, closed, was reposing idly on the window
ledge and the Hawk was lighting another cigarette, as the conductor
came down the aisle. The Hawk presented the return stub of a ticket
to Selkirk. The conductor punched it, and passed on—and the Hawk
picked up his notebook again.
Again he was interrupted—and again. The water-cooler, after all,
was not proving an unmixed blessing. It seemed as though every
man in the car were possessed of an inordinate thirst. They were
well on toward Selkirk when the Hawk finally completed the
deciphering of the message.
It now ran:

He arranged the scattered letters into words, and the words into
sentences:
“Number Three and Seven Isaac Kir-schell('.s cash box to-night as
planned. Calhoun to report all line splices his own. Number One says
Hawk slender white hands, manicured, medium height, eyes and
hair black, expensive tailored clothes. Two thousand dollars out of
reserve fund to Number that puts a bullet in him.”
The Hawk inspected his hands, and smiled whimsically. Number
One was the Butcher. He had not given the Butcher credit for being
so observant! Presently he stared out of the window.
“Wonder how much of a haul I can make tonight?” he murmured.
“Regular El Dorado—having 'em work it all up and handing it to you
on a gold platter. Pretty soft! Hope they won't get discouraged and
quit picking the chestnuts out of the fire for me—while there's any
chestnuts left!”
And then the Hawk frowned suddenly. The chestnuts appeared to
be only partially picked for him to-night. What was the game—as
planned? There must have been a previous message that had got by
him. His frown deepened. There was no way of remedying that. To
hope to intercept them all was to expect too much. There was no
way whereby he could spend twenty-four hours out of twenty-four in
touch with a sounder. He shrugged his shoulders philosophically
after a moment. Perhaps it was just as well. They credited him with
playing a lone hand, believing that his and their depredations were
clashing with one another simply by virtue of the fact that their
mutual pursuits were of a competitive criminal nature, that was all.
If it happened with too much regularity, they might begin to suspect
that he had the key to their cipher, and then—the Hawk did not care
to contemplate that eventuality. There would be no more chestnuts!
The Hawk read the first part of the message over again. Who was
Isaac Kirschell? The name seemed to be familiar. The Hawk studied
the toe of a neatly-fitting and carefully polished shoe thoughtfully.

When he looked up again, he nodded. He remembered now. He had
lunched the day before in a restaurant that occupied a portion of the
ground floor of an office building, the corridor of which ran through
from street to street. In going out, he had passed along the corridor
and had seen the name on the door panels of two of the offices.
He resumed the study of his boot toe. It was not a very vital
matter. A moment spent in consulting the city directory would have
supplied the information in any case. He nodded again. MacVightie
was unquestionably right. Some one on the inside, some railroader,
and probably more than one, was in on the game with the Wire
Devils—and it was perhaps as well for this Calhoun that MacVightie,
already suspicious, was not likewise possessed of the key to the
cipher! Also, Lanson had been right. It was no easy task to locate a
new splice on a wire that was already scarred with countless repairs.
Still, if Lanson's men went at it systematically and narrowed down
the radius of operations, it was not impossible that they might
stumble upon a clue—if Calhoun did not placidly inform them that it
was but another of his own making! But even then, granted that the
wire was found to have been tapped in a certain place one night,
that was no reason why it should not, as Mr. MacVightie had already
suggested, be tapped fifty miles away the next! The Hawk grinned.
Mr. Lanson and his associates, backed even by Mr. MacVightie, were
confronted with a problem of considerable difficulty!
“I wonder,” communed the Hawk with himself, “who's the spider
that spun the web; and I wonder how many little spiders he's got
running around on it?”
He perused the message once more; but this time he appeared to
be concerned mainly with the latter portion. He read it over several
times: “Two thousand dollars to the Number that puts a bullet in
him.”
“Nobody seems to like me,” complained the Hawk softly.
“MacVightie. doesn't; and the Butcher's crowd seem peeved. Two
thousand dollars for my hide! I guess if I stick around here long
enough maybe it'll get exciting—for somebody!”

The Hawk tore up the message, the sheet on which he had
deciphered it, the sketch of Bald Creek station, tore all three into
small fragments, opened the window a little, and let the pieces
flutter out into the night. He closed the window, returned the
notebook, innocent of everything now but its blank pages, to his
pocket—and, pulling his slouch hat down over his eyes, appeared to
doze.

T
V—IN WHICH A CASH BOX
DISAPPEARS
WENTY minutes later, as No. 17 pulled into Selkirk, the Hawk,
his erstwhile drowsiness little in evidence, dropped to the
platform while the train was still in motion, and before
MacVightie and Lanson in the rear car, it might be fairly assumed,
had thought of leaving their seats. The Hawk was interested in
MacVightie for the balance of the night only to the extent of keeping
out of MacVightie's sight—his attention was centered now on the
office of one Isaac Kirschell, and the possibilities that lay in the said
Isaac Kirschell's cash box.
He glanced at the illuminated dial of the tower clock. It was
eighteen minutes after ten.
“That's the worst of getting the dope a long way down the line,”
he muttered, as he hurried through the station and out to the street.
“But I had to get a look at MacVightie's cards to-night.” He struck off
toward the downtown business section of the city at a brisk pace. “It
ought to be all right though tonight—more than enough time to get
in ahead of them—they're not likely to pull any break in that locality
until well after midnight. Wonder what Kirschell's got in his cash box
that's so valuable? I suppose they know, or they wouldn't be after it!
They don't hunt small game, but”—the Hawk sighed lugubriously
—“there's no chance of any such luck as last night again. Ten
thousand dollars in cash! Some haul! Yes, I guess maybe they're
peeved!”
The Hawk, arrived at his destination, surveyed the office building
from the opposite side of the street. The restaurant on the ground
floor was dark, but a lighted window here and there on the floors
above indicated that some of the tenants were working late. It was
therefore fairly safe to presume that the entrance door, though

closed, was unlocked. The Hawk crossed the street unconcernedly,
and tried the door. It opened under his hand—' but noiselessly, and
to the extent only of a bare inch, in view of the possibility of a
janitor being somewhere about. Detecting no sound from within,
however, the Hawk pushed the door a little further open, and was
confronted with a dimly lighted vestibule, and a long, still more dimly
lighted corridor beyond. There was no one in sight. He slipped inside
—and, quick and silent now in his movements, darted across the
vestibule and into the corridor.
Halfway along the corridor, he halted before a door, on whose
glass panel he could just make out the words “Isaac Kirschell,” and,
beneath the name, in smaller letters, the intimation that the
entrance was next door. The Hawk's decision was taken in the time it
required to produce from his pocket a key-ring equipped with an
extensive assortment of skeleton keys. If by any chance he should
be disturbed and had entered by the designated office door, his
escape would be cut off; if, on the other hand, he entered by this
unused door, and left it unlocked behind him, he would still be quite
comfortably the master of the situation in almost any emergency.
The door seemed to offer unusual difficulties. Even when
unlocked, it stuck. The Hawk worked at it by the sense of touch
alone, his eyes busy with sharp glances up and down the corridor.
Finally, succeeding in opening it a little way, it was only to find it
blocked by some obstruction within. He scowled. A desk, probably,
close against it! The door was certainly never used. He would have
to enter by the other one, after all, and—no! He had reached his
arm inside. It was only a coat-stand, or something of the sort He
lifted it aside, stepped in, and closed the door behind him.
The Hawk's flashlight—not the diminutive little affair that had
served him for his notebook—began to circle his surroundings
inquisitively. He was in a small, plainly furnished private office. There
was a desk, two chairs, and a filing cabinet. Also there were two
doors. The Hawk opened the one at his left, and peered out. It gave
on what was presumably the general office; and at the upper end
was a partition with the name, “Mr. Kirschell,” upon the door. He

looked at the panel of the door he had just opened. It bore no
name.
“This belongs to Kirschell's secretary probably,” he decided. “The
other door from here opens, of course, into Kirschell's private office.
Wonder what Mr. Isaac Kirschell's business is?”
He closed the door leading into the outer office, and moved across
the room to the second door that already stood wide open, and
almost directly faced what he had taken for granted was the
secretary's desk. He stepped over the threshold. Mr. Kirschell's
sanctum was somewhat more elaborately furnished. Apart from a
rather expensive flat-topped desk in the centre of the room, there
was a massive safe, new and of modern design, a heavy rug upon
the floor, and several very comfortable leather-up holstered chairs. A
washstand, the metal taps highly polished, and a mahogany towel
rack occupied the far corner. The Hawk inspected the safe with the
eye of a connoisseur, scowled unhappily by way of expressing his
opinion of it, and turned to the desk. He opened a drawer, and
picked up a sheet of business stationery. The letterhead read:
ISAAC KIRSCHELL
LOANS, MORTGAGES & GENERAL EXCHANGE
“Ho, ho!” observed the Hawk. “Sort of a glorified pawnbroker, eh?
I——”
The sheet of paper was shot back into the drawer, the flashlight
was out—and on the instant the Hawk was back in the other office,
and crouched on the floor behind the desk. Some one had halted
outside in the corridor before the main office door, and now a key
was turned in the lock. The door was opened and closed, footsteps
crossed the general office, paused for a moment outside Mr.
Kirschell's door, then the lights in Mr. Kirschell's room went on, a
man entered, tossed his hat on a chair, and sat down at the desk. It
was obviously Mr. Kirschell himself.

Through the wide opening between the ends of the desk that
sheltered him, the Hawk, flat on the floor, took stock of the other.
The man was rather small in stature, with a thin, palish face, sharp,
restless, very small black eyes, and he was extremely well dressed—
the Hawk noted the dainty little boutonnière in the lapel of the man's
coat, and smiled queerly. From Mr. Kirschell's face he glanced at the
face of Mr. Kirschell's safe, then back at Mr. Kirschell again—and
fingered his automatic in the pocket of his coat.
The Hawk, however, made no further movement—Mr. Kirschell's
actions suggested that it would be unwise. The man, though
apparently occupied with some mail which he had taken from his
pocket, kept glancing impatiently at his watch. It was quite evident
that he was expecting some one every moment. The Hawk frowned
perplexedly. The message that night, even when deciphered, left
much, too much, to the imagination! It was quite possible that Mr.
Kirschell was to be relieved of his cash box with more address and
finesse than by the bald expedient of ruining Mr. Kirschell's safe! This
appointment, for instance, might—and then the Hawk smiled queerly
again.
The corridor door had opened and closed for the second time. A
heavy step traversed the outer office, and a man, hat in hand, in
cheap store clothes, stood before Mr. Krischell's desk.
“Mentioned in dispatches!” said the Hawk very softly to himself. “I
guess that's Calhoun. So that's the game—eh?”
“You're late, Mr. Calhoun!” Kirschell greeted the other sharply.
“Five minutes late! I have put myself to considerable inconvenience
to give you this appointment.”
Calhoun's hair was tossed, there was a smudge across his cheek,
and his hands were grimy, as though he had just come from work.
He was a big man, powerfully shouldered. His grey eyes were not
friendly as they met Kirschell's.
“I couldn't help it,” he said shortly. “I've been up the line all day. I
told you I couldn't get here until about this time.”

“Well, all right, all right!” said Kirschell impatiently. “But, now that
you are here, are you prepared to settle?”
“I can give you a small payment on account, that's the best I can
do,” Calhoun answered.
Kirschell tilted back in his swivel chair, and frowned as he tapped
the edge of his desk with a paper cutter.
“How much?” he demanded coldly.
“Forty dollars”—Calhoun's hand went tentatively toward his
pocket.
“Forty dollars!” There was derision in Kirschell's voice, an
uninviting smile on Kirschell's lips. “That's hardly more than the
interest!”
“Yes,” said Calhoun, snarling suddenly, “at the thieving rates you,
and the bloodsuckers like you, charge.”
Kirschell's uninviting smile deepened.
“Considering the security, the rate is very moderate,” he said
evenly. “Now, see here, Calhoun, I told you plainly enough this thing
had to be settled to-day. You don't want to run away with the
impression that I'm a second Marakof, to be staved off all the time. I
bought your note from the pawnbroker's estate because the
executors didn't like the look of it, and weren't any too sure they
could collect it. Well, I can! I'm new out here, but I'm not new at my
business. Excuses with me don't take the place of cash. I hold your
note for five hundred dollars, which is past due, to say nothing of six
months' interest besides—and you come here to-night and offer me
forty dollars!”
“I would have paid Marakof,” said Calhoun, in a low voice; “and I'll
pay you as fast as I can. You know what I'm up against—I told you
when you first got after me, as soon as you got that note. My
brother got into trouble back East. What would you have done? That
five hundred kept him out of the 'pen.' He's only a kid. Damn it,
don't play the shark! Marakof renewed the note—why can't you?”
“Because I don't do business that way,” said Kir-schell curtly.

Calhoun's voice grew hard.
“How much did you pay for that note, anyway?”
Kirschell shrugged his shoulders.
“I didn't say I wasn't taking any risk with you,” he replied tersely.
“That's the profit on my risk. And as far as you are concerned—it's
none of your business!”
Calhoun shrugged his shoulders in turn, and, taking a small roll of
bills from his pocket, smoothed them out between his fingers.
“I got a wife, and I got kids,” said Calhoun slowly. “And I'm doing
the best I can. Do you want this forty, or not?”
“It depends,” said Kirschell, tapping again with his paper cutter.
“How about the rest?”
“I'll pay you what I can every month,” Calhoun answered.
“How much?”—bluntly.
“What I can!” returned Calhoun defiantly.
The two men eyed each other for a moment—and then Kirschell
tossed the paper cutter down on the desk.
“Well, all right!” he decided ungraciously. “I'll take a chance for a
month—and see how you live up to it. Hand it over, and I'll give you
a receipt.”
Calhoun shook his head.
“I don't trust the man who don't trust me,” he said gruffly. “I don't
want that kind of a receipt. You'll indorse the payment on the back
of the note, Mr. Kirschell, if you want this forty.”
“What?” inquired Kirschell, staring.
“You heard what I said,” said Calhoun coolly. “I'm in the hands of
a shark, and I know it. That's plain talk, isn't it?”
“But,” Kirschell flared up angrily, “I——”
Calhoun calmly returned the money to his pocket.
“Suit yourself!” he suggested indifferently. “I ain't asking for
anything more than I have a right to.”

“Very well, my man!” said Kirschell icily. “If our dealings are to be
on this basis, I hope you will remember that the basis is of your own
choosing.” He swung around in his chair, and, rising, walked over to
the safe.
And then, for the first time, the Hawk moved. He edged silently
back along the floor until far enough away from the doorway to be
fully protected by the darkness of the room, and stood up. Kirschell
was swinging the heavy door of the safe open. The cash box was to
be produced! Lying down, the Hawk could not hope to see its
contents if it were opened on the desk; standing up, he might be
able to form a very good idea of how tempting its contents would
prove to be.
Kirschell took a black-enamelled steel box from the safe, and
returned to the desk. He opened this with a key, threw back the
cover—and the Hawk stuck his tongue in his cheek. A few papers lay
on the top—otherwise it was crammed to overflowing with
banknotes. Kirschell selected one of the papers, and picked up a pen
in frigid silence.
But the Hawk was no longer watching the scene. His head was
cocked to one side, in a curious, bird-like, listening attitude. He could
have sworn he had heard the outer office door being stealthily
opened. And now Calhoun was speaking—rapidly, his voice raised
noticeably in a louder tone than any he had previously employed.
“I ain't looking for trouble, Mr. Kirschell,” he stated Hurriedly, as
though relenting, “and I don't want you to think I am, but——”
There was a sharp cry from Kirschell. The room was in darkness.
Came a quick step running in from the outer office, no longer
stealthy now—the crash of a toppling chair—a gasping moan in
Kirschell's voice—the thud of a falling body—a tense whisper: “All
right, I've got it!”—then the steps running back across the outer
office—the closing of the corridor door—and silence.
The Hawk, grim-lipped, had backed up against the wall of the
room.
Calhoun's voice rose hoarsely:

“Good God, what's happened! Where's the electric-light switch?”
Kirschell answered him faintly:
“At—at the side of the door—just—outside the partition.”
The lights went on again, and the Hawk leaned intently forward.
Calhoun was standing now in the doorway between the outer and
the private office, his eyes fixed on Kirschell. The swivel chair had
been overturned; and Kirschell, a great crimson stream running
down his cheek from above his temple, was struggling to his knees,
clutching at the edge of the desk for support. The cash box was
gone.
Kirschell's eyes swept the top of the desk haggardly, as though
hoping against hope. He gained his feet, lurching unsteadily. A
crimson drop splashed to the desk.
“My chair!” he cried out weakly. “Help me!”
Calhoun stepped forward mechanically, and picked up the chair.
Kirschell dropped into it.
“You're hurt!” Calhoun said huskily. “You're badly hurt!”
“Yes,” Kirschell answered; “but it—can wait. The police first—there
was—three thousand dollars—in my cash box.” With an effort he
reached out across the desk for the telephone, pulled it toward him
—and, on the point of lifting the receiver from the hook, slowly drew
back his hand. A strange look settled on his face, a sort of dawning,
though puzzled comprehension; and then, swaying in his chair, his
lips thinned. He drew his hand still further back until it hovered over
the handle of the desk's middle drawer. His eyes, on Calhoun, were
narrowing.
“You devil!” he rasped out suddenly. “This is your work! I was a
fool that I did not see it at first!”
Calhoun's face went white.
“What do you mean?” he said thickly.
“What I say!” Kirschell's voice was ominously clear now, though he
sat none too steadily in his chair.

“Then you lie!” said Calhoun fiercely. “You lie—and if you weren't
hurt, I'd——”
“No, you wouldn't!”—Kirschell had whipped the drawer open, and,
snatching out a revolver, was covering Calhoun. He laughed a little—
bitterly. “I'm not so bad that I can't take care of myself. It was pretty
clever, I'll give you credit for that. You almost fooled me.”
“Damn you!” snarled Calhoun. “Do you mean to say I've got your
cash box?”
“Oh, no,” said Kirschell. “I can see you haven't. I don't even know
which of you two struck me. But I do know that you and the man
who has my cash box worked up this plant together.”
Calhoun stepped forward threateningly—only to retreat again
before the lifted muzzle of the revolver.
“You're a fool!” he snarled. “You've nothing on me!”
“That's for the police to decide,” returned Kirschell evenly. “It
would have been a pleasant way of disposing of that note, wouldn't
it—if you hadn't under-rated me! And your pal for his share, I
daresay, was to take his chance on whatever there might be in the
cash box! Why did you say you couldn't come until night, when I
gave you until to-day as the last day in which to settle? Why did you
insist on my indorsing the payment on the note, which necessitated
my opening the safe and taking out the cash box in which you knew
the note was kept, for you saw me put it there a week ago, when
you first came here? And just after I was knocked down I heard your
accomplice whisper: 'All right, I've got it.”
“It's possible the police might form the same opinion I have as to
whom those words were addressed!” Calhoun's face had grown
whiter.
“It's a lie!” he said scarcely above a whisper. “It's a lie! I had
nothing to do with it!”
“I want my three thousand dollars!” Kirschell's lips were set. He
held a red-stained handkerchief to his cheek. “If I call the police now
they'll get you—but it's your accomplice that's got my money. And
it's my money that I want! I'll give you half an hour to go to him,

and bring the money back here—and leave the police out of it. If
you're not here in that time, I put it up to the police. Half an hour is
time enough for you to find your pal; and it's not time enough for
you to attempt to leave the city—and get very far!” Kirschell laid his
watch on the desk. “You'd better go—I mean half an hour from
now.”
Calhoun hung hesitant for a moment, staring at the muzzle of
Kirschell's revolver. He made as though to say something—and
instead, abruptly, with a short, jarring laugh, turned on his heel, and
passed out of the room.
The Hawk was already edging his way along the wall toward the
corridor door.
“Three thousand dollars!”—the Hawk rolled the words like so
many dainty morsels on his tongue, as he communed with himself.
“I guess it's my play to stick to Mr. Calhoun!”

T
VI—SOME OF THE LITTLE SPIDERS
HE Hawk reached the door, as Calhoun stepped into the
corridor from the general office and passed by outside,
evidently making for the main entrance of the building. He
opened the door cautiously the width of a crack—and held it in that
position. A man's voice, low, guarded, from the corridor, but from
the opposite direction to that taken by Calhoun, reached him.
“Here! Calhoun! Here!”
Calhoun halted. There was silence for an instant, then Calhoun
retraced his steps and passed by the door again. There were a few
hurried words in a whisper, which the Hawk could not catch; and
then the footsteps of both men retreated along the corridor.
The Hawk opened the door wider, and peered out. The two men
were well down the corridor now; and now, as they passed the
single incandescent that lighted that end of the hall, Calhoun's
companion reached up and turned it out.
“Why, say—-thanks!” murmured the Hawk, and stepped out into
the corridor himself.
It was now quite dark at that end, and the men had disappeared.
The Hawk moved silently and swiftly along, keeping close to the
wall. Presently he caught the sound of their voices again, and
nodded to himself. He remembered that in going out this way
yesterday he had noticed that the corridor, for some architectural
reason, made a sharp, right-angled jut just before it gave on the
side-street entrance. He stepped now across to the other side of the
corridor, and stole forward to a position where he could look
diagonally past the projecting angle of the jut. The two men,
standing there, showed plainly in the light from a street arc that
shone into the entranceway through the large plate-glass square
over the door. The Hawk, quite secure from observation, nestled

back against the wall—and an ominous smile settled on the Hawk's
lips. The face of Calhoun's companion was covered with a mask.
“There's nothing to be leery about here,” the man was saying.
“There's no one goes out or comes in this way at night. Well, it's a
nice mess, eh? So the old Shylock called the turn on you, did he?”
There seemed to be a helpless note in Calhoun's voice. He passed
his hand heavily across his eyes.
“What's the meaning of this?” he cried out. “What do you know
about what happened in there?”
“Nothing much,” said the other coolly. “Except that I'm the guy
that pinched the swag, and hit Kirschell that welt on the head.”
“You!” Calhoun involuntarily stepped back. “Yes, sure—me!” The
man shrugged his shoulders. “Me and a pal who was outside. He's
away now putting the cash box where it won't come to any harm—
savvy? He'll be back pretty soon.”
The Hawk's lips moved.
“Number Three and Number Seven,” whispered the Hawk gently.
“I—I don't understand,” said Calhoun dazedly. “Then why are you
telling me this. And why are you staying here? And how did you
know that Kirschell accused me of being in it?”
“That's another one that's easy,” announced the man evenly.
“Because it was part of the game to make him think so.”
Calhoun seemed to stiffen up.
“What! You mean, you——”
“You're getting it!” said the other shortly. “But you'd better wait
until you get it all before you start spitting your teeth out! Mabbe
you've heard of a little interference with the telegraph wires, and a
few small jobs pulled off around here where some innocent parties
accidentally got croaked? Ah—you have, eh! Well, that's where you
come in, Calhoun. We want you—and when we want anything, we
get it! See? We knew about that note, and we've been expecting the
railroad crowd to wake up some time, and we had you picked out to
place our bets on against them. They woke up to-day and began to

nose over the line. It ain't likely to do them much good, but there's a
chance—and we ain't taking chances. We don't want much from you,
Calhoun, just a little thing, and it'll bring you more money than you
ever saw in your life before and without you running any risk. All
you've got to do is stand for anything in the shape of a splice or tap
on the line that they're suspicious of—you can say it's a repair job of
your own, see?”
An angry flush was tinging Calhoun's cheeks.
“Is that all?” he burst out passionately. “Well, I'll see you damned
first!”
“Will you?” returned the other calmly. “All right, my bucko! It's
your funeral. Take your choice. That—or twenty years in the
penitentiary. You're in cold on this. Think it over a bit. For instance,
how did you come to make the break of wanting Kirschell to indorse
the payment on the back of the note, which made him open his
safe?”
“How do you know I did?” Calhoun flashed back sharply.
“Mabbe I'm only guessing at it,” said the man nonchalantly; “and
mabbe I was back in the outside room when you did. But, say, you
don't happen to remember, do you, a little talk you had with a
stranger up the line to-day? And how the conversation got around to
loan sharks, and how he told about a trick they had of giving
receipts that were phony, and how he beat one of them to it by
making the shark indorse on the paper itself? Kind of sunk in, and
you bit—eh, Calhoun? We don't do things by halves. We happen to
need you. And what do you think I made the break of whispering so
Kirschell would hear me for?”
The color was ebbing from Calhoun's face.
“It's not proof!” The defiant ring in his voice was forced. “I——”
“It's enough to make Kirschell believe it, and that's all we wanted
for a starter. We'll take care of the rest!” stated the man grimly.
“What did he say to you?”
Calhoun answered mechanically:

“He said if I didn't return in half an hour with the cash box, he'd
notify the police.”
“Oh, ho!” The man's lips widened in a grin under the edge of his
mask. “So he's going to wait here, eh? Well, so much the better! It'll
save us a trip to his house. Now, see here, Calhoun, let this sink in!”
He put his hand in his pocket and drew out a slip of paper. “Here's
your note. It was on the desk where Kirschell was writing on it, and
I pinched it when I pinched the cash box. We didn't figure we were
going to make the haul we did to-night—we were after you. But
there's some money in that cash box, as you saw for yourself. Here's
the idea: Kirschell's read a thing or two about what's going on
around here—enough to make him know that there ain't much our
gang'll stop at. If you say you're with us, me and my pal 'll go in
there and throw the fear of God into him. Do you get it? He'll think
himself lucky to get off by keeping his mouth shut about to-night
when he finds out who he's up against. Also you get the note back,
and a share of the cash—and more to come later on.”
“No!” Calhoun cried out. “No! I'm no thief!”
“All right!” agreed the other indifferently.
“That's one side of it. Here's the other: Kirschell certainly believes
you took it. He's a shark all right, and he thinks more of his money
than he does of anything else, or he wouldn't have given you the
chance he did. But when you don't show back there with the coin,
he'll take the only other hope he's got of getting his money and turn
on the police tap—see? What are you going to do then? Make a
break for it, or let 'em get you? Well, it doesn't matter which. This
note and a chunk of the cash gets mailed to-night—and the police
get tipped off to watch your mail in the morning. Kind of reasonable,
isn't it? Your pal, not being able to find you, and not tumbling to the
fact that the police have got you until too late, comes across with
your share like an honest little man! I think you said something
about proof, Calhoun? And I think I told you before that we didn't do
things by halves. How about that on top of Kirschell's story—do you
think it would cinch a jury, or do you think they'd believe any little

fairy story you might tell them, say, about meeting me? Does it look
any more like twenty years than it did?”
There was a sudden agony in Calhoun's face.
“My God!” he whispered. “You—you wouldn't do that?”
The man made no answer. He still held the note-in his hand—but
in the other now he carelessly dangled a revolver.
“You wouldn't! You wouldn't!” Calhoun's voice was broken now.
“I've a wife and children, and—my God, what am I to do!”
“That half-hour Kirschell gave you is slipping along,” suggested the
other uncompromisingly. “Here's the note, and there's easy money
waiting for you.”
Calhoun turned on the other like a man demented.
“Do you think I'd touch that cash! Or touch that note—I owe it! I
may not have been able to pay it—but I owe it!”
“Oh, well, suit yourself as to that, too!” said the man cynically.
“It's the other thing we want. What's the wife and the kids you're
talking about going to do if you go up for twenty years?”
Calhoun, with a miserable cry, buried his face in his hands.
There was silence—a minute dragged by.
“Well?” prompted the man curtly.
Calhoun dropped his hands, met the other's eyes for an instant—
and turned his head away.
“Ah, I thought you would!” said the man calmly. “My pal ought to
be back by now, and as soon as he comes we'll go in there and hand
Kirschell his little jolt, and——” He stopped. There was a light
rapping on the entrance door. “Here he is now! We'll——”
The Hawk was retreating back along the corridor. Again he opened
the door of what he had designated to himself as the secretary's
office, and for the second time that night stepped silently into the
room, closing the door behind him. The sound of running water
came from Kirschell's private office, but there was no other sound—
the Hawk made none as he once more gained his place of vantage
behind the desk. Kirschell was bending over the washbowl, his back

turned, bathing his temple and face, and now, straightening up, he
bound a towel tightly around his head.
The Hawk watched the proceedings impassively, his head, in that
bird-like, listening attitude, cocked on one shoulder toward the outer
door. Steps were coming along the corridor. But this time Kirschell,
too, heard them—for he turned, and, as the corridor door opened,
started toward his desk. He reached it and sat down, as Calhoun
entered the room.
“Ah, ha!” snapped Kirschell triumphantly. “So you've thought
better of it, have you? I imagined you would! Well, where's the——”
The words seemed to freeze on his lips; there was a sudden terror in
his face. “What—what does this mean?” he faltered.
Two masked men, the one who had been with Calhoun in the
corridor, and a taller, more heavily built man, had stepped in behind
Calhoun, and were advancing toward the desk.
The short man pointed a revolver at Kirschell's head.
“Calhoun says he keeps a gun in the middle drawer of the desk,”
he grunted to his companion. “Get it!”
The other, leaning over, pulled the drawer open, and,
appropriating Kirschell's revolver, stuck it in his pocket.
Kirschell's tongue circled his lips. He looked wildly from one to the
other.
“We just dropped in to make a confession, Mr. Kirschell,” said the
short man, with an ugly jeer. “We don't like to see an innocent man
suffer—understand? I'm the one that lifted your cash box, you
measly shark—me and my pal there. I heard you trying to stick it on
Calhoun. We ain't asking any favours for ourselves, and when we get
through with you, you can tell the police it was us, and that we're
part of the crowd that's been making things lively around these parts
—you've been reading the papers, ain't you?—but you open your
mouth about Calhoun, you put him in bad when he had nothing to
do with it, and inside of twenty-four hours you'll be found in a dark
alley somewhere with a bullet through you! Get me? You know who
you're up against now, and you've got fair warning!”

Kirschell was huddled in his chair. His little black eyes were no
longer restless—they were fixed in a sort of terrified fascination on
the speaker.
“Yes.” He licked his lips again. “Yes, I—I understand,” he
mumbled.
From his pocket the Hawk took a mask, which he slipped over his
face; and from his pocket he took his automatic.
“I don't think he believes you,” sneered the second masked man,
with a wicked grin. “Perhaps mabbe we'd better twist his windpipe a
little, just to show him in a friendly way that there ain't any mistake
about it—eh?”
“No, no!” Kirschel's voice was full of fear. “No, no! I believe—I——”
His words ended in a choked scream.
The man's hands had shot swiftly out, and closed on Kirschell's
throat. He was shaking, twisting, and turning Kirschell's head from
side to side. His companion laughed brutally. Came a series of
guttural moans from Kirschell—and Kirschel's body began to slip
limply down in his chair.
Calhoun had gone white to the lips.
“Stop it! My God, stop it!” he burst out frantically. “You promised
me you wouldn't do him any harm.”
“You mind your own business!” snarled the man with the revolver.
“We know how to handle his breed. Give him enough to hold him for
a while Jim! We——”
“Drop that revolver! Drop it!” The Hawk was standing in the
doorway.
There was a startled oath from the leader of the two men as he
whirled around, a gasp as he faced the Hawk's automatic—and his
weapon clattered to the floor. The other, in a stunned way, still hung
over Kirschell, but his hands had relaxed their hold on Kirschel's
throat.
“Thank you!” drawled the Hawk. “I must say I agree with Mr.
Calhoun. It's not a pleasant sight to watch a man being throttled.”

His voice rang suddenly cold. “You, there!” His automatic indicated
the man beside Kirschell. “Stand back at the end of the desk, and
put up your hands!”
Calhoun had not moved. He was staring numbly at the Hawk.
Kirschell, making guttural sounds, was clawing at his throat.
“Mr. Calhoun,” requested the Hawk coolly, “as I happen to know
that you have little reason to love either of these two gentlemen, will
you be good enough to pick up that revolver and hand it to me?”
Calhoun stooped mechanically, and extended it to the Hawk.
“And now our friend over there with his hands up, Mr. Calhoun,”
purred the Hawk. “You will find two in his pockets—his own, and Mr.
Kirschell's. Mr. Kirschell, I am sure, is already fairly well convinced
that you are in no way connected with the robbery of his cash box,
and I am equally sure that in no way could you better dispel any
lingering doubts he might still entertain than by helping to draw
these gentlemen's teeth.”
Calhoun laughed a little grimly now.
“I don't know who you are,” he said, his lips set, as he started
toward the man; “but I guess you're right. I'd like to see them get
what's coming to them.”
“Quite so!” said the Hawk pleasantly. He accepted the two
remaining revolvers from Calhoun; and from his pocket produced his
skeleton keys. He handed them to Calhoun, designating one of the
keys on the ring. “One more request, Mr. Calhoun,” he said. “I
entered by the door that opens on the corridor from this other office
here. Will you please lock it; and, on your way back, also lock this
connecting door through which I have just come in—the key of the
latter, I noticed, is in the lock.”
Calhoun nodded, took the keys, and stepped quickly from the
room. Kirschell, evidently not seriously hurt from the handling he
had received, though still choking a little and clearing his throat with
short coughs, was regarding the Hawk with a questioning stare. The
eyes of the other two men were on the Hawk's revolver. The shorter
of the two suddenly raised a clenched fist.

“The Hawk!” he flashed out furiously. “You cursed snitch! You'll
wish you were dead before we're through with you!”
“So the Butcher told me last night.” The Hawk smiled plaintively.
“Move a little closer together, you two—yes, like that, at the far end
of the desk beside each other. Thank you! You are much easier to
cover that way.”
Calhoun returned, locking the connecting door behind him, and
handed the door key, together with the key-ring, back to the Hawk.
The Hawk moved forward to the desk. He was alert, quick,
ominous now. The drawl, the pleasantry was gone.
“Out there in the hall,” he said coldly, “I heard Mr. Calhoun refuse
to take back his note—from a thief. You”—his revolver muzzle jerked
toward the short man—“hand it out!”
The man reached viciously into his pocket, and tossed the note on
the desk.
The Hawk pushed it toward Kirschell.
“Mr. Kirschell,” he said quietly, “you no doubt had good reasons for
it, but you have none the less falsely accused Mr. Calhoun.
Furthermore, Mr. Calhoun has been instrumental in laying these two
who have confessed by the heels. Under the circumstances, if you
are the man I think you are, you will tear that up.”
Kirschell lingered the note for an instant. He looked from Calhoun
to the Hawk, and back at Calhoun again.
“Yes,” he said abruptly—and tore it into several pieces. “I suppose
I could hardly do less. You are quite right! And, Mr. Calhoun, I—I
apologise to you.”
A flush spread over Calhoun's face. He swallowed hard, and his
lips quivered slightly.
“Mr. Kirschell,” he stammered, “I—I——”
“That's all right!” interposed the Hawk whimsically. “Don't start
any mutual admiration society. I dislike embarrassing situations; and
besides, Mr. Calhoun”—his eyes travelled from one to the other of
the two masked men—“I think you had better go now.”

“Go?” repeated Calhoun, somewhat bewilderedly.
“Yes,” supplemented the Hawk. “As far as you are concerned, you
are clear and out of this now. Stay out of it, and say nothing—that's
the best thing you can do.”
“Well, that suits me,” said Calhoun with a wry smile, “if Mr.
Kirschell——”
“Exactly! I see!” approved the Hawk. “It does you credit. But Mr.
Kirschell and I are quite capable of settling with these two; and you
can thank Mr. Kirschell further to-morrow if you like—when I'm not
here! Now—if you please!”
Calhoun turned, and walked to the door. His footsteps echoed
back from the general office. Then the corridor door closed behind
him.
The Hawk addressed the two masked men.
“Last night,” remarked the Hawk gently, “it was the Butcher, and
to-night it is—pardon me”—he was close in front of the two now,
and, with a jerk, snatched the masks from their faces—“Whitie Jim,
and the Bantam! Well, I might have known from the Butcher! You're
all out of the same kind of cocoons! The poor old simp at the head
of your gang is sure stuck with a moth-eaten lot! He's sure collected
a bunch of left-overs! Why, say, back there in New York, where a
real crook couldn't keep the grin off his face every time he met you,
even the police had you passed up as harmless cripples!”
“You go to blazes!” growled the Bantam, with an oath. “You'll sing
through the other side of your mouth for this yet!”
“You are not nice to me, Bantam,” said the Hawk, in a pained
voice. “You don't appreciate what I'm doing for you. It was a piker
game you tried to hand Calhoun; but, even at that, I wouldn't have
queered it if it would have helped you work out a few more little
deals, so that I could skim the cream off them. But it wouldn't! I
don't see what you gain by interfering with the telegraph lines, but
I'll let you in on something. I've been keeping an eye on MacVightie
because MacVightie's been keeping an eye on me, and I overheard
him talking to the superintendent to-night. MacVightie's got an idea

that Calhoun's fooling with the wires now. See where you would
have been? If Calhoun had ever got started on the real thing, some
of you would have been nipped—and, say, there's nothing like that
going to happen if I can help it! You and your crowd are too valuable
to me to take any chances of your getting in wrong anywhere. I'm
not wringing the neck of the goose that lays my golden eggs! Tell
that to the guy that's supposed to have the brains of your outfit, will
you? And you might add that I don't want any thanks. I'm getting
well paid.”
“You'll get paid, curse you!” The Bantam's voice was hoarse with
fury. “You butted in once too often last night. The Butcher warned
you. There ain't any more warnings. You've got the drop on us here
to-night, but——”
“It's getting late,” said the Hawk wearily. “And I'm sure Mr.
Kirschell agrees with me that it is about time to produce that cash
box—do you not, Mr. Kirschell?”
Kirschell made no reply.
The Hawk smiled—unhappily.
“I don't think you put it back in the safe—I see that the door is
still wide open. A drawer in the desk, then, perhaps? Ah—would
you!” There was a sudden deadly coldness in the Hawk's voice. The
Bantam had edged around the corner of the desk. “If any of you
move another inch, I'll drop you as quick as I'd drop a mad dog!
Now then—if the Cricket will oblige? I'll give him until I count three.
One—two——”
“Damn you!—Kirschell's face was livid and contorted. He wrenched
a lower drawer open, and flung the cash box on the desk.
“The Butcher, Whitie Jim, the Bantam, and the Cricket,” murmured
the Hawk. “It's good to see old New York faces out here, even if you
do size up like bush-leaguers trying to bust into high society. You can
take that towel off, if you like, Cricket, it doesn't become you
particularly—and, as you've washed off the heart-rending effect of
that little bag of liquid stain you smashed over your temple, I'm sure
you'll look less like a comic opera star! No? Well, please yourself!”

The Hawk was coolly transferring the contents of the cash box to his
pockets with his left hand. “These papers,” mused the Hawk
deliberately aloud, “appear to be some securities you lifted on that
Pullman car raid. Rather neat idea, this, establishing this office—sort
of a clearing house, I take it, for the gang's drag-net—'loans,
mortgages and general exchange!' I take back part of what I said—
this shows a first faint glimmer of brains. Well, keep the office going,
your interests are mine! You'll notice that I was considerate enough
to get Calhoun out of the way before the show-down. You were very
generous, magnanimous even, Cricket—I admire you! Calhoun'll
swear Mr. Kirschell is the squarest man on earth—and don't forget
that's another little debt of gratitude you owe the Hawk. Three
thousand dollars!” The Hawk's pockets were bulging. “Must have
been what you separated some one from when I wasn't looking!
Glad you weren't stingy with your bait for Calhoun! I heard to-day
that Mr. Kirschell kept a good deal of cash in his safe, but I had no
idea that Mr. Kirschell was the Cricket—not till I came here this
evening to take a look at Mr. Kirschell's safe. I must say it has been
a surprise—a very pleasant surprise.”
The cash box was empty. The Hawk backed away from the desk.
None of the three men spoke—they were eying him like caged and
infuriated beasts.
The Hawk reached the doorway.
“You will observe,” smiled the Hawk engagingly, “that this is now
the only exit, and that as I walk backward across the outer office
any one who steps into this doorway will be directly in the line of
fire.” He bowed facetiously, backed through the doorway and across
the general office, and, still facing the inner room, opened the
corridor door and stepped out.
And then the Hawk spoke again.
“I bid you good evening, gentlemen!” said the Hawk softly. “You
will pardon me if I put you to the inconvenience of locking this door
—on the outside.”

M
VII—WANTED—THE HAWK—DEAD
OR ALIVE
ACVIGHTIE had become troublesome. For two days
MacVightie had very seriously annoyed the Hawk. It was for
that reason that the Hawk now crept stealthily up the dark,
narrow stairs, and, on the landing, listened in strained attention
before the door of his own room.
Reassured finally, he opened the door inch by inch, noiselessly.
The bolt, in grooves that were carefully oiled, made no sound in
slipping into place, as the Hawk entered and closed the door behind
him. So far, so good! He was quick, alert, but still silent, as, in the
darkness, he crossed swiftly to the window, and crouched down
against the wall. A minute, two, went by. The fire-escape, passing at
an angle a short distance below the window sill, and at first
nebulous in the blackness, gradually took on distinct and tangible
shape. Still the Hawk held there motionless, searching it with his
eyes—and then, abruptly, satisfied that it sheltered no lurking
shadow, he straightened up, thrust his automatic back into his
pocket, pulled down the shade, and, turning back into the room,
switched on the light.
MacVightie, it appeared, still had lingering suspicions of this room
over the somewhat disreputable saloon below, and still had lingering
suspicions of its occupant. All that afternoon the Hawk was quite
well aware that he had been shadowed—but the result had been
rather in his favour than in Mac-Vightie's. From the moment he had
discovered that he was being followed, he had devoted his time to
making applications for a job—for MacVightie's benefit—that being
the reason he had given MacVightie for his presence in Selkirk. Later
on, when it had grown dark, having business of his own, he had left
MacVightie's satellite standing on a street corner somewhat puzzled

just which way to turn! That, however, had no bearing on the watch
that had been, or might be at the present moment, set upon this
room.
The Hawk, in apparent abstraction, was flipping a coin up in the
air and catching it. There was a slight frown on the Hawk's face.
MacVightie's suspicions were still lingering for the simple reason that
MacVightie, utterly at sea, was clutching at the only straw in sight,
unless—the coin slipped through the Hawk's fingers and fell beside
his trunk. He stooped to pick it up—yes, not only had the room been
searched, but the trunk had been opened! The single strand of hair,
almost indiscernible against the brass and quite innocently caught in
the lock, was broken. Well, he had not finished that mental
sentence. Unless—what?
He tucked the coin into his pocket, and, standing up, yawned and
stretched himself. With the toe of his boot he lazily pushed a chair
out from the wall. The chair fell over. The Hawk picked it up, and
quite casually set it down—near the door. He took off his coat, and
flung it over the back of the chair.
The Hawk's face was greyer now, as it set in rigid lines, but there
was no tremor in the hand that inserted the key in the lock of the
trunk. He flung back the lid—and his eyes, for an instant, searched
the room again sharply. The window shade was securely drawn; the
coat over the back of the chair completely screened the keyhole of
the door. He laughed a little then—mirthlessly. Well, the trunk had
been opened! Had MacVightie found all—or nothing?
His fingers were working swiftly, deftly now around the inside
edges of the lid. He was either caught here, cornered, at bay—or
MacVightie, once for all, would be satisfied, and, as far as
MacVightie was concerned, the coast would hereafter be clear. The
Hawk's dark eyes narrowed, the square under jaw crept out and set
doggedly. It had been a close call, perilously close, that other night
when he had taken the ten thousand dollars from the paymaster's
safe, and MacVightie had followed him here to this room. He had
pulled the wool over MacVightie's eyes for the moment—but
MacVightie had returned to the old trail again. Well, the cards were

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