SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                 Schedule 13D**

                    Under the Securities Exchange Act of 1934
                               (Amendment No. 37)*

                               Mesa Royalty Trust
                                (Name of Issuer)

                          Units of Beneficial Interest
                         (Title of Class of Securities)

                                    590660106
                                 (Cusip Number)

                               J. Taylor Crandall
                           201 Main Street, Suite 3100
                             Fort Worth, Texas 76102
                                 (817) 390-8400
                  (Name, Address and Telephone Number of Person
                Authorized to Receive Notices and Communications)

                                November 19, 2004
             (Date of Event which Requires Filing of this Statement)

     If the filing person has previously filed a statement on Schedule 13G to
report the acquisition which is the subject of this Schedule 13D, and is filing
this schedule because of Rule 13d-1(e), 13d-1(f) or 13d-1(g), check the
following box [_].

*The remainder of this cover page shall be filled out for a reporting person's
initial filing on this form with respect to the subject class of securities, and
for any subsequent amendment containing information which would alter
disclosures provided in a prior cover page.

The information required on the remainder of this cover page shall not be deemed
to be "filed" for the purpose of Section 18 of the Securities Exchange Act of
1934 ("Act") or otherwise subject to the liabilities of that section of the Act
but shall be subject to all other provisions of the Act (however, see the
Notes).

**The total number of units reported herein is 271,316 units, which constitutes
approximately 14.6% of the total number of units outstanding. All ownership
percentages set forth herein assume that there are 1,863,590 units outstanding.



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1.     Name of Reporting Person:

       Alpine Capital, L.P.
--------------------------------------------------------------------------------

2.     Check the Appropriate Box if a Member of a Group:               (a)  [_]
                                                                       (b)  [X]
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3.     SEC Use Only

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4.     Source of Funds: WC

--------------------------------------------------------------------------------
5.     Check box if Disclosure of Legal Proceedings is Required Pursuant 
       to Items 2(d) or 2(e):                                               [_]

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6.     Citizenship or Place of Organization: Texas

--------------------------------------------------------------------------------
                       7.     Sole Voting Power:        271,316(1)
Number of
Units                 ----------------------------------------------------------
Beneficially           8.     Shared Voting Power:      -0-
Owned By
Each                  ----------------------------------------------------------
Reporting              9.     Sole Dispositive Power:   271,316(1)
Person                 
With                  ----------------------------------------------------------
                      10.     Shared Dispositive Power: -0-

--------------------------------------------------------------------------------
11.     Aggregate Amount Beneficially Owned by Each Reporting Person:

        271,316
--------------------------------------------------------------------------------
12.     Check Box if the Aggregate Amount in Row (11) Excludes 
        Certain Units:                                                      [_]

--------------------------------------------------------------------------------
13.     Percent of Class Represented by Amount in Row (11):  

        14.6%
--------------------------------------------------------------------------------
14.     Type of Reporting Person: 

        PN
--------------------------------------------------------------------------------

------------
(1)  Power is exercised through its two general partners, Robert W. Bruce III
     and Algenpar, Inc.



--------------------------------------------------------------------------------
1.     Name of Reporting Person:

       Robert W. Bruce III
--------------------------------------------------------------------------------

2.     Check the Appropriate Box if a Member of a Group:               (a)  [_]
                                                                       (b)  [X]
--------------------------------------------------------------------------------
3.     SEC Use Only

--------------------------------------------------------------------------------
4.     Source of Funds: Not Applicable

--------------------------------------------------------------------------------
5.     Check box if Disclosure of Legal Proceedings is Required Pursuant 
       to Items 2(d) or 2(e):                                               [_]

--------------------------------------------------------------------------------
6.     Citizenship or Place of Organization: USA

--------------------------------------------------------------------------------
                       7.     Sole Voting Power:        -0-
Number of
Units                 ----------------------------------------------------------
Beneficially           8.     Shared Voting Power:      271,316(1)
Owned By
Each                  ----------------------------------------------------------
Reporting              9.     Sole Dispositive Power:   -0-
Person                 
With                  ----------------------------------------------------------
                      10.     Shared Dispositive Power: 271,316(1)

--------------------------------------------------------------------------------
11.     Aggregate Amount Beneficially Owned by Each Reporting Person:

        271,316(1)
--------------------------------------------------------------------------------
12.     Check Box if the Aggregate Amount in Row (11) Excludes 
        Certain Units:                                                      [_]

--------------------------------------------------------------------------------
13.     Percent of Class Represented by Amount in Row (11):  

        14.6%
--------------------------------------------------------------------------------
14.     Type of Reporting Person: 

        IN
--------------------------------------------------------------------------------

----------
(1)  Solely in his capacity as one of two general partners of Alpine 
     Capital, L.P.



--------------------------------------------------------------------------------
1.     Name of Reporting Person:

       Algenpar, Inc.
--------------------------------------------------------------------------------

2.     Check the Appropriate Box if a Member of a Group:               (a)  [_]
                                                                       (b)  [X]
--------------------------------------------------------------------------------
3.     SEC Use Only

--------------------------------------------------------------------------------
4.     Source of Funds: Not Applicable

--------------------------------------------------------------------------------
5.     Check box if Disclosure of Legal Proceedings is Required Pursuant 
       to Items 2(d) or 2(e):                                               [_]

--------------------------------------------------------------------------------
6.     Citizenship or Place of Organization: Texas

--------------------------------------------------------------------------------
                       7.     Sole Voting Power:        -0-
Number of
Units                 ----------------------------------------------------------
Beneficially           8.     Shared Voting Power:      271,316(1)(2)
Owned By
Each                  ----------------------------------------------------------
Reporting              9.     Sole Dispositive Power:   -0-
Person                 
With                  ----------------------------------------------------------
                      10.     Shared Dispositive Power: 271,316(1)(2)

--------------------------------------------------------------------------------
11.     Aggregate Amount Beneficially Owned by Each Reporting Person:

        271,316(2)
--------------------------------------------------------------------------------
12.     Check Box if the Aggregate Amount in Row (11) Excludes 
        Certain Units:                                                      [_]

--------------------------------------------------------------------------------
13.     Percent of Class Represented by Amount in Row (11):  

        14.6%
--------------------------------------------------------------------------------
14.     Type of Reporting Person: 

        CO
--------------------------------------------------------------------------------

------------
(1)  Power is exercised through its President, J. Taylor Crandall.
(2)  Solely in its capacity as one of two general partners of Alpine 
     Capital, L.P.



--------------------------------------------------------------------------------
1.     Name of Reporting Person:

       J. Taylor Crandall
--------------------------------------------------------------------------------

2.     Check the Appropriate Box if a Member of a Group:               (a)  [_]
                                                                       (b)  [X]
--------------------------------------------------------------------------------
3.     SEC Use Only

--------------------------------------------------------------------------------
4.     Source of Funds: Not Applicable

--------------------------------------------------------------------------------
5.     Check box if Disclosure of Legal Proceedings is Required Pursuant 
       to Items 2(d) or 2(e):                                               [_]

--------------------------------------------------------------------------------
6.     Citizenship or Place of Organization: USA

--------------------------------------------------------------------------------
                       7.     Sole Voting Power:        -0-
Number of
Units                 ----------------------------------------------------------
Beneficially           8.     Shared Voting Power:      271,316(1)
Owned By
Each                  ----------------------------------------------------------
Reporting              9.     Sole Dispositive Power:   -0-
Person                 
With                  ----------------------------------------------------------
                      10.     Shared Dispositive Power: 271,316(1)

--------------------------------------------------------------------------------
11.     Aggregate Amount Beneficially Owned by Each Reporting Person:

        271,316(1)
--------------------------------------------------------------------------------
12.     Check Box if the Aggregate Amount in Row (11) Excludes 
        Certain Units:                                                      [_]

--------------------------------------------------------------------------------
13.     Percent of Class Represented by Amount in Row (11):  

        14.6%
--------------------------------------------------------------------------------
14.     Type of Reporting Person: 

        IN
--------------------------------------------------------------------------------

------------
(1)  Solely in his capacity as President and sole stockholder of Algenpar,
     Inc., which is one of two general partners of Alpine Capital, L.P.



Pursuant to Rule 13d-2(a) of Regulation 13D-G of the General Rules and
Regulations under the Securities Exchange Act of 1934, as amended (the "Act"),
the undersigned hereby amend their Schedule 13D Statement dated June 25, 1993,
as amended by Amendment No. 1 dated February 24, 1994, Amendment No. 2 dated
August 30, 1994, Amendment No. 3 dated August 10, 1995, Amendment No. 4 dated
November 10, 1995, Amendment No. 5 dated December 15, 1995, Amendment No. 6
dated January 8, 1996, Amendment No. 7 dated February 22, 1996, Amendment No. 8
dated May 15, 1996, Amendment No. 9 dated July 19, 1996 Amendment No. 10 dated
October 22, 1998, Amendment No. 11 dated November 9, 1998, Amendment No. 12
dated March 22, 1999, Amendment No. 13 dated June 16, 1999, Amendment No. 14
dated July 29, 1999, Amendment No. 15 dated February 2, 2000, Amendment No. 16
dated March 28, 2001, Amendment No. 17 dated April 23, 2001, Amendment No. 18
dated May 1, 2001, Amendment No. 19 dated September 27, 2001, Amendment No. 20
dated January 30, 2002, Amendment No. 21 dated February 18, 2003, Amendment No.
22 dated August 19, 2003, Amendment No. 23 dated October 30, 2003, Amendment No.
24 dated January 29, 2004, Amendment No. 25 dated April 28, 2004, Amendment No.
26 dated August 4, 2004, Amendment No. 27 dated October 5, 2004, Amendment No.
28 dated October 6, 2004, Amendment No. 29 dated October 7, 2004, Amendment No.
30 dated October 8, 2004, Amendment No. 31 dated October 12, 2004, Amendment No.
32 dated October 15, 2004, Amendment No. 33 dated October 22, 2004, Amendment
No. 34 dated November 1, 2004, Amendment No. 35 dated November 9, 2004 and
Amendment No. 36 dated November 12, 2004 (as amended, the "Schedule 13D"),
relating to the Units of Beneficial Interest (the "Units") of Mesa Royalty Trust
(the "Issuer").

Item 1.     SECURITY AND ISSUER.

No material change.

Item 2.     IDENTITY AND BACKGROUND.

No material change.

Item 3.     SOURCE AND AMOUNT OF FUNDS OR OTHER CONSIDERATION.

Item 3 is hereby amended in its entirety as follows:

The source and amount of the funds used by the Reporting Persons to purchase
Units are as follows:

REPORTING PERSON        SOURCE OF FUNDS         AMOUNT OF FUNDS
----------------        ---------------         ---------------
Alpine                  Working Capital(1)      $31,487,017.16(2) 
Bruce                   Not Applicable          Not Applicable
Algenpar                Not Applicable          Not Applicable 



REPORTING PERSON        SOURCE OF FUNDS         AMOUNT OF FUNDS
----------------        ---------------         ---------------
Crandall                Not Applicable          Not Applicable

(1)      As used herein, the term "Working Capital" includes income from the
business operations of the entity plus sums borrowed from banks and brokerage
firm margin accounts to operate such business in general. None of the funds
reported herein as "Working Capital" were borrowed or otherwise obtained for the
specific purpose of acquiring, handling, trading or voting the Stock.

(2)      This figure represents the total amount expended by Alpine for all
purchases of the Stock without subtracting sales; therefore, such figure does
not represent Alpine's net investment in the Stock. Alpine's net investment in
the Stock is $77,091.35.

Item 4.     PURPOSE OF TRANSACTION.

No material change.

Item 5.     INTEREST IN SECURITIES OF THE ISSUER.

Item 5 is hereby amended and restated in its entirety as follows:

(a)

ALPINE

The aggregate number of Units that Alpine owns beneficially, pursuant to Rule
13d-3(d)(1)(i) of the Act, is 271,316, which constitutes approximately 14.6% of
the outstanding Units.

BRUCE

Because of his position as one of two general partners of Alpine, Bruce may,
pursuant to Rule 13d-3 of the Act, be deemed to be the beneficial owner of
271,316 Units, which constitutes approximately 14.6% of the outstanding Units.

ALGENPAR

Because of its position as one of two general partners of Alpine, Algenpar may,
pursuant to Rule 13d-3 of the Act, be deemed to be the beneficial owner of
271,316 Units, which constitutes approximately 14.6% of the outstanding Units.

CRANDALL

Because of his positions as President and sole stockholder of Algenpar, one of
two general partners of Alpine, Crandall may, pursuant to Rule 13d-3 of the Act,
be deemed to be the beneficial owner of 271,316 Units, which constitutes
approximately 14.6% of the outstanding Units.

To the best of the knowledge of each of the Reporting Persons, other than as set
forth



above, none of the persons named in Item 2 herein is the beneficial owner
of any Units.







(b)

ALPINE

Acting through its two general partners, Alpine has the sole power to vote or to
direct the vote and to dispose or to direct the disposition of 271,316 Units.

BRUCE

As one of two general partners of Alpine, Bruce has shared power to vote or to
direct the vote and to dispose or to direct the disposition of 271,316 Units.

ALGENPAR

As one of two general partners of Alpine, Algenpar has shared power to vote or
to direct the vote and to dispose or to direct the disposition of 271,316 Units.

CRANDALL

As the President and sole stockholder of Algenpar, which is one of two general
partners of Alpine, Crandall has shared power to vote or to direct the vote and
to dispose or to direct the disposition of 271,316 Units.

(c)      Since the most recent filing on Schedule 13D, Alpine has sold Units in
open market transactions on the New York Stock Exchange as follows:

REPORTING PERSON      DATE            NO. OF UNITS SOLD         PRICE PER UNIT
----------------      ----            -----------------         --------------

Alpine                11/12/04               6,100              $ 68.44
Alpine                11/15/04               3,700              $ 68.45
Alpine                11/16/04               3,200              $ 68.45
Alpine                11/17/04               2,700              $ 68.45
Alpine                11/18/04               1,600              $ 68.45
Alpine                11/19/04               1,900              $ 68.45


Except as set forth in this paragraph (c), to the best of the knowledge of each
of the Reporting Persons, none of the persons named in response to paragraph (a)
has effected any transactions in the Units since the most recent filing on
Schedule 13D.

(d)      Each of the Reporting Persons affirms that no person other than such
Reporting Person has the right to receive or the power to direct the receipt of
dividends from, or the proceeds from the sale of, the Units owned by such
Reporting Person.

(e)      Not applicable.

Item 6.     CONTRACTS, ARRANGEMENTS, UNDERSTANDINGS OR



            RELATIONSHIPS WITH RESPECT TO SECURITIES OF THE ISSUER.

No material change.

Item 7.     MATERIAL TO BE FILED AS EXHIBITS.

Exhibit 99.1 -- Agreement pursuant to Rule 13d-1(k)(1)(iii), filed herewith.






After reasonable inquiry and to the best of my knowledge and belief, I certify
that the information set forth in this statement is true, complete and correct.

DATED:  November 22, 2004


                                        ALPINE CAPITAL, L.P.


                                        By: /s/ Robert W. Bruce III
                                            -------------------------
                                        Robert W. Bruce III,
                                        Manager


                                        /s/ Robert W. Bruce III
                                        -----------------------------
                                        ROBERT W. BRUCE III


                                        
                                        ALGENPAR, INC.


                                        By: /s/ Kevin G. Levy
                                            -------------------------
                                        Kevin G. Levy,
                                        Vice President


                                        /s/ Kevin G. Levy
                                        -----------------------------
                                        KEVIN G. LEVY,
                                        Attorney-in-Fact for:
                                        J. TAYLOR CRANDALL (1)


(1)  A Power of Attorney authorizing Kevin G. Levy, et al., to act on behalf
     of J. Taylor Crandall previously has been filed with the Securities and
     Exchange Commission.




                                   EXHIBIT INDEX


EXHIBIT           DESCRIPTION

99.1              Agreement pursuant to Rule 13d-1(k)(1)(iii), filed herewith.